THE ALGEBRO-GEOMETRIC INITIAL VALUE PROBLEM FOR THE ABLOWITZ–LADIK HIERARCHY

FRITZ GESZTESY, HELGE HOLDEN, JOHANNA MICHOR, AND GERALD TESCHL

Dedicated with great pleasure to Percy Deift on the occasion of his 60th birthday

Abstract. We discuss the algebro-geometric initial value problem for the Ablowitz–Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz–Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz–Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz–Ladik hierarchy.

The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz–Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz–Ladik hierarchy but applies generally to (1 + 1)-dimensional completely integrable soliton equations of differential-difference type.

1. Introduction

The principal aim of this paper is an explicit construction of unique global solutions of the algebro-geometric initial value problem for the Ablowitz–Ladik hierarchy for a general class of initial data. However, to put this circle of ideas into a proper perspective, we first very briefly recall the origins of this subject: In the mid-seventies, Ablowitz and Ladik, in a series of papers [3]–[6] (see also [1], [2, Sect. 3.2.2], [7, Ch. 3]), used inverse scattering methods to analyze certain integrable differential-difference systems. One of their integrable variants of such systems included a discretization of the celebrated AKNS-ZS system, the pair of

Date: May 14, 2009. 2000 Mathematics Subject Classification. Primary 37K10, 37K20, 47B36; Secondary 35Q58,

37K60.

Key words and phrases. Ablowitz–Ladik hierarchy, complex-valued solutions, initial value problem.

Research supported in part by the Research Council of Norway, the US National Science Foundation under Grant No. DMS-0405526, and the Austrian Science Fund (FWF) under Grants No. Y330, J2655.

To appear in Discrete Contin. Dyn. Syst.

coupled nonlinear differential-difference equations,

\begin{aligned}−iα_{t} − (1 − αβ)(α^{−} + α^{+}) + 2α & = 0, \\ −iβ_{t} + (1 − αβ)(β^{−} + β^{+}) − 2β & = 0 \end{aligned}  (1.1)

with α = α(n, t), β = β(n, t), (n, t) ∈ ℤ × ℝ. Here we used the notation f^{±}(n) = f(n ± 1), n ∈ ℤ, for complex-valued sequences f = {f(n)}_{n∈ℤ}. In particular, Ablowitz and Ladik [4] (see also [7, Ch. 3]) showed that in the focusing case, where β = −\overline{α}, and in the defocusing case, where β = \overline{α}, (1.1) yields the discrete analog of the nonlinear Schrödinger equation

− iα_{t} − (1 ± |α|^{2})(α^{−} + α^{+}) + 2α = 0.  (1.2)

Since then there has been an enormous activity in this area and we refer, for instance, to [7, Ch. 3], [27], [31], [32], [33], [36], [37], [40], [43], [44], [45], [50] and the extensive literature cited therein, for developments leading up to current research in this particular area of completely integrable differential-difference systems. Particularly relevant to this paper are algebro-geometric (and periodic) solutions of the AL system (1.1) and its associated hierarchy of integrable equations. The first systematic and detailed treatment of algebro-geometric solutions of the AL system (1.1) was performed by Miller, Ercolani, Krichever, and Levermore [40] (see also [9], [10], [14], [15], [39], [52]). Algebro-geometric solutions of the AL hierarchy were discussed in great detail in [32] (see also [25], [26], [53]). The initial value problem for the half-infinite discrete linear Schrödinger equation and the Schur flow were discussed by Common [17] (see also [18]) using a continued fraction approach. The corresponding nonabelian cases on a finite interval were studied by Gekhtman [24]. In addition to these developments within integrable systems and their applications to fields such as nonlinear optics, the study of AL systems recently gained considerable momentum due to its connections with the theory of orthogonal polynomials. Especially, the particular defocusing case β = \overline{α} and the associated CMV matrices and orthogonal polynomials on the unit circle attracted great interest. In this context we refer the interested reader to the two-volume treatise by Simon [47] (see also [48]) and the survey by Deift [19] and the detailed reference lists therein.

Returning to the principal scope of this paper, we intend to describe a solution of the following problem: Given \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2} \ \{(0, 0)\}, \underline{r} ∈ ℕ_{0}^{2}, assume α^{(0)}, β^{(0)} to be solutions of the \underline{p}th stationary Ablowitz–Ladik system s-AL\underline{p}(α, β) = 0 associated with a prescribed hyperelliptic curve K_{p} of genus p = p_{−} + p_{+} − 1 (with nonsingular affine part). We want to construct a unique global solution α = α(t\underline{r}), β = β(t\underline{r}) of the \underline{r}th AL flow AL\underline{r}(α, β) = 0 with α(t_{0,\underline{r}}) = α^{(0)}, β(t_{0,\underline{r}}) = β^{(0)} for some t_{0,\underline{r}} ∈ ℝ. Thus, we seek the unique global solution of the initial value problem

\begin{matrix}AL\underline{r}(α, β) = 0, \\ (α, β)|_{t\underline{r}=t_{0,\underline{r}}} = (α^{(0)}, β^{(0)}), \end{matrix}  (1.3)
s-AL\underline{p} (α^{(0)}, β^{(0)}) = 0,  (1.4)

where α = α(n, t\underline{r}), β = β(n, t\underline{r}) satisfy the conditions in (2.2).

Given the particularly familiar case of real-valued algebro-geometric solutions of the Toda hierarchy (see, e.g., [16], [29, Sect. 1.3], [51, Sect. 8.3] and the extensive literature cited therein), the actual solution of this algebro-geometric initial value problem, naively, might consist of the following two-step procedure:^{1}

(i) An algorithm that constructs admissible (cf. Section 4) nonspecial divisors D_{\underline{μ̂}(n)} ∈ Sym^{p} K_{p} for all n ∈ ℤ, starting from a nonspecial initial Dirichlet divisor D_{\underline{μ̂}(n_{0})} ∈ Sym^{p} K_{p}. “Trace formulas” of the type (3.30) and (3.31) (the latter requires prior construction of the Neumann divisor D\underline{ν̂} from the Dirichlet divisor D\underline{μ̂}, though, cf. (4.3)) should then construct the stationary solutions α^{(0)}, β^{(0)} of s-AL\underline{p}(α, β) = 0.

(ii) The first-order Dubrovin-type system of differential equations (5.43), augmented by the initial divisor D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} = D_{\underline{μ̂}(n_{0})} (cf. step (i)) together with the analogous “trace formulas” (3.30), (3.31) in the time-dependent context should then yield unique global solutions α = α(t\underline{r}), β = β(t\underline{r}) of the \underline{r}th AL flow AL\underline{r}(α, β) = 0 satisfying α(t_{0,\underline{r}}) = α^{(0)}, β(t_{0,\underline{r}}) = β^{(0)}.

However, this approach can be expected to work only if the Dirichlet divisors D_{\underline{μ̂}(n,t\underline{r})} ∈ Sym^{p} K_{p}, \underline{μ̂}(n, t\underline{r}) = (μ̂_{1}(n, t\underline{r}), …, μ̂_{p}(n, t\underline{r})), yield pairwise distinct Dirichlet eigenvalues μ_{j}(n, t\underline{r}), j = 1, …, p, for fixed (n, t\underline{r}) ∈ ℤ × ℝ, such that formula (5.43) for the time-derivatives μ_{j,t\underline{r}}, j = 1, …, p, is well-defined. Analogous considerations apply to the Neumann divisors D\underline{ν̂} ∈ Sym^{p} K_{p}.

Unfortunately, this scenario of pairwise distinct Dirichlet eigenvalues is not realistic and “collisions” between them can occur at certain values of (n, t\underline{r}) ∈ ℤ × ℝ. Thus, the stationary algorithm in step (i) as well as the Dubrovin-type first-order system of differential equations (5.43) in step (ii) above breaks down at such values of (n, t\underline{r}). A priori, one has no control over such collisions, especially, it is not possible to identify initial conditions D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} at some (n_{0}, t_{0,\underline{r}}) ∈ ℤ × ℝ, which avoid collisions for all (n, t\underline{r}) ∈ ℤ × ℝ. We solve this problem head on by explicitly permitting collisions in the stationary as well as the time-dependent context from the outset. In the stationary context, we introduce an appropriate algorithm alluded to in step (i) by employing a general interpolation formalism (cf. Appendix B) for polynomials, going beyond the usual Lagrange interpolation formulas. In the time-dependent context we replace the first-order system of Dubrovin-type equations (5.43), augmented with the initial divisor D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})}, by a different first-order system of differential equations (6.15), (6.23), and (6.24) with initial conditions (6.25) which focuses on symmetric functions of μ_{1}(n, t\underline{r}), …, μ_{p}(n, t\underline{r}) rather than individual Dirichlet eigenvalues μ_{j}(n, t\underline{r}), j = 1, …, p. In this manner it will be shown that collisions of Dirichlet eigenvalues no longer pose a problem.

In addition, there is an additional complication: In general, it cannot be guaranteed that μ_{j}(n, t\underline{r}) and ν_{j}(n, t\underline{r}), j = 1, …, p, stay finite and nonzero for all (n, t\underline{r}) ∈ ℤ × ℝ. We solve this particular problem in the stationary as well as the time-dependent case by properly restricting the initial Dirichlet and Neumann divisors D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})}, D_{\underline{ν̂}(n_{0},t_{0,\underline{r}})} ∈ Sym^{p} K_{p} to a dense set of full measure.

Summing up, we offer a new algorithm to solve the inverse algebro-geometric spectral problem for general Ablowitz–Ladik Lax operators, starting from a properly chosen dense set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the Dubrovin-type equations), this yields the construction of global algebrogeometric solutions of the time-dependent Ablowitz–Ladik hierarchy.

1We freely use the notation of divisors of degree p as introduced in Appendix A (see also the beginning of Section 3).

We emphasize that the approach described in this paper is not limited to the Ablowitz–Ladik hierarchy but applies universally to constructing algebro-geometric solutions of (1+1)-dimensional integrable soliton equations. In particular, it applies to the Toda lattice hierarchy as discussed in [30]. Moreover, the principal idea of replacing Dubrovin-type equations by a first-order system of the type (6.15), (6.23), and (6.24) is also relevant in the context of general (non-self-adjoint) Lax operators for the continuous models in (1 + 1)-dimensions. In particular, the models studied in detail in [28] can be revisited from this point of view. (However, the fact that the set in (6.67) is of measure zero relies on the fact that n varies in the countable set ℤ and hence is not applicable to continuous models in 1 + 1-dimensions.) We also note that while the periodic case with complex-valued α, β is of course included in our analysis, we throughout consider the more general algebro-geometric case (in which α, β need not be quasi-periodic).

Finally we briefly describe the content of each section. Section 2 presents a quick summary of the basics of the Ablowitz–Ladik hierarchy, its recursive construction, Lax pairs, and zero-curvature equations. The stationary algebro-geometric Ablowitz–Ladik hierarchy solutions, the underlying hyperelliptic curve, trace formulas, etc., are the subject of Section 3. A new algorithm solving the algebrogeometric inverse spectral problem for general Ablowitz–Ladik Lax operators is presented in Section 4. In Section 5 we briefly summarize the properties of algebrogeometric time-dependent solutions of the Ablowitz–Ladik hierarchy and formulate the algebro-geometric initial value problem. Uniqueness and existence of global solutions of the algebro-geometric initial value problem as well as their explicit construction are then presented in our final and principal Section 6. Appendix A reviews the basics of hyperelliptic Riemann surfaces of the Ablowitz–Ladik-type and sets the stage of much of the notation used in this paper. Finally, various interpolation formulas of fundamental importance to our stationary inverse spectral algorithm developed in Section 4 are summarized in Appendix B. These appendices support our intention to make this paper reasonably self-contained.

2. The Ablowitz–Ladik Hierarchy in a Nutshell

We briefly review the recursive construction of the Ablowitz–Ladik hierarchy and zero-curvature equations following [31] and [33].

Throughout this section we suppose the following hypothesis.

Hypothesis 2.1. In the stationary case we assume that α, β satisfy

α, β ∈ ℂ^{ℤ}, α(n)β(n) ∉ {0, 1}, n ∈ ℤ.  (2.1)

In the time-dependent case we assume that α, β satisfy

\begin{matrix}α( · , t), β( · , t) ∈ ℂ^{ℤ}, t ∈ ℝ, α(n, ·), β(n, ·) ∈ C^{1}(ℝ), n ∈ ℤ, \\ α(n, t)β(n, t) ∉ \{0, 1\}, (n, t) ∈ ℤ × ℝ. \end{matrix}  (2.2)

Here ℂ^{ℤ} denotes the set of complex-valued sequences indexed by ℤ. For a discussion of assumptions (2.1) and (2.2) we refer to Remark 3.4.

We denote by S^{±} the shift operators acting on complex-valued sequences f = {f(n)}_{n∈ℤ} ∈ ℂ^{ℤ} according to

(S^{±}f)(n) = f(n ± 1), n ∈ ℤ.  (2.3)

Moreover, we will frequently use the notation

f^{±} = S^{±}f, f ∈ ℂ^{ℤ}.  (2.4)

Next, we define sequences {f_{ℓ,±}}_{ℓ∈ℕ_{0}}, {g_{ℓ,±}}_{ℓ∈ℕ_{0}}, and {h_{ℓ,±}}_{ℓ∈ℕ_{0}} recursively by

g_{0,+} = \frac{1}{2}c_{0,+}, f_{0,+} = −c_{0,+}α^{+}, h_{0,+} = c_{0,+}β,  (2.5)
g_{ℓ+1,+} − g_{ℓ+1,+}^{−} = αh_{ℓ,+}^{−} + βf_{ℓ,+}, ℓ ∈ ℕ_{0},  (2.6)
f_{ℓ+1,+}^{−} = f_{ℓ,+} − α(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}), ℓ ∈ ℕ_{0},  (2.7)
h_{ℓ+1,+} = h_{ℓ,+}^{−} + β(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}), ℓ ∈ ℕ_{0},  (2.8)

and

g_{0,−} = \frac{1}{2}c_{0,−}, f_{0,−} = c_{0,−}α, h_{0,−} = −c_{0,−}β^{+},  (2.9)
g_{ℓ+1,−} − g_{ℓ+1,−}^{−} = αh_{ℓ,−} + βf_{ℓ,−}^{−}, ℓ ∈ ℕ_{0},  (2.10)
f_{ℓ+1,−} = f_{ℓ,−}^{−} + α(g_{ℓ+1,−} + g_{ℓ+1,−}^{−}), ℓ ∈ ℕ_{0},  (2.11)
h_{ℓ+1,−}^{−} = h_{ℓ,−} − β(g_{ℓ+1,−} + g_{ℓ+1,−}^{−}), ℓ ∈ ℕ_{0}.  (2.12)

Here c_{0,±} ∈ ℂ are given constants. For later use we also introduce

f_{−1,±} = h_{−1,±} = 0.  (2.13)

Remark 2.2. The sequences {f_{ℓ,+}}_{ℓ∈ℕ_{0}}, {g_{ℓ,+}}_{ℓ∈ℕ_{0}}, and {h_{ℓ,+}}_{ℓ∈ℕ_{0}} can be computed recursively as follows: Assume that f_{ℓ,+}, g_{ℓ,+}, and h_{ℓ,+} are known. Equation (2.6) is a first-order difference equation in g_{ℓ+1,+} that can be solved directly and yields a local lattice function that is determined up to a new constant denoted by c_{ℓ+1,+} ∈ ℂ. Relations (2.7) and (2.8) then determine f_{ℓ+1,+} and h_{ℓ+1,+}, etc. The sequences {f_{ℓ,−}}_{ℓ∈ℕ_{0}}, {g_{ℓ,−}}_{ℓ∈ℕ_{0}}, and {h_{ℓ,−}}_{ℓ∈ℕ_{0}} are determined similarly.

Upon setting

γ = 1 − αβ,  (2.14)

one explicitly obtains

\begin{aligned}f_{0,+} & = c_{0,+}(−α^{+}), f_{1,+} = c_{0,+}( − γ^{+}α^{++} + (α^{+})^{2}β) + c_{1,+}(−α^{+}), \\ g_{0,+} & = \frac{1}{2}c_{0,+}, g_{1,+} = c_{0,+}(−α^{+}β) + \frac{1}{2}c_{1,+}, \\ h_{0,+} & = c_{0,+}β, h_{1,+} = c_{0,+}(γβ^{−} − α^{+}β^{2}) + c_{1,+}β, \\ f_{0,−} & = c_{0,−}α, f_{1,−} = c_{0,−}(γα^{−} − α^{2}β^{+}) + c_{1,−}α, \\ g_{0,−} & = \frac{1}{2}c_{0,−}, g_{1,−} = c_{0,−}(−αβ^{+}) + \frac{1}{2}c_{1,−}, \\ h_{0,−} & = c_{0,−}(−β^{+}), h_{1,−} = c_{0,−}( − γ^{+}β^{++} + α(β^{+})^{2}) + c_{1,−}(−β^{+}), etc. \end{aligned} (2.15)

Here {c_{ℓ,±}}_{ℓ∈ℕ} denote summation constants which naturally arise when solving the difference equations for g_{ℓ,±} in (2.6), (2.10). Subsequently, it will also be useful to work with the corresponding homogeneous coefficients f̂_{ℓ,±}, ĝ_{ℓ,±}, and ĥ_{ℓ,±}, defined by the vanishing of all summation constants c_{k,±} for k = 1, …, ℓ, and choosing c_{0,±} = 1,

f̂_{0,+} = −α^{+}, f̂_{0,−} = α, f̂_{ℓ,±} = f_{ℓ,±}|_{c_{0,±}=1, c_{j,±}=0,j=1,...,ℓ}, ℓ ∈ ℕ,  (2.16)
ĝ_{0,±} = \frac{1}{2}, ĝ_{ℓ,±} = g_{ℓ,±}|_{c_{0,±}=1, c_{j,±}=0,j=1,...,ℓ}, ℓ ∈ ℕ,  (2.17)
ĥ_{0,+} = β, ĥ_{0,−} = −β^{+}, ĥ_{ℓ,±} = h_{ℓ,±}|_{c_{0,±}=1, c_{j,±}=0,j=1,...,ℓ}, ℓ ∈ ℕ. (2.18)

By induction one infers that

f_{ℓ,±} = ∑_{k=0}^{ℓ} c_{ℓ−k,±}f̂_{k,±}, g_{ℓ,±} = ∑_{k=0}^{ℓ} c_{ℓ−k,±}ĝ_{k,±}, h_{ℓ,±} = ∑_{k=0}^{ℓ} c_{ℓ−k,±}ĥ_{k,±}. (2.19)

Next we define the 2 × 2 zero-curvature matrices

U(z) = (\begin{matrix}z & α \\ zβ & 1 \end{matrix})  (2.20)

and

V\underline{p}(z) = i (\begin{matrix}G− ; \underline{p}(z) & −F − ; \underline{p}(z) \\ H− ; \underline{p}(z) & −K− ; \underline{p}(z) \end{matrix}) , \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2},  (2.21)

for appropriate Laurent polynomials F\underline{p}(z), G\underline{p}(z), H\underline{p}(z), and K\underline{p}(z) in the spectral parameter z ∈ ℂ\{0} to be determined shortly. By postulating the stationary zerocurvature relation,

0 = UV\underline{p} − V + ; \underline{p}U,  (2.22)

one concludes that (2.22) is equivalent to the following relations

z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p} = 0,  (2.23)
zβF − ; \underline{p} + αH\underline{p} − K\underline{p} + K− ; \underline{p} = 0,  (2.24)
−F\underline{p} + zF − ; \underline{p} + α(G\underline{p} + K− ; \underline{p}) = 0,  (2.25)
zβ(G− ; \underline{p} + K\underline{p}) − zH\underline{p} + H− ; \underline{p} = 0.  (2.26)

In order to make the connection between the zero-curvature formalism and the recursion relations (2.5)–(2.12), we now define Laurent polynomials F\underline{p}, G\underline{p}, H\underline{p}, and K\underline{p} by^{2}

F\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}z^{ℓ},  (2.27)

G\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ},  (2.28)

H\underline{p}(z) = ∑_{ℓ=0}^{p_{−}−1} h_{p_{−}−1−ℓ,−}z^{−ℓ} + ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}z^{ℓ},  (2.29)

K\underline{p}(z) = ∑_{ℓ=0}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=1}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ} = G\underline{p}(z) + g_{p_{−},−} − g_{p_{+},+}.  (2.30)

The stationary zero-curvature relation (2.22), 0 = UV\underline{p}−V + ; \underline{p}U, is then equivalent to

−α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} = 0,  (2.31)
β(g_{p_{+},+}^{−} + g_{p_{−},−}) + h_{p_{+}−1,+}^{−} − h_{p_{−}−1,−} = 0.  (2.32)

2In this paper, a sum is interpreted as zero whenever the upper limit in the sum is strictly less than its lower limit.

Thus, varying p_{±} ∈ ℕ_{0}, equations (2.31) and (2.32) give rise to the stationary Ablowitz–Ladik (AL) hierarchy which we introduce as follows

\begin{matrix}s-AL\underline{p}(α, β) = (\begin{matrix}−α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} \\ β(g_{p_{+},+}^{−} + g_{p_{−},−}) + h_{p_{+}−1,+}^{−} − h_{p_{−}−1,−} \end{matrix}) = 0, \\ \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}. \end{matrix}  (2.33)

Explicitly (recalling γ = 1 − αβ and taking p_{−} = p_{+} for simplicity),

s-AL_{(0,0)}(α, β) = (\begin{matrix}−c_{(0,0)}α \\ c_{(0,0)}β \end{matrix}) = 0,
s-AL_{(1,1)}(α, β) = (\begin{matrix}−γ(c_{0,−}α^{−} + c_{0,+}α^{+}) − c_{(1,1)}α \\ γ(c_{0,+}β^{−} + c_{0,−}β^{+}) + c_{(1,1)}β \end{matrix}) = 0,

s-AL_{(2,2)}(α, β) = (\begin{matrix}−γ(c_{0,+}α^{++}γ^{+} + c_{0,−}α^{−−}γ^{−} − α(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −β(c_{0,−}(α^{−})^{2} + c_{0,+}(α^{+})^{2})) \end{matrix} ; \begin{matrix}γ(c_{0,−}β^{++}γ^{+} + c_{0,+}β^{−−}γ^{−} − β(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −α(c_{0,+}(β^{−})^{2} + c_{0,−}(β^{+})^{2})) \end{matrix})

+ (\begin{matrix}−γ(c_{1,−}α^{−} + c_{1,+}α^{+}) − c_{(2,2)}α \\ γ(c_{1,+}β^{−} + c_{1,−}β^{+}) + c_{(2,2)}β \end{matrix}) = 0, etc.,  (2.34)

represent the first few equations of the stationary Ablowitz–Ladik hierarchy. Here we introduced

c\underline{p} = (c_{p_{−},−} + c_{p_{+},+})/2, p_{±} ∈ ℕ_{0}.  (2.35)

By definition, the set of solutions of (2.33), with \underline{p} ranging in ℕ_{0}^{2} and c_{ℓ,±} ∈ ℂ, ℓ ∈ ℕ_{0}, represents the class of algebro-geometric Ablowitz–Ladik solutions.

In the following we will frequently assume that α, β satisfy the \underline{p}th stationary Ablowitz–Ladik system supposing a particular choice of summation constants c_{ℓ,±} ∈ ℂ, ℓ = 0, …, p_{±}, p_{±} ∈ ℕ_{0}, has been made.

In accordance with our notation introduced in (2.16)–(2.18), the corresponding homogeneous stationary Ablowitz–Ladik equations are defined by

s-ÂL\underline{p}(α, β) = s-AL\underline{p}(α, β)|_{c_{0,±}=1, c_{ℓ,±}=0, ℓ=1,...,p_{±}}, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}. (2.36)

One can show (cf. [31, Lemma 2.2]) that g_{p_{+},+} = g_{p_{−},−} up to a lattice constant which can be set equal to zero without loss of generality. Thus, we will henceforth assume that

g_{p_{+},+} = g_{p_{−},−},  (2.37)

which in turn implies that

K\underline{p} = G\underline{p}  (2.38)

and hence renders V\underline{p} in (2.21) traceless in the stationary context. (We note that equations (2.37) and (2.38) cease to be valid in the time-dependent context, though.)

Next, still assuming (2.1) and taking into account (2.38), one infers by taking determinants in the stationary zero-curvature equation (2.22) that the quantity

R\underline{p} = G2 ; \underline{p} − F\underline{p}H\underline{p}  (2.39)

is a lattice constant, that is, R\underline{p} − R− ; \underline{p} = 0. Hence, R\underline{p}(z) only depends on z, and assuming in addition to (2.1) that

c_{0,±} ∈ ℂ \ {0},  (2.40)

one may write R\underline{p} as^{3}

R\underline{p}(z) = (c_{0,+}^{2}/4)z^{−2p_{−}} ∏_{m=0}^{2p+1}(z − E_{m}), \{E_{m}\}_{m=0}^{2p+1} ⊂ ℂ \ \{0\},  (2.41)

where

p = p_{−} + p_{+} − 1.  (2.42)

In addition, we note that the summation constants c_{1,±}, …, c_{p_{±},±} in (2.33) can be expressed as symmetric functions in the zeros E_{0}, …, E_{2p+1} of the associated Laurent polynomial R\underline{p} in (2.41). In fact, one can prove (cf. [31]) that

c_{ℓ,±} = c_{0,±}c_{ℓ}(\underline{E}^{±1}), ℓ = 0, …, p_{±},  (2.43)

where

c_{0}(\underline{E}^{±1}) = 1,

c_{k}(\underline{E}^{±1})  (2.44)

= − ∑_{\begin{matrix}j_{0},...,j_{2p+1}=0 \\ j_{0}+···+j_{2p+1}=k \end{matrix}}^{k}\frac{(2j_{0})! ⋯ (2j_{2p+1})!}{2^{2k}(j_{0}!)^{2} ⋯ (j_{2p+1}!)^{2}(2j_{0} − 1) ⋯ (2j_{2p+1} − 1)}E_{0}^{±j_{0}} ⋯ E_{2p+1}^{±j_{2p+1}},

k ∈ ℕ,

are symmetric functions of \underline{E}^{±1} = (E_{0}^{±1}, …, E_{2p+1}^{±1}).

Next we turn to the time-dependent Ablowitz–Ladik hierarchy. For that purpose the coefficients α and β are now considered as functions of both the lattice point and time. For each system in the hierarchy, that is, for each \underline{p} ∈ ℕ_{0}^{2}, we introduce a deformation (time) parameter t\underline{p} ∈ ℝ in α, β, replacing α(n), β(n) by α(n, t\underline{p}), β(n, t\underline{p}). Moreover, the definitions (2.20), (2.21), and (2.27)–(2.30) of U, V\underline{p}, and F\underline{p}, G\underline{p}, H\underline{p}, K\underline{p}, respectively, still apply. Imposing the zero-curvature relation

U_{t\underline{p}} + UV\underline{p} − V + ; \underline{p}U = 0, \underline{p} ∈ ℕ_{0}^{2},  (2.45)

then results in the equations

α_{t\underline{p}} = i(zF − ; \underline{p} + α(G\underline{p} + K− ; \underline{p}) − F\underline{p}),  (2.46)
β_{t\underline{p}} = −i(β(G− ; \underline{p} + K\underline{p}) − H\underline{p} + z^{−1}H− ; \underline{p}),  (2.47)
0 = z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p},  (2.48)
0 = zβF − ; \underline{p} + αH\underline{p} + K− ; \underline{p} − K\underline{p}.  (2.49)

Varying \underline{p} ∈ ℕ_{0}^{2}, the collection of evolution equations

\begin{matrix}AL\underline{p}(α, β) = (\begin{matrix}−iα_{t\underline{p}} − α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} \\ −iβ_{t\underline{p}} + β(g_{p_{+},+}^{−} + g_{p_{−},−}) − h_{p_{−}−1,−} + h_{p_{+}−1,+}^{−} \end{matrix}) = 0, \\ t\underline{p} ∈ ℝ, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}, \end{matrix} (2.50)

then defines the time-dependent Ablowitz–Ladik hierarchy. Explicitly,

AL_{(0,0)}(α, β) = (\begin{matrix}−iα_{t_{(0,0)}} − c_{(0,0)}α \\ −iβ_{t_{(0,0)}} + c_{(0,0)}β \end{matrix}) = 0,

3We use the convention that a product is to be interpreted equal to 1 whenever the upper limit of the product is strictly less than its lower limit.

AL_{(1,1)}(α, β) = (\begin{matrix}−iα_{t_{(1,1)}} − γ(c_{0,−}α^{−} + c_{0,+}α^{+}) − c_{(1,1)}α \\ −iβ_{t_{(1,1)}} + γ(c_{0,+}β^{−} + c_{0,−}β^{+}) + c_{(1,1)}β \end{matrix}) = 0,

AL_{(2,2)}(α, β)  (2.51)

= (\begin{matrix}−iα_{t_{(2,2)}} − γ(c_{0,+}α^{++}γ^{+} + c_{0,−}α^{−−}γ^{−} − α(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −β(c_{0,−}(α^{−})^{2} + c_{0,+}(α^{+})^{2})) \end{matrix} ; \begin{matrix}−iβ_{t_{(2,2)}} + γ(c_{0,−}β^{++}γ^{+} + c_{0,+}β^{−−}γ^{−} − β(c_{0,+}α^{+}β^{−} + c_{0,−}α^{−}β^{+}) \\ −α(c_{0,+}(β^{−})^{2} + c_{0,−}(β^{+})^{2})) \end{matrix})

+ (\begin{matrix}−γ(c_{1,−}α^{−} + c_{1,+}α^{+}) − c_{(2,2)}α \\ γ(c_{1,+}β^{−} + c_{1,−}β^{+}) + c_{(2,2)}β \end{matrix}) = 0, etc.,

represent the first few equations of the time-dependent Ablowitz–Ladik hierarchy. Here we recall the definition of c\underline{p} in (2.35).

The special case \underline{p} = (1, 1), c_{0,±} = 1, and c_{(1,1)} = −2, that is,

(\begin{matrix}−iα_{t_{1}} − γ(α^{−} + α^{+}) + 2α \\ −iβ_{t_{1}} + γ(β^{−} + β^{+}) − 2β \end{matrix}) = 0,  (2.52)

represents the Ablowitz–Ladik system (1.1).

By (2.50), (2.6), and (2.10), the time derivative of γ = 1 − αβ is given by

γ_{t\underline{p}} = iγ((g_{p_{+},+} − g_{p_{+},+}^{−}) − (g_{p_{−},−} − g_{p_{−},−}^{−})).  (2.53)

The corresponding homogeneous equations are then defined by

ÂL\underline{p}(α, β) = AL\underline{p}(α, β)|_{c_{0,±}=1, c_{ℓ,±}=0, ℓ=1,...,p_{±}} = 0, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}. (2.54)

Remark 2.3. From (2.23)–(2.26) and the explicit computations of the coefficients f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±}, one concludes that the zero-curvature equation (2.45) and hence the Ablowitz–Ladik hierarchy is invariant under the scaling transformation

α → α_{c} = {c α(n)}_{n∈ℤ}, β → β_{c} = {β(n)/c}_{n∈ℤ}, c ∈ ℂ \ {0}.  (2.55)

In particular, solutions α, β of the stationary and time-dependent AL equations are determined only up to a multiplicative constant.

Remark 2.4. (i) The special choices β = ±\overline{α}, c_{0,±} = 1 lead to the discrete nonlinear Schrödinger hierarchy. In particular, choosing c_{(1,1)} = −2 yields the discrete nonlinear Schrödinger equation in its usual form (see, e.g., [7, Ch. 3] and the references cited therein),

− iα_{t} − (1 ∓ |α|^{2})(α^{−} + α^{+}) + 2α = 0,  (2.56)

as its first nonlinear element. The choice β = \overline{α} is called the defocusing case, β = −\overline{α} represents the focusing case of the discrete nonlinear Schrödinger hierarchy. (ii) The alternative choice β = \overline{α}, c_{0,±} = ∓i, leads to the hierarchy of Schur flows. In particular, choosing c_{(1,1)} = 0 yields

α_{t} − (1 − |α|^{2})(α^{+} − α^{−}) = 0  (2.57)

as the first nonlinear element of this hierarchy (cf. [11], [22], [23], [34], [41], [49]).

Finally, we briefly recall the Lax pair (L, P\underline{p}) for the Ablowitz–Ladik hierarchy and refer to [33] for detailed discussions of this topic.

In the standard basis {δ_{m}}_{m∈ℤ} in ℓ^{2}(ℤ) defined by

δ_{m} = \{δ_{m,n}\}_{n∈ℤ}, m ∈ ℤ, δ_{m,n} = \begin{cases}1, & m = n, \\ 0, & m ≠ n, \end{cases}  (2.58)

the underlying Lax difference expression L is given by

L = ( \begin{matrix}... & ... & ... & ... & ... & 0 \\ 0 & −α(0)ρ(−1) & −β(−1)α(0) & −α(1)ρ(0) & ρ(0)ρ(1) \\ ρ(−1)ρ(0) & β(−1)ρ(0) & −β(0)α(1) & β(0)ρ(1) & 0 \\ 0 & −α(2)ρ(1) & −β(1)α(2) & −α(3)ρ(2) & ρ(2)ρ(3) \\ 0 & ρ(1)ρ(2) & β(1)ρ(2) & −β(2)α(3) & β(2)ρ(3) & 0 \\ ... & ... & ... & ... & ... \end{matrix} ) (2.59)

= ρ^{−}ρ δ_{even} S^{−−} + (β^{−}ρ δ_{even} − α^{+}ρ δ_{odd})S^{−} − βα^{+}

+ (βρ^{+} δ_{even} − α^{++}ρ^{+} δ_{odd})S^{+} + ρ^{+}ρ^{++} δ_{odd} S^{++},  (2.60)

where δ_{even} and δ_{odd} denote the characteristic functions of the even and odd integers,

δ_{even} = χ_{2ℤ}, δ_{odd} = 1 − δ_{even} = χ_{2ℤ+1}.  (2.61)

In particular, terms of the form −β(n)α(n+1) represent the diagonal (n, n)-entries, n ∈ ℤ, in the infinite matrix (2.59). In addition, we used the abbreviation

ρ = γ^{1/2} = (1 − αβ)^{1/2}.  (2.62)

Next, let T be a bounded operator in the Hilbert space ℓ^{2}(ℤ) (with scalar product denoted by ( · , ·)). Given the standard basis (2.58) in ℓ^{2}(ℤ), we represent T by

T = (T (m, n))_{(m,n)∈ℤ^{2}}, T (m, n) = (δ_{m}, T δ_{n}), (m, n) ∈ ℤ^{2}.  (2.63)

Actually, for our purpose below, it is sufficient that T is an N-diagonal matrix for some N ∈ ℕ. Moreover, we introduce the upper and lower triangular parts T_{±} of T by

T_{±} = (T_{±}(m, n))_{(m,n)∈ℤ^{2}}, T_{±}(m, n) = \begin{cases}T (m, n), & ±(n − m) > 0, \\ 0, & otherwise. \end{cases} (2.64)

Then, the finite difference expression P\underline{p} is given by

\begin{aligned}P\underline{p} & = \frac{i}{2} ∑_{ℓ=1}^{p_{+}} c_{p_{+}−ℓ,+}((L^{ℓ})_{+} − (L^{ℓ})_{−}) − \frac{i}{2} ∑_{ℓ=1}^{p_{−}} c_{p_{−}−ℓ,−}((L^{−ℓ})_{+} − (L^{−ℓ})_{−}) \\ & − \frac{i}{2}c\underline{p} Q_{d}, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}, \end{aligned} (2.65)

with Q_{d} denoting the doubly infinite diagonal matrix

Q_{d} = ((−1)^{k}δ_{k,ℓ})_{k,ℓ∈ℤ}  (2.66)

and c\underline{p} = (c_{p_{−},−} + c_{p_{+},+})/2. The commutator relations [P\underline{p}, L] = 0 and L_{t\underline{p}} − [P\underline{p}, L] = 0 are then equivalent to the stationary and time-dependent Ablowitz– Ladik equations (2.33) and (2.50), respectively.

3. Properties of Stationary Algebro-Geometric Solutions of the Ablowitz–Ladik Hierarchy

In this section we present a quick review of properties of algebro-geometric solutions of the stationary Ablowitz–Ladik hierarchy. We refer to [31] and [32] for detailed presentations.

We recall the hyperelliptic curve K_{p} of genus p, where

\begin{matrix}K_{p}: F_{p}(z, y) = y^{2} − 4c_{0,+}^{−2}z^{2p_{−}}R\underline{p}(z) = y^{2} − ∏_{m=0}^{2p+1}(z − E_{m}) = 0, \\ R\underline{p}(z) = (\frac{c_{0,+}}{2z^{p_{−}}} )^{2} ∏_{m=0}^{2p+1}(z − E_{m}), \{E_{m}\}_{m=0}^{2p+1} ⊂ ℂ \ \{0\}, p = p_{−} + p_{+} − 1. \end{matrix} (3.1)

Throughout this section we make the assumption:

Hypothesis 3.1. Suppose that

α, β ∈ ℂ^{ℤ} and α(n)β(n) ∉ {0, 1} for all n ∈ ℤ.  (3.2)

In addition, assume that the affine part of the hyperelliptic curve K_{p} in (3.1) is nonsingular, that is, suppose that

E_{m} ≠ E_{m′} for m ≠ m′, m, m′ = 0, 1, …, 2p + 1.  (3.3)

The curve K_{p} is compactified by joining two points P_{∞_{±}}, P_{∞_{+}} ≠ P_{∞_{−}}, but for notational simplicity the compactification is also denoted by K_{p}. Points P on K_{p}\{P_{∞_{+}}, P_{∞_{−}}} are represented as pairs P = (z, y), where y( ·) is the meromorphic function on K_{p} satisfying F_{p}(z, y) = 0. The complex structure on K_{p} is then defined in the usual way, see Appendix A. Hence, K_{p} becomes a two-sheeted hyperelliptic Riemann surface of genus p in a standard manner.

We also emphasize that by fixing the curve K_{p} (i.e., by fixing E_{0}, …, E_{2p+1}), the summation constants c_{1,±}, …, c_{p_{±},±} in f_{p_{±},±}, g_{p_{±},±}, and h_{p_{±},±} (and hence in the corresponding stationary s-AL\underline{p} equations) are uniquely determined as is clear from (2.44) which establishes the summation constants c_{ℓ,±} as symmetric functions of E_{0}^{±1}, …, E_{2p+1}^{±1}.

For notational simplicity we will usually tacitly assume that p ∈ ℕ.

We denote by \{μ_{j}(n)\}_{j=1,...,p} and \{ν_{j}(n)\}_{j=1,...,p} the zeros of ( ·)^{p_{−}}F\underline{p}( · , n) and ( ·)^{p_{−}−1}H\underline{p}( · , n), respectively. Thus, we may write

F\underline{p}(z) = −c_{0,+}α^{+}z^{−p_{−}} ∏_{j=1}^{p}(z − μ_{j}),  (3.4)

H\underline{p}(z) = c_{0,+}βz^{−p_{−}+1} ∏_{j=1}^{p}(z − ν_{j}),  (3.5)

and we recall that (cf. (2.39))

R\underline{p} − G2 ; \underline{p} = −F\underline{p}H\underline{p}.  (3.6)

The next step is crucial; it permits us to “lift” the zeros μ_{j} and ν_{j} from the complex plane ℂ to the curve K_{p}. From (3.6) one infers that

R\underline{p}(z) − G\underline{p}(z)^{2} = 0, z ∈ \{μ_{j}, ν_{k}\}_{j,k=1,...,p}.  (3.7)

We now introduce {μ̂_{j}}_{j=1,...,p} ⊂ K_{p} and {ν̂_{j}}_{j=1,...,p} ⊂ K_{p} by

μ̂_{j}(n) = (μ_{j}(n), (2/c_{0,+})μ_{j}(n)^{p_{−}}G\underline{p}(μ_{j}(n), n)), j = 1, …, p, n ∈ ℤ,  (3.8)

and

ν̂_{j}(n) = (ν_{j}(n), −(2/c_{0,+})ν_{j}(n)^{p_{−}}G\underline{p}(ν_{j}(n), n)), j = 1, …, p, n ∈ ℤ. (3.9)

We also introduce the points P_{0,±} by

P_{0,±} = (0, ±(c_{0,−}/c_{0,+})) ∈ K_{p}, \frac{c_{0,−}^{2}}{c_{0,+}^{2}} = ∏_{m=0}^{2p+1} E_{m}.  (3.10)

We emphasize that P_{0,±} and P_{∞_{±}} are not necessarily on the same sheet of K_{p}. Moreover,

y(P) =_{ζ→0} \begin{cases}∓ζ^{−2p}(1 + O(ζ)), & P → P_{∞_{±}}, ζ = 1/z, \\ ±(c_{0,−}/c_{0,+}) + O(ζ), & P → P_{0,±}, ζ = z. \end{cases}  (3.11)

Next we introduce the fundamental meromorphic function on K_{p} by

ϕ(P, n) = \frac{(c_{0,+}/2)z^{−p_{−}}y + G\underline{p}(z, n)}{F\underline{p}(z, n)}  (3.12)

= \frac{−H\underline{p}(z, n)}{(c_{0,+}/2)z^{−p_{−}}y − G\underline{p}(z, n)},  (3.13)
P = (z, y) ∈ K_{p}, n ∈ ℤ,

with divisor (ϕ( · , n)) of ϕ( · , n) given by

(ϕ( · , n)) = D_{P_{0,−}\underline{ν̂}(n)} − D_{P_{∞_{−}} \underline{μ̂}(n)},  (3.14)

using (3.4) and (3.5). Here we abbreviated

\underline{μ̂} = \{μ̂_{1}, …, μ̂_{p}\}, \underline{ν̂} = \{ν̂_{1}, …, ν̂_{p}\} ∈ Sym^{p}(K_{p}).  (3.15)

For brevity, and in close analogy to the Toda hierarchy, we will frequently refer to \underline{μ̂} and \underline{ν̂} as the Dirichlet and Neumann divisors, respectively.

Given ϕ( · , n), the meromorphic stationary Baker–Akhiezer vector Ψ( · , n, n_{0}) on K_{p} is then defined by

Ψ(P, n, n_{0}) = \binom{ψ_{1}(P, n, n_{0})}{ψ_{2}(P, n, n_{0})},

ψ_{1}(P, n, n_{0}) = \begin{cases}∏_{n′=n_{0}+1}^{n} (z + α(n′)ϕ^{−}(P, n′)), & n ≥ n_{0} + 1, \\ 1, & n = n_{0}, \\ ∏_{n′=n+1}^{n_{0}} (z + α(n′)ϕ^{−}(P, n′))^{−1}, & n ≤ n_{0} − 1, \end{cases}  (3.16)

ψ_{2}(P, n, n_{0}) = ϕ(P, n_{0}) \begin{cases}∏_{n′=n_{0}+1}^{n} (zβ(n′)ϕ^{−}(P, n′)^{−1} + 1), & n ≥ n_{0} + 1, \\ 1, & n = n_{0}, \\ ∏_{n′=n+1}^{n_{0}} (zβ(n′)ϕ^{−}(P, n′)^{−1} + 1)^{−1}, & n ≤ n_{0} − 1. \end{cases} (3.17)

Basic properties of ϕ and Ψ are summarized in the following result.

Lemma 3.2 ([31]). Suppose that α, β satisfy (3.2) and the \underline{p}th stationary Ablowitz– Ladik system (2.33). Moreover, assume (3.1) and (3.3) and let P = (z, y) ∈ K_{p} \ {P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}}, (n, n_{0}) ∈ ℤ^{2}. Then ϕ satisfies the Riccati-type equation

αϕ(P)ϕ^{−}(P) − ϕ^{−}(P) + zϕ(P) = zβ,  (3.18)

as well as

ϕ(P)ϕ(P^{∗}) = \frac{H\underline{p}(z)}{F\underline{p}(z)},  (3.19)

ϕ(P) + ϕ(P^{∗}) = 2\frac{G\underline{p}(z)}{F\underline{p}(z)},  (3.20)
ϕ(P) − ϕ(P^{∗}) = c_{0,+}z^{−p_{−}} \frac{y(P)}{F\underline{p}(z)}.  (3.21)

The vector Ψ satisfies

U(z)Ψ^{−}(P) = Ψ(P),  (3.22)
V\underline{p}(z)Ψ^{−}(P) = −(i/2)c_{0,+}z^{−p_{−}}yΨ^{−}(P),  (3.23)
ψ_{2}(P, n, n_{0}) = ϕ(P, n)ψ_{1}(P, n, n_{0}),  (3.24)
ψ_{1}(P, n, n_{0})ψ_{1}(P^{∗}, n, n_{0}) = z^{n−n_{0}} \frac{F\underline{p}(z, n)}{F\underline{p}(z, n_{0})}Γ(n, n_{0}),  (3.25)

ψ_{2}(P, n, n_{0})ψ_{2}(P^{∗}, n, n_{0}) = z^{n−n_{0}} \frac{H\underline{p}(z, n)}{F\underline{p}(z, n_{0})}Γ(n, n_{0}),  (3.26)
ψ_{1}(P, n, n_{0})ψ_{2}(P^{∗}, n, n_{0}) + ψ_{1}(P^{∗}, n, n_{0})ψ_{2}(P, n, n_{0})  (3.27)

= 2z^{n−n_{0}} \frac{G\underline{p}(z, n)}{F\underline{p}(z, n_{0})}Γ(n, n_{0}),

ψ_{1}(P, n, n_{0})ψ_{2}(P^{∗}, n, n_{0}) − ψ_{1}(P^{∗}, n, n_{0})ψ_{2}(P, n, n_{0})  (3.28)

= −c_{0,+}z^{n−n_{0}−p_{−}} \frac{y}{F\underline{p}(z, n_{0})}Γ(n, n_{0}),

where we used the abbreviation

Γ(n, n_{0}) = \begin{cases}∏_{n′=n_{0}+1}^{n} γ(n′) & n ≥ n_{0} + 1, \\ 1 & n = n_{0}, \\ ∏_{n′=n+1}^{n_{0}} γ(n′)^{−1} & n ≤ n_{0} − 1. \end{cases}  (3.29)

Combining the Laurent polynomial recursion approach of Section 2 with (3.4) and (3.5) readily yields trace formulas for f_{ℓ,±} and h_{ℓ,±} in terms of symmetric functions of the zeros μ_{j} and ν_{k} of ( ·)^{p_{−}}F\underline{p} and ( ·)^{p_{−}−1}H\underline{p}, respectively. For simplicity we just record the simplest cases.

Lemma 3.3 ([31]). Suppose that α, β satisfy (3.2) and the \underline{p}th stationary Ablowitz– Ladik system (2.33). Then,

\frac{α}{α^{+}} = (−1)^{p+1} \frac{c_{0,+}}{c_{0,−}} ∏_{j=1}^{p} μ_{j},  (3.30)

\frac{β^{+}}{β} = (−1)^{p+1} \frac{c_{0,+}}{c_{0,−}} ∏_{j=1}^{p} ν_{j},  (3.31)

∑_{j=1}^{p} μ_{j} = α^{+}β − γ^{+} \frac{α^{++}}{α^{+}} − \frac{c_{1,+}}{c_{0,+}} ,  (3.32)

∑_{j=1}^{p} ν_{j} = α^{+}β − γ \frac{β^{−}}{β} − \frac{c_{1,+}}{c_{0,+}} .  (3.33)

Remark 3.4. The trace formulas in Lemma 3.3 illustrate why we assumed the condition α(n)β(n) ≠ 0 for all n ∈ ℕ throughout this paper. Moreover, the following section shows that this condition is intimately connected with admissible divisors D\underline{μ̂}, D\underline{ν̂} avoiding the exceptional points P_{∞_{±}}, P_{0,±}. On the other hand, as is clear from the matrix representation (2.59) of the Lax difference expression L, if α(n_{0})β(n_{0}) = 1 for some n_{0} ∈ ℕ, and hence ρ(n_{0}) = 0, the infinite matrix L splits into a direct sum of two half-line matrices L_{±}(n_{0}) (in analogy to the familiar singular case of infinite Jacobi matrices aS^{+} + a^{−}S^{−} + b on ℤ with a(n_{0}) = 0). This explains why we assumed α(n)β(n) ≠ 1 for all n ∈ ℕ throughout this paper.

Since nonspecial divisors and the linearization property of the Abel map when applied to D\underline{μ̂} and D\underline{ν̂} will play a fundamental role later on, we also recall the following facts.

Lemma 3.5 ([31], [32]). Suppose that α, β satisfy (3.2) and the \underline{p}th stationary Ablowitz–Ladik system (2.33). Moreover, assume (3.1) and (3.3) and let n ∈ ℤ. Let D\underline{μ̂}, \underline{μ̂} = \{μ̂_{1}, …, μ̂_{p}\}, and D\underline{ν̂}, \underline{ν̂} = \{\underline{ν̂}_{1}, …, \underline{ν̂}_{p}\}, be the pole and zero divisors of degree p, respectively, associated with α, β, and ϕ defined according to (3.8) and (3.9), that is,

\begin{aligned}μ̂_{j}(n) & = (μ_{j}(n), (2/c_{0,+})μ_{j}(n)^{p_{−}}G\underline{p}(μ_{j}(n), n)), j = 1, …, p, \\ ν̂_{j}(n) & = (ν_{j}(n), −(2/c_{0,+})ν_{j}(n)^{p_{−}}G\underline{p}(ν_{j}(n), n)), j = 1, …, p. \end{aligned}  (3.34)

Then D_{\underline{μ̂}(n)} and D_{\underline{ν̂}(n)} are nonspecial for all n ∈ ℤ. Moreover, the Abel map linearizes the auxiliary divisors D\underline{μ̂} and D\underline{ν̂} in the sense that

\underline{α}_{Q_{0}}(D_{\underline{μ̂}(n)}) = \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n_{0})}) + (n − n_{0})\underline{A}_{P_{0,−}}(P_{∞_{+}}),  (3.35)
\underline{α}_{Q_{0}}(D_{\underline{ν̂}(n)}) = \underline{α}_{Q_{0}}(D_{\underline{ν̂}(n_{0})}) + (n − n_{0})\underline{A}_{P_{0,−}}(P_{∞_{+}}),  (3.36)

where Q_{0} ∈ K_{p} is a given base point. In addition,

\underline{α}_{Q_{0}}(D_{ˆ\underline{ν}(n)}) = \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n)}) + \underline{A}_{P_{0,−}}(P_{∞_{−}}).  (3.37)

For theta function representations of α and β we refer to [32] and the references cited therein. These theta function representations also show that γ(n) ∉ {0, 1} for all n ∈ ℤ, that is, the second condition in (3.2) is satisfied for the stationary algebro-geometric AL solutions discussed in this section provided the associated Dirichlet and Neumann divisors are admissible.

4. The Stationary Algorithm

The aim of this section is to derive an algorithm that enables one to construct algebro-geometric solutions for the stationary Ablowitz–Ladik hierarchy for general initial data. Equivalently, we offer a solution of the inverse algebro-geometric spectral problem for general Lax operators L in (2.60), starting with initial divisors in general position.

Up to the end of Section 3 the material was based on the assumption that α, β ∈ ℂ^{ℤ} satisfy the \underline{p}th stationary AL system (2.33). Now we embark on the corresponding inverse problem consisting of constructing a solution of (2.33) given certain initial data. More precisely, we seek to construct solutions α, β ∈ ℂ^{ℤ} satisfying the \underline{p}th stationary Ablowitz–Ladik system (2.33) starting from a properly restricted set M_{0} of admissible nonspecial Dirichlet divisor initial data D_{\underline{μ̂}(n_{0})} at some fixed n_{0} ∈ ℤ,

\begin{aligned}\underline{μ̂}(n_{0}) & = \{μ̂_{1}(n_{0}), …, μ̂_{p}(n_{0})\} ∈ M_{0}, M_{0} ⊂ Sym^{p}(K_{p}), \\ μ̂_{j}(n_{0}) & = (μ_{j}(n_{0}), (2/c_{0,+})μ_{j}(n_{0})^{p_{−}}G\underline{p}(μ_{j}(n_{0}), n_{0})), j = 1, …, p. \end{aligned}  (4.1)

For convenience we will frequently use the phrase that α, β blow up in this manuscript whenever one of the divisors D\underline{μ̂} or D\underline{ν̂} hits one of the points P_{∞_{±}}, P_{0,±}.

Of course we would like to ensure that the sequences α, β obtained via our algorithm do not blow up. To investigate when this happens, we study the image of our divisors under the Abel map. A key ingredient in our analysis will be (3.35) which yields a linear discrete dynamical system on the Jacobi variety J(K_{p}). In particular, we will be led to investigate solutions D\underline{μ̂}, D\underline{ν̂} of the discrete initial value problem

\begin{matrix}\underline{α}_{Q_{0}}(D_{\underline{μ̂}(n)}) = \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n_{0})}) + (n − n_{0})\underline{A}_{P_{0,−}}(P_{∞_{+}}), \\ \underline{μ̂}(n_{0}) = \{μ̂_{1}(n_{0}), …, μ̂_{p}(n_{0})\} ∈ Sym^{p}(K_{p}), \end{matrix}  (4.2)

respectively

\begin{matrix}\underline{α}_{Q_{0}}(D_{\underline{ν̂}(n)}) = \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n_{0})}) + \underline{A}_{P_{0,−}}(P_{∞_{−}}) + (n − n_{0})\underline{A}_{P_{0,−}}(P_{∞_{+}}), \\ \underline{ν̂}(n_{0}) = \{ν̂_{1}(n_{0}), …, ν̂_{p}(n_{0})\} ∈ Sym^{p}(K_{p}), \end{matrix}  (4.3)

where Q_{0} ∈ K_{p} is a given base point. Eventually, we will be interested in solutions D\underline{μ̂}, D\underline{ν̂} of (4.2), (4.3) with initial data D_{\underline{μ̂}(n_{0})} satisfying (4.1) and M_{0} to be specified as in (the proof of) Lemma 4.1.

Before proceeding to develop the stationary Ablowitz–Ladik algorithm, we briefly analyze the dynamics of (4.2).

Lemma 4.1. Let n ∈ ℤ and suppose that D_{\underline{μ̂}(n)} is defined via (4.2) for some divisor D_{\underline{μ̂}(n_{0})} ∈ Sym^{p}(K_{p}).

(i) If D_{\underline{μ̂}(n)} is nonspecial and does not contain any of the points P_{0,±}, P_{∞_{±}}, and D_{\underline{μ̂}(n+1)} contains one of the points P_{0,±}, P_{∞_{±}}, then D_{\underline{μ̂}(n+1)} contains P_{0,−} or P_{∞_{−}} but not P_{∞_{+}} or P_{0,+}. (ii) If D_{\underline{μ̂}(n)} is nonspecial and D_{\underline{μ̂}(n+1)} is special, then D_{\underline{μ̂}(n)} contains at least one of the points P_{∞_{+}}, P_{∞_{−}} and one of the points P_{0,+}, P_{0,−}. (iii) Item (i) holds if n + 1 is replaced by n − 1, P_{∞_{+}} by P_{∞_{−}}, and P_{0,+} by P_{0,−}. (iv) Items (i)–(iii) also hold for D_{\underline{ν̂}(n)}.

Proof. (i) Suppose one point in D_{\underline{μ̂}(n+1)} equals P_{∞_{+}} and denote the remaining ones by D_{˜\underline{μ}(n+1)}. Then (4.2) implies that \underline{α}_{Q_{0}}(D_{˜\underline{μ}(n+1)}) + \underline{A}_{Q_{0}}(P_{∞_{+}}) = \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n)}) + \underline{A}_{P_{0,−}}(P_{∞_{+}}). Since D_{\underline{μ̂}(n)} is assumed to be nonspecial one concludes D_{\underline{μ̂}(n)} = D_{˜\underline{μ}(n+1)} + D_{P_{0,−}}, contradicting our assumption on D_{\underline{μ̂}(n)}. The statement for P_{0,+} follows similarly; here we choose Q_{0} to be a branch point of K_{p} such that \underline{A}_{Q_{0}}(P^{∗}) = −\underline{A}_{Q_{0}}(P). (ii) Next, we choose Q_{0} to be a branch point of K_{p}. If D_{\underline{μ̂}(n+1)} is special, then it contains a pair of points (Q, Q^{∗}) whose contribution will cancel under the Abel map, that is, \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n+1)}) = \underline{α}_{Q_{0}}(D_{\underline{η̂}(n+1)}) for some D_{\underline{η̂}(n+1)} ∈ Sym^{p−2}(K_{p}). Invoking (4.2) then shows that \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n)}) = \underline{α}_{Q_{0}}(D_{\underline{η̂}(n+1)}) + \underline{A}_{Q_{0}}(P_{∞_{−}}) + \underline{A}_{Q_{0}}(P_{0,−}). As D_{\underline{μ̂}(n)} was assumed to be nonspecial, this shows that D_{\underline{μ̂}(n)} = D_{\underline{η̂}(n+1)} + D_{P_{∞_{−}}} + D_{P_{0,−}}, as claimed. (iii) This is proved as in item (i). (iv) Since D_{\underline{ν̂}(n)} satisfies the same equation as D_{\underline{μ̂}(n)} in (4.2) (cf. (3.36)), items (i)–(iii) also hold for D_{\underline{ν̂}(n)}. □

We also note the following result:

Lemma 4.2. Let n ∈ ℤ and assume that D_{\underline{μ̂}(n)} and D_{\underline{ν̂}(n)} are nonspecial. Then D_{\underline{μ̂}(n)} contains P_{0,−} if and only if D_{\underline{ν̂}(n)} contains P_{∞_{−}}. Moreover, D_{\underline{μ̂}(n)} contains P_{∞_{+}} if and only if D_{\underline{ν̂}(n)} contains P_{0,+}.

Proof. Suppose a point in D_{\underline{μ̂}(n)} equals P_{0,−} and denote the remaining ones by D_{˜\underline{μ}(n)}. By (3.37),

\underline{α}_{Q_{0}}(D_{\underline{ν̂}(n)}) = \underline{α}_{Q_{0}}(D_{˜\underline{μ}(n)}) + \underline{A}_{Q_{0}}(P_{0,−}) + \underline{A}_{P_{0,−}}(P_{∞_{−}}) = \underline{α}_{Q_{0}}(D_{˜\underline{μ}(n)}) + \underline{A}_{Q_{0}}(P_{∞_{−}}).

(4.4)

Since D_{\underline{ν̂}(n)} is nonspecial, D_{\underline{ν̂}(n)} contains P_{∞_{−}}, and vice versa. The second statement follows similarly. □

Let us call the points P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, and P_{0,−} exceptional points. Then Lemma 4.1 yields the following behavior of D_{\underline{μ̂}(n)} assuming one starts with some nonspecial initial divisor D_{\underline{μ̂}(n_{0})} without exceptional points: As n increases, D_{\underline{μ̂}(n)} stays nonspecial as long as it does not include exceptional points. If an exceptional point appears, D_{\underline{μ̂}(n)} is still nonspecial and contains P_{0,−} or P_{∞_{−}} at least once (but not P_{0,+} and P_{∞_{+}}). Further increasing n, all instances of P_{0,−} and P_{∞_{−}} will be rendered into P_{0,+} and P_{∞_{+}}, until we have again a nonspecial divisor that has the same number of P_{0,+} and P_{∞_{+}} as the first one had of P_{0,−} and P_{∞_{−}}. Generically, one expects the subsequent divisor to be nonspecial without exceptional points again.

Next we show that most initial divisors are well-behaved in the sense that their iterates stay away from P_{∞_{±}}, P_{0,±}. Since we want to show that this set is of full measure, it will be convenient to identify Sym^{p}(K_{p}) with the Jacobi variety J(K_{p}) via the Abel map and take the Haar measure on J(K_{p}). Of course, the Abel map is only injective when restricted to the set of nonspecial divisors, but these are the only ones we are interested in.

Lemma 4.3. The set M_{0} ⊂ Sym^{p}(K_{p}) of initial divisors D_{\underline{μ̂}(n_{0})} for which D_{\underline{μ̂}(n)} and D_{\underline{ν̂}(n)}, defined via (4.2) and (4.3), are admissible (i.e., do not contain the points P_{∞_{±}}, P_{0,±}) and hence are nonspecial for all n ∈ ℤ, forms a dense set of full measure in the set Sym^{p}(K_{p}) of positive divisors of degree p.

Proof. Let M_{∞,0} be the set of divisors in Sym^{p}(K_{p}) for which (at least) one point is equal to P_{∞_{±}} or P_{0,±}. The image \underline{α}_{Q_{0}}(M_{∞,0}) of M_{∞,0} is then contained in the following set,

\underline{α}_{Q_{0}}(M_{∞,0}) ⊆ ⋃_{P ∈\{P_{0,±},P_{∞_{±}} \}} (\underline{A}_{Q_{0}}(P) + \underline{α}_{Q_{0}}(Sym^{p−1}(K_{p}))) ⊂ J(K_{p}). (4.5)

Since the (complex) dimension of Sym^{p−1}(K_{p}) is p−1, its image must be of measure zero by Sard’s theorem (see, e.g., [8, Sect. 3.6]). Similarly, let M_{sp} be the set of special divisors, then its image is given by

\underline{α}_{Q_{0}}(M_{sp}) = \underline{α}_{Q_{0}}(Sym^{p−2}(K_{p})),  (4.6)

assuming Q_{0} to be a branch point. In particular, we conclude that \underline{α}_{Q_{0}}(M_{sp}) ⊂ \underline{α}_{Q_{0}}(M_{∞,0}) and thus \underline{α}_{Q_{0}}(M_{sing}) = \underline{α}_{Q_{0}}(M_{∞,0}) has measure zero, where

M_{sing} = M_{∞,0} ∪ M_{sp}.  (4.7)

Hence,

S_{μ} = ⋃_{n∈ℤ} (\underline{α}_{Q_{0}}(M_{sing}) + n\underline{A}_{P_{0,−}}(P_{∞_{+}})) and S_{ν} = S_{μ} + \underline{A}_{P_{0,−}}(P_{∞_{−}}) (4.8)

are of measure zero as well. But the set S_{μ} ∪ S_{ν} contains all initial divisors for which D_{\underline{μ̂}(n)} or D_{\underline{ν̂}(n)} will hit P_{∞_{±}} or P_{0,±}, or become special at some n ∈ ℤ. We denote by M_{0} the inverse image of the complement of the set S_{μ} ∪ S_{ν} under the Abel map,

M_{0} = \underline{α}_{Q_{0}}^{−1}(Sym^{p}(K_{p}) \ (S_{μ} ∪ S_{ν})).  (4.9)

Since M_{0} is of full measure, it is automatically dense in Sym^{p}(K_{p}). □

Next, we describe the stationary Ablowitz–Ladik algorithm. Since this is a somewhat lengthy affair, we will break it up into several steps. The Stationary Ablowitz–Ladik Algorithm:

We prescribe the following data

(i) The coefficient α(n_{0}) ∈ ℂ \ {0} and the constant c_{0,+} ∈ ℂ \ {0}. (ii) The set

{E_{m}}_{m=0}^{2p+1} ⊂ ℂ \ {0}, E_{m} ≠ E_{m′} for m ≠ m′, m, m′ = 0, …, 2p + 1, (4.10)

for some fixed p ∈ ℕ. Given {E_{m}}_{m=0}^{2p+1}, we introduce the function

R\underline{p}(z) = (\frac{c_{0,+}}{2z^{p_{−}}} )^{2} ∏_{m=0}^{2p+1}(z − E_{m})  (4.11)

and the hyperelliptic curve K_{p} with nonsingular affine part as in (3.1). (iii) The nonspecial divisor

D_{\underline{μ̂}(n_{0})} ∈ Sym^{p}(K_{p}),  (4.12)

where \underline{μ̂}(n_{0}) is of the form

\underline{μ̂}(n_{0}) = \{\underbrace{μ̂_{1}(n_{0}), …, μ̂_{1}(n_{0})}_{p_{1}(n_{0}) times}, …, \underbrace{μ̂_{q(n_{0})}, …, μ̂_{q(n_{0})}}_{p_{q(n_{0})}(n_{0}) times}\}  (4.13)

with

μ̂_{k}(n_{0}) = (μ_{k}(n_{0}), y(μ̂_{k}(n_{0}))), μ_{k}(n_{0}) ≠ μ_{k′}(n_{0}) for k ≠ k′, k, k′ = 1, …, q(n_{0}), (4.14) and

p_{k}(n_{0}) ∈ ℕ, k = 1, …, q(n_{0}), ∑_{k=1}^{q(n_{0})} p_{k}(n_{0}) = p.  (4.15)

With \{E_{m}\}_{m=0}^{2p+1}, D_{\underline{μ̂}(n_{0})}, α(n_{0}), and c_{0,+} prescribed, we next introduce the following quantities (for z ∈ ℂ \ {0}):

α^{+}(n_{0}) = α(n_{0})( ∏_{m=0}^{2p+1} E_{m})^{1/2} ∏_{k=1}^{q(n_{0})} μ_{k}(n_{0})^{−p_{k}(n_{0})},  (4.16)

c_{0,−}^{2} = c_{0,+}^{2} ∏_{m=0}^{2p+1} E_{m},  (4.17)

F\underline{p}(z, n_{0}) = −c_{0,+}α^{+}(n_{0})z^{−p_{−}} ∏_{k=1}^{q(n_{0})}(z − μ_{k}(n_{0}))^{p_{k}(n_{0})},  (4.18)

G\underline{p}(z, n_{0}) = \frac{1}{2}(\frac{1}{α(n_{0})} − \frac{z}{α^{+}(n_{0})})F\underline{p}(z, n_{0})  (4.19)

− \frac{z}{2α^{+}(n_{0})}F\underline{p}(z, n_{0}) ∑_{k=1}^{q(n_{0})} ∑_{ℓ=0}^{p_{k}(n_{0})−1} \frac{(d^{ℓ}(ζ^{−1}y(P))/dζ^{ℓ})|_{P =(ζ,η)=μ̂_{k}(n_{0})}}{ℓ!(p_{k}(n_{0}) − ℓ − 1)!}

× (\frac{d^{p_{k}(n_{0})−ℓ−1}}{dζ^{p_{k}(n_{0})−ℓ−1}} ((z − ζ)^{−1} ∏_{k′=1, k′≠k}^{q(n_{0})}(ζ − μ_{k′}(n_{0}))^{−p_{k′} (n_{0})}))|_{ζ=μ_{k}(n_{0})}.

Here the sign of the square root is chosen according to (4.14).

Next we record a series of facts: (I) By construction (cf. Lemma B.1),

\begin{matrix}\frac{d^{ℓ}(G\underline{p}(z, n_{0})^{2})}{dz^{ℓ}} |_{z=μ_{k}(n_{0})} = \frac{d^{ℓ}R\underline{p}(z)}{dz^{ℓ}} |_{z=μ_{k}(n_{0})}, \\ z ∈ ℂ \ \{0\}, ℓ = 0, …, p_{k}(n_{0}) − 1, k = 1, …, q(n_{0}). \end{matrix}  (4.20)

(II) Since D_{\underline{μ̂}(n_{0})} is nonspecial by hypothesis, one concludes that

p_{k}(n_{0}) ≥ 2 implies R\underline{p}(μ_{k}(n_{0})) ≠ 0, k = 1, …, q(n_{0}).  (4.21)

(III) By (4.19) and (4.20) one infers that F\underline{p} divides G2 ; \underline{p} − R\underline{p}. (IV) By (4.11) and (4.19) one verifies that

G\underline{p}(z, n_{0})^{2} − R\underline{p}(z) =_{z→∞} O(z^{2p_{+}−1}),  (4.22)
G\underline{p}(z, n_{0})^{2} − R\underline{p}(z) =_{z→0} O(z^{−2p_{−}+1}).  (4.23)

By (III) and (IV) we may write

G\underline{p}(z, n_{0})^{2} − R\underline{p}(z) = F\underline{p}(z, n_{0})Ȟ_{q,r}(z, n_{0}), z ∈ ℂ \ \{0\},  (4.24)

for some q ∈ {0, …, p_{−} − 1}, r ∈ {0, …, p_{+}}, where Ȟ_{q,r}(z, n_{0}) is a Laurent polynomial of the form c_{−q}z^{−q} + ⋯ + c_{r}z^{r}. If, in fact, Ȟ_{0,0} = 0, then R\underline{p}(z) = G\underline{p}(z, n_{0})^{2} would yield double zeros of R\underline{p}, contradicting our basic hypothesis (4.10).

Thus we conclude that in the case r = q = 0, Ȟ_{0,0} cannot vanish identically and hence we may break up (4.24) in the following manner

\begin{matrix}ϕ̌(P, n_{0}) = \frac{G\underline{p}(z, n_{0}) + (c_{0,+}/2)z^{−p_{−}}y}{F\underline{p}(z, n_{0})} = \frac{Ȟ_{q,r}(z, n_{0})}{G\underline{p}(z, n_{0}) − (c_{0,+}/2)z^{−p_{−}}y} , \\ P = (z, y) ∈ K_{p}. \end{matrix} (4.25)

Next we decompose

Ȟ_{q,r}(z, n_{0}) = Cz^{−q} ∏_{j=1}^{r+q}(z − ν_{j}(n_{0})), z ∈ ℂ \ {0},  (4.26)

where C ∈ ℂ \ {0} and {ν_{j}(n_{0})}_{j=1}^{r+q} ⊂ ℂ (if r = q = 0 we replace the product in (4.26) by 1). By inspection of the local zeros and poles as well as the behavior near P_{0,±}, P_{∞_{±}} of the function ϕ̌( · , n_{0}) using (3.11), its divisor, (ϕ̌( · , n_{0})), is given by

(ϕ̌( · , n_{0})) = D_{P_{0,−} ˆ\underline{ν}(n_{0})} − D_{P_{∞_{−}} ˆ \underline{μ}(n_{0})},  (4.27)

where

ˆ\underline{ν}(n_{0}) = \{\underbrace{P_{0,−}, …, P_{0,−}}_{p_{−}−1−q times}, ν̂_{1}(n_{0}), …, ν̂_{r+q}(n_{0}), \underbrace{P_{∞_{+}}, …, P_{∞_{+}}}_{p_{+}−r times}\}.  (4.28)

In the following we call a positive divisor of degree p admissible if it does not contain any of the points P_{∞_{±}}, P_{0,±}.

Hence,

D_{ˆ\underline{ν}(n_{0})} is an admissible divisor if and only if r = p_{+} and q = p_{−} − 1. (4.29)

We note that

\underline{α}_{Q_{0}}(D_{ˆ\underline{ν}(n_{0})}) = \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n_{0})}) + \underline{A}_{P_{0,−}}(P_{∞_{−}}),  (4.30)

in accordance with (3.37). (V) Assuming that (4.22), (4.23) are precisely of order z^{±(2p_{±}−1)}, that is, assuming r = p_{+} and q = p_{−} − 1 in (4.24), we rewrite (4.24) in the more appropriate manner

G\underline{p}(z, n_{0})^{2} − R\underline{p}(z) = F\underline{p}(z, n_{0})H\underline{p}(z, n_{0}), z ∈ ℂ \ \{0\}.  (4.31)

(We will later discuss conditions which indeed guarantee that q = p_{−}−1 and r = p_{+}, cf. (4.29) and the discussion in step (X) below.) By construction, H\underline{p}( · , n_{0}) is then of the type

H\underline{p}(z, n_{0}) = c_{0,+}β(n_{0})z^{−p_{−}+1} ∏_{k=1}^{ℓ(n_{0})}(z − ν_{k}(n_{0}))^{s_{k}(n_{0})}, ∑_{k=1}^{ℓ(n_{0})} s_{k}(n_{0}) = p,

ν_{k}(n_{0}) ≠ ν_{k′}(n_{0}) for k ≠ k′, k, k′ = 1, …, ℓ(n_{0}), z ∈ ℂ \ {0}, (4.32)

where we introduced the coefficient β(n_{0}). We define

ν̂_{k}(n_{0}) = (ν_{k}(n_{0}), −(2/c_{0,+})ν_{k}(n_{0})^{p_{−}}G\underline{p}(ν_{k}(n_{0}), n_{0})), k = 1, …, ℓ(n_{0}). (4.33)

An explicit computation of β(n_{0}) then yields

α^{+}(n_{0})β(n_{0}) = −\frac{1}{2} ∑_{k=1}^{q(n_{0})} \frac{(d^{p_{k}(n_{0})−1}(ζ^{−1}y(P))/dζ^{p_{k}(n_{0})−1})|_{P =(ζ,η)=μ̂_{k}(n_{0})}}{(p_{k}(n_{0}) − 1)!}

× ∏_{k′=1, k′≠k}^{q(n_{0})}(μ_{k}(n_{0}) − μ_{k′}(n_{0}))^{−p_{k}(n_{0})}

+ \frac{1}{2}(\frac{α^{+}(n_{0})}{α(n_{0})} + ∑_{k=1}^{q(n_{0})} p_{k}(n_{0})μ_{k}(n_{0}) − \frac{1}{2} ∑_{m=0}^{2p+1} E_{m}).  (4.34)

The result (4.34) is obtained by inserting the expressions (4.18), (4.19), and (4.32) for F\underline{p}( · , n_{0}), G\underline{p}( · , n_{0}), and H\underline{p}( · , n_{0}) into (4.31) and collecting all terms of order z^{2p_{+}−1}. (VI) Introduce

β^{+}(n_{0}) = β(n_{0}) ∏_{k=1}^{ℓ(n_{0})} ν_{k}(n_{0})^{s_{k}(n_{0})}( ∏_{m=0}^{2p+1} E_{m})^{−1/2}.  (4.35)

(VII) Using G\underline{p}(z, n_{0}), H\underline{p}(z, n_{0}), F\underline{p}(z, n_{0}), β(n_{0}), α^{+}(n_{0}), and β^{+}(n_{0}), we next construct the n_{0} ± 1 terms from the following equations:

F − ; \underline{p} = \frac{1}{zγ} (α^{2}H\underline{p} − 2αG\underline{p} + F\underline{p}),  (4.36)
H− ; \underline{p} = \frac{z}{γ} (β^{2}F\underline{p} − 2βG\underline{p} + H\underline{p}),  (4.37)
G− ; \underline{p} = \frac{1}{γ} ((1 + αβ)G\underline{p} − αH\underline{p} − βF\underline{p}),  (4.38)

respectively,

F + ; \underline{p} = \frac{1}{zγ^{+}} ((α^{+})^{2}H\underline{p} + 2α^{+}zG\underline{p} + z^{2}F\underline{p}),  (4.39)
H+ ; \underline{p} = \frac{1}{zγ^{+}} ((β^{+}z)^{2}F\underline{p} + 2β^{+}zG\underline{p} + H\underline{p}),  (4.40)
G+ ; \underline{p} = \frac{1}{zγ^{+}} ((1 + α^{+}β^{+})zG\underline{p} + α^{+}H\underline{p} + β^{+}z^{2}F\underline{p}).  (4.41)

Moreover,

(G− ; \underline{p})^{2} − F − ; \underline{p}H− ; \underline{p} = R\underline{p}, (G+ ; \underline{p})^{2} − F + ; \underline{p}H+ ; \underline{p} = R\underline{p}.  (4.42)

Inserting (4.18), (4.19), and (4.32) in (4.36)–(4.38) one verifies

F − ; \underline{p}(z, n_{0}) =_{z→∞} −c_{0,+}α(n_{0})z^{p_{+}−1} + O(z^{p_{+}−2}),  (4.43)
H− ; \underline{p}(z, n_{0}) =_{z→∞} O(z^{p_{+}}),  (4.44)
F − ; \underline{p}(z, n_{0}) =_{z→0} O(z^{−p_{−}}),  (4.45)
H− ; \underline{p}(z, n_{0}) =_{z→0} −c_{0,−}β(n_{0})z^{−p_{−}+1} + O(z^{−p_{−}+2}),  (4.46)
G− ; \underline{p}(z, n_{0}) = \frac{1}{2}c_{0,−}z^{−p_{−}} + ⋯ + \frac{1}{2}c_{0,+}z^{p_{+}}.  (4.47)

The last equation implies

G\underline{p}(z, n_{0} − 1)^{2} − R\underline{p}(z) =_{z→∞} O(z^{2p_{+}−1}),  (4.48)
G\underline{p}(z, n_{0} − 1)^{2} − R\underline{p}(z) =_{z→0} O(z^{−2p_{−}+1}),  (4.49)

so we may write

G\underline{p}(z, n_{0} −1)^{2} −R\underline{p}(z) = F̌_{s,p_{+}−1}(z, n_{0} −1)Ȟ_{p_{−}−1,r}(z, n_{0} −1), z ∈ ℂ\\{0\}, (4.50)

THE ABLOWITZ–LADIK HIERARCHY INITIAL VALUE PROBLEM 21

for some s ∈ {1, …, p_{−}}, r ∈ {1, …, p_{+}}, where

F̌_{s,p_{+}−1}(n_{0} − 1) = c_{−s}z^{−s} + ⋯ − c_{0,+}α(n_{0})z^{p_{+}−1},
Ȟ_{p_{−}−1,r}(n_{0} − 1) = −c_{0,−}β(n_{0})z^{−p_{−}+1} + ⋯ + c_{r}z^{r}.

The right-hand side of (4.50) cannot vanish identically (since otherwise R\underline{p}(z) = G\underline{p}(z, n_{0} − 1)^{2} would yield double zeros of R\underline{p}(z)), and hence,

ϕ̌(P, n_{0} − 1) = \frac{G\underline{p}(z, n_{0} − 1) + (c_{0,+}/2)z^{−p_{−}}y}{F̌_{s,p_{+}−1}(z, n_{0} − 1)} = \frac{Ȟ_{p_{−}−1,r}(z, n_{0} − 1)}{G\underline{p}(z, n_{0} − 1) − (c_{0,+}/2)z^{−p_{−}}y} ,

P = (z, y) ∈ K_{p}. (4.51)

Next, we decompose

F̌_{s,p_{+}−1}(z, n_{0} − 1) = −c_{0,+}α(n_{0})z^{−s} ∏_{j=1}^{p_{+}−1+s}(z − μ_{j}(n_{0} − 1)),  (4.52)
Ȟ_{p_{−}−1,r}(z, n_{0} − 1) = Cz^{−p_{−}+1} ∏_{j=1}^{p_{−}−1+r}(z − ν_{j}(n_{0} − 1)),  (4.53)

where C ∈ ℂ \ {0} and {μ_{j}(n_{0} − 1)}_{j=1}^{p_{+}−1+s} ⊂ ℂ, {ν_{j}(n_{0} − 1)}_{j=1}^{p_{−}−1+r} ⊂ ℂ. The divisor of ϕ̌( · , n_{0} − 1) is then given by

(ϕ̌( · , n_{0} − 1)) = D_{P_{0,−} ˆ\underline{ν}(n_{0}−1)} − D_{P_{∞_{−}} ˆ \underline{μ}(n_{0}−1)},  (4.54)

where

ˆ\underline{μ}(n_{0} − 1) = \{\underbrace{P_{0,+}, …, P_{0,+}}_{p_{−}−s times}, μ̂_{1}(n_{0} − 1), …, μ̂_{p_{+}−1+s}(n_{0} − 1)\},  (4.55)
ˆ\underline{ν}(n_{0} − 1) = \{ν̂_{1}(n_{0} − 1), …, ν̂_{p_{−}−1+r}(n_{0} − 1), \underbrace{P_{∞_{+}}, …, P_{∞_{+}}}_{p_{+}−r times}\}.  (4.56)

In particular,

D_{ˆ\underline{μ}(n_{0}−1)} is an admissible divisor if and only if s = p_{−},  (4.57)
D_{ˆ\underline{ν}(n_{0}−1)} is an admissible divisor if and only if r = p_{+}.  (4.58)

(VIII) Assuming that (4.48), (4.49) are precisely of order z^{±(2p_{±}−1)}, that is, assuming s = p_{−} and r = p_{+} in (4.51), we rewrite (4.51) as

G\underline{p}(z, n_{0} − 1)^{2} − R\underline{p}(z) = F\underline{p}(z, n_{0} − 1)H\underline{p}(z, n_{0} − 1), z ∈ ℂ \ \{0\}.  (4.59)

By construction, F\underline{p}( · , n_{0} − 1) and H\underline{p}( · , n_{0} − 1) are then of the type

F\underline{p}(z, n_{0} − 1) = −c_{0,+}α(n_{0})z^{−p_{−}} ∏_{k=1}^{q(n_{0}−1)}(z − μ_{j}(n_{0} − 1))^{p_{k}(n_{0}−1)},

∑_{k=1}^{q(n_{0}−1)} p_{k}(n_{0} − 1) = p,  (4.60)

μ_{k}(n_{0} − 1) ≠ μ_{k′}(n_{0} − 1) for k ≠ k′, k, k′ = 1, …, q(n_{0} − 1), z ∈ ℂ \ {0},

H\underline{p}(z, n_{0} − 1) = c_{0,+}β(n_{0} − 1)z^{−p_{−}+1} ∏_{k=1}^{ℓ(n_{0}−1)}(z − ν_{k}(n_{0} − 1))^{s_{k}(n_{0}−1)},

∑_{k=1}^{ℓ(n_{0}−1)} s_{k}(n_{0} − 1) = p,  (4.61)
ν_{k}(n_{0} − 1) ≠ ν_{k′}(n_{0} − 1) for k ≠ k′, k, k′ = 1, …, ℓ(n_{0} − 1), z ∈ ℂ \ {0},

where we introduced the coefficient β(n_{0} − 1). We define

\begin{matrix}μ̂_{k}(n_{0} − 1) = (μ_{k}(n_{0} − 1), (2/c_{0,+})μ_{k}(n_{0} − 1)^{p_{−}}G\underline{p}(μ_{k}(n_{0} − 1), n_{0} − 1)), \\ k = 1, …, q(n_{0} − 1), \\ ν̂_{k}(n_{0} − 1) = (ν_{k}(n_{0} − 1), −(2/c_{0,+})ν_{k}(n_{0} − 1)^{p_{−}}G\underline{p}(ν_{k}(n_{0} − 1), n_{0} − 1)), \\ k = 1, …, ℓ(n_{0} − 1). \end{matrix} (4.62)

(IX) At this point one can iterate the procedure step by step to construct F\underline{p}( · , n), G\underline{p}( · , n), H\underline{p}( · , n), α(n), β(n), μ_{j}(n), ν_{j}(n), etc., for n ∈ (−∞, n_{0}] ∩ ℤ, subject to the following assumption (cf. (4.57), (4.58)) at each step:

D_{\underline{μ̂}(n−1)} is an admissible divisor (and hence α(n − 1) ≠ 0)  (4.63)
for all n ∈ (−∞, n_{0}] ∩ ℤ,
D_{\underline{ν̂}(n−1)} is an admissible divisor (and hence β(n − 1) ≠ 0)  (4.64)
for all n ∈ (−∞, n_{0}] ∩ ℤ.

The formalism is symmetric with respect to n_{0} and can equally well be developed for n ∈ (−∞, n_{0}] ∩ ℤ subject to the analogous assumption

D_{\underline{μ̂}(n+1)} is an admissible divisor (and hence α(n + 2) ≠ 0)  (4.65)
for all n ∈ [n_{0}, ∞) ∩ ℤ,
D_{\underline{ν̂}(n+1)} is an admissible divisor (and hence β(n + 2) ≠ 0)  (4.66)
for all n ∈ [n_{0}, ∞) ∩ ℤ.

(X) Choosing the initial data D_{\underline{μ̂}(n_{0})} such that

D_{\underline{μ̂}(n_{0})} ∈ M_{0},  (4.67)

where M_{0} ⊂ Sym^{p}(K_{p}) is the set of admissible initial divisors introduced in Lemma 4.3, then guarantees that assumptions (4.63)–(4.66) are satisfied for all n ∈ ℤ.

Equations (4.36)–(4.41) (for arbitrary n ∈ ℤ) are equivalent to s-AL\underline{p}(α, β) = 0. At this stage we have verified the basic hypotheses of Section 3 (i.e., (3.2) and

the assumption that α, β satisfy the \underline{p}th stationary AL system (2.33)) and hence all results of Section 3 apply.

In summary, we proved the following result:

Theorem 4.4. Let n ∈ ℤ, suppose the set {E_{m}}_{m=0}^{2p+1} ⊂ ℂ satisfies E_{m} ≠ E_{m′} for m ≠ m′, m, m′ = 0, …, 2p + 1, and introduce the function R\underline{p} and the hyperelliptic curve K_{p} as in (3.1). Choose α(n_{0}) ∈ ℂ \ {0}, c_{0,+} ∈ ℂ \ {0}, and a nonspecial divisor D_{\underline{μ̂}(n_{0})} ∈ M_{0}, where M_{0} ⊂ Sym^{p}(K_{p}) is the set of admissible initial divisors introduced in Lemma 4.3. Then the stationary (complex) Ablowitz–Ladik algorithm as outlined in steps (I )–(X ) produces solutions α, β of the \underline{p}th stationary Ablowitz– Ladik system,

\begin{matrix}s-AL\underline{p}(α, β) = (\begin{matrix}−α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} \\ β(g_{p_{+},+}^{−} + g_{p_{−},−}) + h_{p_{+}−1,+}^{−} − h_{p_{−}−1,−} \end{matrix}) = 0, \\ \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}, \end{matrix}  (4.68)

satisfying (3.2) and

α(n) = ( ∏_{m=0}^{2p+1} E_{m})^{(n−n_{0})/2}A(n, n_{0}) α(n_{0}),  (4.69)

β(n) = ( − \frac{1}{2} ∑_{k=1}^{q(n)} \frac{(d^{p_{k}(n)−1}(ζ^{−1}y(P))/dζ^{p_{k}(n)−1})|_{P =(ζ,η)=μ̂_{k}(n)}}{(p_{k}(n) − 1)!}

× ∏_{k′=1, k′≠k}^{q(n)}(μ_{k}(n) − μ_{k′}(n))^{−p_{k}(n)}

+ \frac{1}{2}(( ∏_{m=0}^{2p+1} E_{m})^{1/2} ∏_{k=1}^{q(n)} μ_{k}(n)^{−p_{k}(n)} + ∑_{k=1}^{q(n)} p_{k}(n)μ_{k}(n) − \frac{1}{2} ∑_{m=0}^{2p+1} E_{m}))

× ( ∏_{m=0}^{2p+1} E_{m})^{−(n+1−n_{0})/2}A(n + 1, n_{0})^{−1} α(n_{0})^{−1},  (4.70)

where

A(n, n_{0}) = \begin{cases}∏_{n′=n_{0}}^{n−1} ∏_{k=1}^{q(n′)} μ_{k}(n′)^{−p_{k}(n′)}, & n ≥ n_{0} + 1, \\ 1, & n = n_{0}, \\ ∏_{n′=n}^{n_{0}−1} ∏_{k=1}^{q(n′)} μ_{k}(n′)^{p_{k}(n′)}, & n ≤ n_{0} − 1. \end{cases}  (4.71)

Moreover, Lemmas 3.2–3.5 apply.

Finally, we briefly illustrate some aspects of this analysis in the special case \underline{p} = (1, 1) (i.e., the case where (3.1) represents an elliptic Riemann surface) in more detail.

Example 4.5. The case \underline{p} = (1, 1). In this case one has

F_{(1,1)}(z, n) = −c_{0,+}α(n + 1)z^{−1}(z − μ_{1}(n)),
G_{(1,1)}(z, n) = \frac{1}{2}(\frac{1}{α(n)} − \frac{z}{α(n + 1)})F_{(1,1)}(z, n) + R_{(1,1)}(μ̂_{1}(n))^{1/2},  (4.72)

R_{(1,1)}(z) = (\frac{c_{0,+}α^{+}}{z} )^{2} ∏_{m=0}^{3}(z − E_{m}),

and hence a straightforward calculation shows that

G_{(1,1)}(z, n)^{2} − R_{(1,1)}(z) = −c_{0,+}^{2}α(n + 1)β(n)z^{−1}(z − μ_{1}(n))(z − ν_{1}(n))

= −\frac{c_{0,+}^{2}}{2z} (z − μ_{1}(n))(( − \frac{y(μ̂_{1}(n))}{μ_{1}(n)} + \frac{Ẽ^{1/2}}{μ_{1}(n)} + μ_{1}(n) − \frac{Ê^{+}}{2} )z (4.73)

− \frac{Ẽ}{μ_{1}(n)}( − \frac{1}{Ẽ^{1/2}} \frac{y(μ̂_{1}(n))}{μ_{1}(n)} + \frac{μ_{1}(n)}{Ẽ^{1/2}} + \frac{1}{μ_{1}(n)} − \frac{Ê^{−}}{2} )),

where

Ê^{±} = ∑_{m=0}^{3} E_{m}^{±1}, Ẽ = ∏_{m=0}^{3} E_{m}.  (4.74)

Solving for ν_{1}(n) one then obtains

ν_{1}(n) = \frac{Ẽ}{μ_{1}(n)} \frac{−\frac{y(μ̂_{1}(n))}{μ_{1}(n)} + \frac{Ẽ^{1/2}}{μ_{1}(n)} + μ_{1}(n) − \frac{Ê^{+}}{2}}{−\frac{1}{Ẽ^{1/2}} \frac{y(μ̂_{1}(n))}{μ_{1}(n)} + \frac{μ_{1}(n)}{Ẽ^{1/2}} + \frac{1}{μ_{1}(n)} − \frac{Ê^{−}}{2}} .  (4.75)

Thus, ν_{1}(n_{0}) could be 0 or ∞ even if μ_{1}(n_{0}) ≠ 0, ∞.

5. Properties of Algebro-Geometric Solutions of the Time-Dependent Ablowitz–Ladik Hierarchy

In this section we present a quick review of properties of algebro-geometric solutions of the time-dependent Ablowitz–Ladik hierarchy. Again we omit all proofs and refer to [29], [31], and [32] for details.

For most of this section we assume the following hypothesis.

Hypothesis 5.1. (i) Suppose that α, β satisfy

\begin{matrix}α( · , t), β( · , t) ∈ ℂ^{ℤ}, t ∈ ℝ, α(n, ·), β(n, ·) ∈ C^{1}(ℝ), n ∈ ℤ, \\ α(n, t)β(n, t) ∉ \{0, 1\}, (n, t) ∈ ℤ × ℝ. \end{matrix}  (5.1)

(ii) Assume that the hyperelliptic curve K_{p} satisfies (3.1) and (3.3).

In order to briefly analyze algebro-geometric solutions of the time-dependent Ablowitz–Ladik hierarchy we proceed as follows. Given \underline{p} ∈ ℕ_{0}^{2}, consider a complexvalued solution α^{(0)}, β^{(0)} of the \underline{p}th stationary Ablowitz–Ladik system s-AL\underline{p}(a, b) = 0, associated with K_{p} and a given set of summation constants {c_{ℓ,±}}_{ℓ=1,...,p_{±}} ⊂ ℂ. Next, let \underline{r} ∈ ℕ_{0}^{2}; we intend to consider solutions α = α(t\underline{r}), β = β(t\underline{r}) of the \underline{r}th AL flow AL\underline{r}(α, β) = 0 with α(t_{0,\underline{r}}) = α^{(0)}, β(t_{0,\underline{r}}) = β^{(0)} for some t_{0,\underline{r}} ∈ ℝ. To emphasize that the summation constants in the definitions of the stationary and the time-dependent Ablowitz–Ladik equations are independent of each other, we indicate this by adding a tilde on all the time-dependent quantities. Hence we shall employ the notation Ṽ\underline{r}, F̃\underline{r}, G̃\underline{r}, H̃\underline{r}, K̃\underline{r}, f̃_{s,±}, g̃_{s,±}, h̃_{s,±}, c̃_{s,±}, in order to distinguish them from V\underline{p}, F\underline{p}, G\underline{p}, H\underline{p}, K\underline{p}, f_{ℓ,±}, g_{ℓ,±}, h_{ℓ,±}, c_{ℓ,±}, in the following. In addition, we will follow a more elaborate notation inspired by Hirota’s τ-function approach and indicate the individual \underline{r}th Ablowitz–Ladik flow by a separate time variable t\underline{r} ∈ ℝ. More precisely, we will review properties of solutions α, β of the time-dependent algebro-geometric initial value problem

\begin{matrix}ÃL\underline{r}(α, β) = (\begin{matrix}−iα_{t\underline{r}} − α(g̃_{r_{+},+} + g̃_{r_{−},−}^{−}) + f̃_{r_{+}−1,+} − f̃_{r_{−}−1,−}^{−} \\ −iβ_{t\underline{r}} + β(g̃_{r_{+},+}^{−} + g̃_{r_{−},−}) − h̃_{r_{−}−1,−} + h̃_{r_{+}−1,+}^{−} \end{matrix}) = 0, \\ (α, β)|_{t\underline{r}=t_{0,\underline{r}}} = (α^{(0)}, β^{(0)}), \end{matrix}  (5.2)

s-AL\underline{p} (α^{(0)}, β^{(0)}) = (\begin{matrix}−α^{(0)}(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} \\ β^{(0)}(g_{p_{+},+}^{−} + g_{p_{−},−}) − h_{p_{−}−1,−} + h_{p_{+}−1,+}^{−} \end{matrix}) = 0 (5.3)

for some t_{0,\underline{r}} ∈ ℝ, where α = α(n, t\underline{r}), β = β(n, t\underline{r}) satisfy (5.1) and a fixed curve K_{p} is associated with the stationary solutions α^{(0)}, β^{(0)} in (5.3). Here,

\underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2} \ \{(0, 0)\}, \underline{r} = (r_{−}, r_{+}) ∈ ℕ_{0}^{2}, p = p_{−} + p_{+} − 1.  (5.4)

In terms of the zero-curvature formulation this amounts to solving

U_{t\underline{r}}(z, t\underline{r}) + U(z, t\underline{r})Ṽ\underline{r}(z, t\underline{r}) − Ṽ + ; \underline{r}(z, t\underline{r})U(z, t\underline{r}) = 0,  (5.5)
U(z, t_{0,\underline{r}})V\underline{p}(z, t_{0,\underline{r}}) − V + ; \underline{p}(z, t_{0,\underline{r}})U(z, t_{0,\underline{r}}) = 0.  (5.6)

One can show (cf. Lemma 6.2) that the stationary Ablowitz–Ladik system (5.6) is actually satisfied for all times t\underline{r} ∈ ℝ. Thus, we impose

U_{t\underline{r}} + UṼ\underline{r} − Ṽ + ; \underline{r}U = 0,  (5.7)
UV\underline{p} − V + ; \underline{p}U = 0,  (5.8)

instead of (5.5) and (5.6). For further reference, we recall the relevant quantities here (cf. (2.20), (2.21), (2.27)–(2.30), (2.38)):

\begin{aligned}U(z) & = (\begin{matrix}z & α \\ zβ & 1 \end{matrix}) , \\ V\underline{p}(z) & = i (\begin{matrix}G− ; \underline{p}(z) & −F − ; \underline{p}(z) \\ H− ; \underline{p}(z) & −G− ; \underline{p}(z) \end{matrix}) , Ṽ\underline{r}(z) = i (\begin{matrix}G̃− ; \underline{r}(z) & −F̃ − ; \underline{r}(z) \\ H̃− ; \underline{r}(z) & −K̃− ; \underline{r}(z) \end{matrix}) , \end{aligned}  (5.9)

and

F\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}z^{ℓ} = −c_{0,+}α^{+}z^{−p_{−}} ∏_{j=1}^{p}(z − μ_{j}),

G\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ},

H\underline{p}(z) = ∑_{ℓ=0}^{p_{−}−1} h_{p_{−}−1−ℓ,−}z^{−ℓ} + ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}z^{ℓ} = c_{0,+}βz^{−p_{−}+1} ∏_{j=1}^{p}(z − ν_{j}),

F̃\underline{r}(z) = ∑_{s=1}^{r_{−}} f̃_{r_{−}−s,−}z^{−s} + ∑_{s=0}^{r_{+}−1} f̃_{r_{+}−1−s,+}z^{s},  (5.10)

G̃\underline{r}(z) = ∑_{s=1}^{r_{−}} g̃_{r_{−}−s,−}z^{−s} + ∑_{s=0}^{r_{+}} g̃_{r_{+}−s,+}z^{s},

H̃\underline{r}(z) = ∑_{s=0}^{r_{−}−1} h̃_{r_{−}−1−s,−}z^{−s} + ∑_{s=1}^{r_{+}} h̃_{r_{+}−s,+}z^{s},

K̃\underline{r}(z) = ∑_{s=0}^{r_{−}} g̃_{r_{−}−s,−}z^{−s} + ∑_{s=1}^{r_{+}} g̃_{r_{+}−s,+}z^{s} = G̃\underline{r}(z) + g̃_{r_{−},−} − g̃_{r_{+},+}

for fixed \underline{p} ∈ ℕ_{0}^{2} \ \{(0, 0)\}, \underline{r} ∈ ℕ_{0}^{2}. Here f_{ℓ,±}, f̃_{s,±}, g_{ℓ,±}, g̃_{s,±}, h_{ℓ,±}, and h̃_{s,±} are defined as in (2.5)–(2.12) with appropriate sets of summation constants c_{ℓ,±}, ℓ ∈ ℕ_{0}, and c̃_{k,±}, k ∈ ℕ_{0}. Explicitly, (5.7) and (5.8) are equivalent to (cf. (2.23)–(2.26), (2.46)–(2.49))

α_{t\underline{r}} = i(zF̃ − ; \underline{r} + α(G̃\underline{r} + K̃− ; \underline{r}) − F̃\underline{r}),  (5.11)

β_{t\underline{r}} = −i(β(G̃− ; \underline{r} + K̃\underline{r}) − H̃\underline{r} + z^{−1}H̃− ; \underline{r}),  (5.12)
0 = z(G̃− ; \underline{r} − G̃\underline{r}) + zβF̃\underline{r} + αH̃− ; \underline{r},  (5.13)
0 = zβF̃ − ; \underline{r} + αH̃\underline{r} + K̃− ; \underline{r} − K̃\underline{r},  (5.14)
0 = z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p},  (5.15)
0 = zβF − ; \underline{p} + αH\underline{p} − G\underline{p} + G− ; \underline{p},  (5.16)
0 = −F\underline{p} + zF − ; \underline{p} + α(G\underline{p} + G− ; \underline{p}),  (5.17)
0 = zβ(G\underline{p} + G− ; \underline{p}) − zH\underline{p} + H− ; \underline{p},  (5.18)

respectively. In particular, (2.39) holds in the present t\underline{r}-dependent setting, that is,

G2 ; \underline{p} − F\underline{p}H\underline{p} = R\underline{p}.  (5.19)

As in the stationary context (3.8), (3.9) we introduce

\begin{aligned}μ̂_{j}(n, t\underline{r}) & = (μ_{j}(n, t\underline{r}), (2/c_{0,+})μ_{j}(n, t\underline{r})^{p_{−}}G\underline{p}(μ_{j}(n, t\underline{r}), n, t\underline{r})) ∈ K_{p}, \\ & j = 1, …, p, (n, t\underline{r}) ∈ ℤ × ℝ, \end{aligned}  (5.20)

and

\begin{aligned}ν̂_{j}(n, t\underline{r}) & = (ν_{j}(n, t\underline{r}), −(2/c_{0,+})ν_{j}(n, t\underline{r})^{p_{−}}G\underline{p}(ν_{j}(n, t\underline{r}), n, t\underline{r})) ∈ K_{p}, \\ & j = 1, …, p, (n, t\underline{r}) ∈ ℤ × ℝ, \end{aligned}  (5.21)

and note that the regularity assumptions (5.1) on α, β imply continuity of μ_{j} and ν_{k} with respect to t\underline{r} ∈ ℝ (away from collisions of these zeros, μ_{j} and ν_{k} are of course C^{∞}).

In analogy to (3.12), (3.13), one defines the following meromorphic function ϕ( · , n, t\underline{r}) on K_{p},

ϕ(P, n, t\underline{r}) = \frac{(c_{0,+}/2)z^{−p_{−}}y + G\underline{p}(z, n, t\underline{r})}{F\underline{p}(z, n, t\underline{r})}  (5.22)

= \frac{−H\underline{p}(z, n, t\underline{r})}{(c_{0,+}/2)z^{−p_{−}}y − G\underline{p}(z, n, t\underline{r})},  (5.23)
P = (z, y) ∈ K_{p}, (n, t\underline{r}) ∈ ℤ × ℝ,

with divisor (ϕ( · , n, t\underline{r})) of ϕ( · , n, t\underline{r}) given by

(ϕ( · , n, t\underline{r})) = D_{P_{0,−}\underline{ν̂}(n,t\underline{r})} − D_{P_{∞_{−}} \underline{μ̂}(n,t\underline{r})}.  (5.24)

The time-dependent Baker–Akhiezer vector is then defined in terms of ϕ by

Ψ(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = (\begin{matrix}ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) \\ ψ_{2}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) \end{matrix}) ,  (5.25)
ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = exp (i ∫_{t_{0,\underline{r}}}^{t\underline{r}} ds(G̃\underline{r}(z, n_{0}, s) − F̃\underline{r}(z, n_{0}, s)ϕ(P, n_{0}, s))) (5.26)

× \begin{cases}∏_{n′=n_{0}+1}^{n} (z + α(n′, t\underline{r})ϕ^{−}(P, n′, t\underline{r})), & n ≥ n_{0} + 1, \\ 1, & n = n_{0}, \\ ∏_{n′=n+1}^{n_{0}} (z + α(n′, t\underline{r})ϕ^{−}(P, n′, t\underline{r}))^{−1}, & n ≤ n_{0} − 1, \end{cases}

ψ_{2}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = exp (i ∫_{t_{0,\underline{r}}}^{t\underline{r}} ds(G̃\underline{r}(z, n_{0}, s) − F̃\underline{r}(z, n_{0}, s)ϕ(P, n_{0}, s))) (5.27)

× ϕ(P, n_{0}, t\underline{r}) \begin{cases}∏_{n′=n_{0}+1}^{n} (zβ(n′, t\underline{r})ϕ^{−}(P, n′, t\underline{r})^{−1} + 1), & n ≥ n_{0} + 1, \\ 1, & n = n_{0}, \\ ∏_{n′=n+1}^{n_{0}} (zβ(n′, t\underline{r})ϕ^{−}(P, n′, t\underline{r})^{−1} + 1)^{−1}, & n ≤ n_{0} − 1, \end{cases}

P = (z, y) ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}\}, (n, t\underline{r}) ∈ ℤ × ℝ.

One observes that

\begin{matrix}ψ_{1}(P, n, n_{0}, t\underline{r}, t̃\underline{r}) = ψ_{1}(P, n_{0}, n_{0}, t\underline{r}, t̃\underline{r})ψ_{1}(P, n, n_{0}, t\underline{r}, t\underline{r}), \\ P = (z, y) ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}\}, (n, n_{0}, t\underline{r}, t̃\underline{r}) ∈ ℤ^{2} × ℝ^{2}. \end{matrix}  (5.28)

The following lemma records basic properties of ϕ and Ψ in analogy to the stationary case discussed in Lemma 3.2.

Lemma 5.2 ([31]). Assume Hypothesis 5.1 and suppose that (5.7), (5.8) hold. In addition, let P = (z, y) ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}\}, (n, n_{0}, t\underline{r}, t_{0,\underline{r}}) ∈ ℤ^{2} × ℝ^{2}. Then ϕ satisfies

αϕ(P)ϕ^{−}(P) − ϕ^{−}(P) + zϕ(P) = zβ,  (5.29)
ϕ_{t\underline{r}}(P) = iF̃\underline{r}ϕ^{2}(P) − i(G̃\underline{r}(z) + K̃\underline{r}(z))ϕ(P) + iH̃\underline{r}(z),  (5.30)
ϕ(P)ϕ(P^{∗}) = \frac{H\underline{p}(z)}{F\underline{p}(z)},  (5.31)

ϕ(P) + ϕ(P^{∗}) = 2\frac{G\underline{p}(z)}{F\underline{p}(z)},  (5.32)
ϕ(P) − ϕ(P^{∗}) = c_{0,+}z^{−p_{−}} \frac{y(P)}{F\underline{p}(z)}.  (5.33)

Moreover, assuming P = (z, y) ∈ K_{p} \ {P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}}, then Ψ satisfies

ψ_{2}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = ϕ(P, n, t\underline{r})ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}),  (5.34)
U(z)Ψ^{−}(P) = Ψ(P),  (5.35)
V\underline{p}(z)Ψ^{−}(P) = −(i/2)c_{0,+}z^{−p_{−}}yΨ^{−}(P),  (5.36)
Ψ_{t\underline{r}}(P) = Ṽ + ; \underline{r}(z)Ψ(P),  (5.37)
ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}})ψ_{1}(P^{∗}, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = z^{n−n_{0}} \frac{F\underline{p}(z, n, t\underline{r})}{F\underline{p}(z, n_{0}, t_{0,\underline{r}})}Γ(n, n_{0}, t\underline{r}), (5.38)

ψ_{2}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}})ψ_{2}(P^{∗}, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = z^{n−n_{0}} \frac{H\underline{p}(z, n, t\underline{r})}{F\underline{p}(z, n_{0}, t_{0,\underline{r}})}Γ(n, n_{0}, t\underline{r}), (5.39)

ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}})ψ_{2}(P^{∗}, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) + ψ_{1}(P^{∗}, n, n_{0}, t\underline{r}, t_{0,\underline{r}})ψ_{2}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}})

= 2z^{n−n_{0}} \frac{G\underline{p}(z, n, t\underline{r})}{F\underline{p}(z, n_{0}, t_{0,\underline{r}})}Γ(n, n_{0}, t\underline{r}),  (5.40)

ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}})ψ_{2}(P^{∗}, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) − ψ_{1}(P^{∗}, n, n_{0}, t\underline{r}, t_{0,\underline{r}})ψ_{2}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}})

= −c_{0,+}z^{n−n_{0}−p_{−}} \frac{y}{F\underline{p}(z, n_{0}, t_{0,\underline{r}})}Γ(n, n_{0}, t\underline{r}),  (5.41)

where

Γ(n, n_{0}, t\underline{r}) = \begin{cases}∏_{n′=n_{0}+1}^{n} γ(n′, t\underline{r}) & n ≥ n_{0} + 1, \\ 1 & n = n_{0}, \\ ∏_{n′=n+1}^{n_{0}} γ(n′, t\underline{r})^{−1} & n ≤ n_{0} − 1. \end{cases}  (5.42)

In addition, as long as the zeros μ_{j}(n_{0}, s) of ( ·)^{p_{−}}F\underline{p}( · , n_{0}, s) are all simple and distinct from zero for s ∈ I_{μ}, I_{μ} ⊆ ℝ an open interval, Ψ( · , n, n_{0}, t\underline{r}, t_{0,\underline{r}}) is meromorphic on K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}\} for (n, t\underline{r}, t_{0,\underline{r}}) ∈ ℤ × I_{μ}^{2}.

The trace formulas recorded in Lemma 3.3 extend to the present time-dependent context without any change as t\underline{r} ∈ ℝ can be viewed as a fixed parameter. Further details are thus omitted.

For completeness we next mention the Dubrovin-type equations for the time variation of the zeros μ_{j} of ( ·)^{p_{−}}F\underline{p} and ν_{j} of ( ·)^{p_{−}−1}H\underline{p} governed by the ÃL\underline{r} flow.

Lemma 5.3 ([31]). Assume Hypothesis 5.1 and suppose that (5.7), (5.8) hold on ℤ × I_{μ} with I_{μ} ⊆ ℝ an open interval. In addition, assume that the zeros μ_{j}, j = 1, …, p, of ( ·)^{p_{−}}F\underline{p}( ·) remain distinct and nonzero on ℤ × I_{μ}. Then \{μ̂_{j}\}_{j=1,...,p}, defined in (5.20), satisfies the following first-order system of differential equations on ℤ × I_{μ},

μ_{j,t\underline{r}} = −iF̃\underline{r}(μ_{j})y(μ̂_{j})(α^{+})^{−1} ∏_{\begin{matrix}k=1 \\ k≠j \end{matrix}}^{p}(μ_{j} − μ_{k})^{−1}, j = 1, …, p,  (5.43)

with

μ̂_{j}(n, ·) ∈ C^{∞}(I_{μ}, K_{p}), j = 1, …, p, n ∈ ℤ.  (5.44)

For the zeros ν_{j}, j = 1, …, p, of ( ·)^{p_{−}−1}H\underline{p}( ·), identical statements hold with μ_{j} and I_{μ} replaced by ν_{j} and I_{ν}, etc. (with I_{ν} ⊆ ℝ an open interval ). In particular, {ν̂_{j}}_{j=1,...,p}, defined in (5.21), satisfies the first-order system on ℤ × I_{ν},

ν_{j,t\underline{r}} = iH̃\underline{r}(ν_{j})y(ν̂_{j})(βν_{j})^{−1} ∏_{\begin{matrix}k=1 \\ k≠j \end{matrix}}^{p}(ν_{j} − ν_{k})^{−1}, j = 1, …, p,  (5.45)

with

ν̂_{j}(n, ·) ∈ C^{∞}(I_{ν}, K_{p}), j = 1, …, p, n ∈ ℤ.  (5.46)

When attempting to solve the Dubrovin-type systems (5.43) and (5.45), they must be augmented with appropriate divisors D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} ∈ Sym^{p} K_{p}, t_{0,\underline{r}} ∈ I_{μ}, and D_{\underline{ν̂}(n_{0},t_{0,\underline{r}})} ∈ Sym^{p} K_{p}, t_{0,\underline{r}} ∈ I_{ν}, as initial conditions.

For the t\underline{r}-dependence of F\underline{p}, G\underline{p}, and H\underline{p} one obtains the following result.

Lemma 5.4 ([31]). Assume Hypothesis 5.1 and suppose that (5.7), (5.8) hold. In addition, let (z, n, t\underline{r}) ∈ ℂ × ℤ × ℝ. Then,

F_{\underline{p},t\underline{r}} = −2iG\underline{p}F̃\underline{r} + i(G̃\underline{r} + K̃\underline{r})F\underline{p},  (5.47)
G_{\underline{p},t\underline{r}} = iF\underline{p}H̃\underline{r} − iH\underline{p}F̃\underline{r},  (5.48)
H_{\underline{p},t\underline{r}} = 2iG\underline{p}H̃\underline{r} − i(G̃\underline{r} + K̃\underline{r})H\underline{p}.  (5.49)

In particular, (5.47)–(5.49) are equivalent to

V_{\underline{p},t\underline{r}} = [Ṽ\underline{r}, V\underline{p}].  (5.50)

It will be shown in Section 6 that Lemma 5.4 yields a first-order system of differential equations for f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±}, that serves as a pertinent substitute for the Dubrovin equations (5.43) even (in fact, especially) when some of the μ_{j} coincide.

Lemma 3.5 on nonspecial divisors and the linearization property of the Abel map extend to the present time-dependent setting. For this fact we need to introduce a particular differential of the second kind, Ω̃(2) ; \underline{r}, defined as follows. Let ω_{P_{∞_{±}} ,q}^{(2)}

and ω_{P_{0,±},q}^{(2)} be the normalized differentials of the second kind with a unique pole at P_{∞_{±}} and P_{0,±}, respectively, and principal parts

ω_{P_{∞_{±}} ,q}^{(2)} =_{ζ→0} (ζ^{−2−q} + O(1))dζ, P → P_{∞_{±}}, ζ = 1/z, q ∈ ℕ_{0},  (5.51)
ω_{P_{0,±},q}^{(2)} =_{ζ→0} (ζ^{−2−q} + O(1))dζ, P → P_{0,±}, ζ = z, q ∈ ℕ_{0},  (5.52)

with vanishing a-periods,

∫_{a_{j}} ω_{P_{∞_{±}} ,q}^{(2)} = ∫_{a_{j}} ω_{P_{0,±},q}^{(2)} = 0, j = 1, …, p.  (5.53)

Moreover, we define

Ω̃(2) ; \underline{r} = \frac{i}{2}( ∑_{s=1}^{r_{−}} sc̃_{r_{−}−s,−}(ω_{P_{0,+},s−1}^{(2)} − ω_{P_{0,−},s−1}^{(2)})

+ ∑_{s=1}^{r_{+}} sc̃_{r_{+}−s,+}(ω_{P_{∞_{+}} ,s−1}^{(2)} − ω_{P_{∞_{−}} ,s−1}^{(2)})),  (5.54)

where c̃_{ℓ,±} are the summation constants in F̃\underline{r}. The corresponding vector of bperiods of Ω̃(2) ; \underline{r}/(2πi) is then denoted by

˜\underline{U} (2) ; \underline{r} = (Ũ_{\underline{r},1}^{(2)}, …, Ũ_{\underline{r},p}^{(2)}), Ũ_{\underline{r},j}^{(2)} = \frac{1}{2πi} ∫_{b_{j}} Ω̃(2) ; \underline{r}, j = 1, …, p.  (5.55)

The time-dependent analog of Lemma 3.5 then reads as follows.

Lemma 5.5 ([31], [32]). Assume Hypothesis 5.1 and suppose that (5.7), (5.8) hold. Moreover, let (n, t\underline{r}) ∈ ℤ × ℝ. Denote by D\underline{μ̂}, \underline{μ̂} = \{μ̂_{1}, …, μ̂_{p}\} and D\underline{ν̂}, \underline{ν̂} = {ν̂_{1}, …, ν̂_{p}}, the pole and zero divisors of degree p, respectively, associated with α, β, and ϕ defined according to (5.20) and (5.21), that is,

μ̂_{j}(n, t\underline{r}) = (μ_{j}(n, t\underline{r}), (2/c_{0,+})μ_{j}(n, t\underline{r})^{p_{−}}G\underline{p}(μ_{j}(n, t\underline{r}), n, t\underline{r})), j = 1, …, p, (5.56)
ν̂_{j}(n, t\underline{r}) = (ν_{j}(n, t\underline{r}), −(2/c_{0,+})ν_{j}(n, t\underline{r})^{p_{−}}G\underline{p}(ν_{j}(n, t\underline{r}), n, t\underline{r})), j = 1, …, p. (5.57)

Then D_{\underline{μ̂}(n,t\underline{r})} and D_{\underline{ν̂}(n,t\underline{r})} are nonspecial for all (n, t\underline{r}) ∈ ℤ × ℝ. Moreover, the Abel map linearizes the auxiliary divisors D_{\underline{μ̂}(n,t\underline{r})} and D_{\underline{ν̂}(n,t\underline{r})} in the sense that

\underline{α}_{Q_{0}}(D_{\underline{μ̂}(n,t\underline{r})}) = \underline{α}_{Q_{0}}(D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})}) + (n − n_{0})\underline{A}_{P_{0,−}}(P_{∞_{+}}) + (t\underline{r} − t_{0,\underline{r}})˜\underline{U} (2) ; \underline{r}, (5.58)

\underline{α}_{Q_{0}}(D_{\underline{ν̂}(n,t\underline{r})}) = \underline{α}_{Q_{0}}(D_{\underline{ν̂}(n_{0},t_{0,\underline{r}})}) + (n − n_{0})\underline{A}_{P_{0,−}}(P_{∞_{+}}) + (t\underline{r} − t_{0,\underline{r}})˜\underline{U} (2) ; \underline{r}, (5.59)

where Q_{0} ∈ K_{p} is a given base point and ˜ \underline{U} (2) ; \underline{r} is the vector of b-periods introduced in (5.55).

Again we refer to [32] (and the references cited therein) for theta function representations of α and β. These theta function representations also show that γ(n, t\underline{r}) ∉ \{0, 1\} for all (n, t\underline{r}) ∈ ℤ × ℝ, that is, the last condition in (5.1) is satisfied for the time-dependent algebro-geometric AL solutions discussed in this section provided the associated Dirichlet and Neumann divisors are admissible.

6. The Algebro-Geometric Ablowitz–Ladik Hierarchy Initial Value Problem

In this section we consider the algebro-geometric Ablowitz–Ladik hierarchy initial value problem (5.2), (5.3) with complex-valued initial data. For a generic set of initial data we will prove unique solvability of the initial value problem globally in time.

As mentioned in the introduction, the strategy of basing the solution of the algebro-geometric initial value problem on the Dubrovin-type equations (5.43), (5.45), and the trace formulas for α and β, meets with serious difficulties as the Dirichlet eigenvalues μ̂_{j} may collide on K_{p} and hence the denominator of (5.43) can blow up. Hence, we will develop an alternative strategy based on the use of elementary symmetric functions of the variables {μ_{j}}_{j=1,...,p} in this section, which can accommodate collisions of μ̂_{j}. In short, our strategy will consist of the following:

(i) Replace the first-order autonomous Dubrovin-type system (5.43) of differential equations in t\underline{r} for the Dirichlet eigenvalues μ_{j}(n, t\underline{r}), j = 1, …, p, augmented by appropriate initial conditions, by the first-order autonomous system (6.57), (6.58) for the coefficients f_{ℓ,±}, h_{ℓ,±}, ℓ = 1, …, p_{±} − 1, and g_{ℓ,±}, ℓ = 1, …, p_{±}, with respect to t\underline{r}. Solve this first-order autonomous system in some time interval (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}) under appropriate initial conditions at (n_{0}, t_{0,\underline{r}}) derived from an initial (nonspecial) Dirichlet divisor D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})}.

(ii) Use the stationary algorithm derived in Section 4 to extend the solution of step (i) from \{n_{0}\} × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}) to ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}) (cf. Lemma 6.2).

(iii) Prove consistency of this approach, that is, show that the discrete algorithm of Section 4 is compatible with the time-dependent Lax and zero-curvature equations in the sense that first solving the autonomous system (6.57), (6.58) and then applying the discrete algorithm, or first applying the discrete algorithm and then solving the autonomous system (6.57), (6.58) yields the same result whenever the same endpoint (n, t\underline{r}) is reached (cf. Lemma 6.3 and Theorem 6.4).

(iv) Prove that there is a dense set of initial conditions of full measure for which this strategy yields global solutions of the algebro-geometric Ablowitz–Ladik hierarchy initial value problem.

To set up this formalism we need some preparations. From the outset we make the following assumption.

Hypothesis 6.1. Suppose that

α, β ∈ ℂ^{ℤ} and α(n)β(n) ∉ {0, 1} for all n ∈ ℤ,  (6.1)

and assume that α, β satisfy the \underline{p}th stationary Ablowitz–Ladik equation (2.33). In addition, suppose that the affine part of the hyperelliptic curve K_{p} in (3.1) is nonsingular.

We introduce a deformation (time) parameter t\underline{r} ∈ ℝ in α = α(t\underline{r}) and β = β(t\underline{r}) and hence obtain t\underline{r}-dependent quantities f_{ℓ} = f_{ℓ}(t\underline{r}), g_{ℓ} = g_{ℓ}(t\underline{r}), F\underline{p}(z) = F\underline{p}(z, t\underline{r}), G\underline{p}(z) = G\underline{p}(z, t\underline{r}), etc. At a fixed initial time t_{0,\underline{r}} ∈ ℝ we require that

(α, β)|_{t\underline{r}=t_{0,\underline{r}}} = (α^{(0)}, β^{(0)}),  (6.2)

where α^{(0)} = α( · , t_{0,\underline{r}}), β^{(0)} = β( · , t_{0,\underline{r}}) satisfy the \underline{p}th stationary Ablowitz–Ladik system (2.33). As discussed in Section 4, in order to guarantee that the stationary solutions (6.2) can be constructed for all n ∈ ℤ one starts from a particular divisor

D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} ∈ M_{0},  (6.3)

where \underline{μ̂}(n_{0}, t_{0,\underline{r}}) is of the form

\underline{μ̂}(n_{0}, t_{0,\underline{r}})  (6.4)

= \{\underbrace{μ̂_{1}(n_{0}, t_{0,\underline{r}}), …, μ̂_{1}(n_{0}, t_{0,\underline{r}})}_{p_{1}(n_{0},t_{0,\underline{r}}) times}, …, \underbrace{μ̂_{q(n_{0},t_{0,\underline{r}})}(n_{0}, t_{0,\underline{r}}), …, μ̂_{q(n_{0},t_{0,\underline{r}})}(n_{0}, t_{0,\underline{r}})}_{p_{q(n_{0},t_{0,\underline{r}})}(n_{0},t_{0,\underline{r}}) times}\}.

Moreover, as in Section 4 we prescribe the data

α(n_{0}, t_{0,\underline{r}}) ∈ ℂ \ \{0\} and c_{0,+} ∈ ℂ \ \{0\},  (6.5)

and of course the hyperelliptic curve K_{p} with nonsingular affine part (cf. (4.10)). In addition, we introduce

α^{+}(n_{0}, t_{0,\underline{r}}) = α(n_{0}, t_{0,\underline{r}})( ∏_{m=0}^{2p+1} E_{m})^{1/2} ∏_{k=1}^{q(n_{0},t_{0,\underline{r}})} μ_{k}(n_{0}, t_{0,\underline{r}})^{−p_{k}(n_{0},t_{0,\underline{r}})}, (6.6)

F\underline{p}(z, n_{0}, t_{0,\underline{r}}) = ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}(n_{0}, t_{0,\underline{r}})z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}(n_{0}, t_{0,\underline{r}})z^{ℓ}

= −c_{0,+}α^{+}(n_{0}, t_{0,\underline{r}})z^{−p_{−}} ∏_{k=1}^{q(n_{0},t_{0,\underline{r}})}(z − μ_{k}(n_{0}, t_{0,\underline{r}}))^{p_{k}(n_{0},t_{0,\underline{r}})}, (6.7)

G\underline{p}(z, n_{0}, t_{0,\underline{r}}) = \frac{1}{2}(\frac{1}{α(n_{0}, t_{0,\underline{r}})} − \frac{z}{α^{+}(n_{0}, t_{0,\underline{r}})})F\underline{p}(z, n_{0}, t_{0,\underline{r}})

− \frac{z}{2α^{+}(n_{0}, t_{0,\underline{r}})}F\underline{p}(z, n_{0}, t_{0,\underline{r}})

× ∑_{k=1}^{q(n_{0},t_{0,\underline{r}})} ∑_{ℓ=0}^{p_{k}(n_{0},t_{0,\underline{r}})−1} \frac{(d^{ℓ}(ζ^{−1}y(P))/dζ^{ℓ})|_{P =(ζ,η)=μ̂_{k}(n_{0},t_{0,\underline{r}})}}{ℓ!(p_{k}(n_{0}, t_{0,\underline{r}}) − ℓ − 1)!}  (6.8)

× (\frac{d^{p_{k}(n_{0},t_{0,\underline{r}})−ℓ−1}}{dζ^{p_{k}(n_{0},t_{0,\underline{r}})−ℓ−1}} ((z − ζ)^{−1}

× ∏_{k′=1, k′≠k}^{q(n_{0},t_{0,\underline{r}})}(ζ − μ_{k′}(n_{0}, t_{0,\underline{r}}))^{−p_{k′} (n_{0},t_{0,\underline{r}})}))|_{ζ=μ_{k}(n_{0},t_{0,\underline{r}})},

in analogy to (4.16).

Our aim is to find an autonomous first-order system of ordinary differential equations with respect to t\underline{r} for f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±} rather than for μ_{j}. We divide the differential equation

F_{\underline{p},t\underline{r}} = −2iG\underline{p}F̃\underline{r} + i(G̃\underline{r} + K̃\underline{r})F\underline{p}  (6.9)

by c_{0,+}z^{−p_{−}}y and rewrite it using Theorem C.1 as

∑_{ℓ=0}^{∞} f̂_{ℓ,+,t\underline{r}}ζ^{ℓ+1} = −2i( ∑_{s=1}^{r_{−}} f̃_{r_{−}−s,−}ζ^{s} + ∑_{s=0}^{r_{+}−1} f̃_{r_{+}−1−s,+}ζ^{−s}) ∑_{ℓ=0}^{∞} ĝ_{ℓ,+}ζ^{ℓ}

+ i(2 ∑_{s=0}^{r_{−}} g̃_{r_{−}−s,−}ζ^{s} + 2 ∑_{s=1}^{r_{+}} g̃_{r_{+}−s,+}ζ^{−s} − g̃_{r_{−},−} + g̃_{r_{+},+}) ∑_{ℓ=0}^{∞} f̂_{ℓ,+}ζ^{ℓ+1}, (6.10)
P → P_{∞_{−}}, ζ = 1/z.

The coefficients of ζ^{−s}, s = 0, …, r_{+} − 1, cancel since

∑_{k=0}^{ℓ} f̃_{ℓ−k,+}ĝ_{k,+} = ∑_{k=0}^{ℓ} g̃_{ℓ−k,+}f̂_{k,+}, ℓ ∈ ℕ_{0}.  (6.11)

In (6.11) we used (2.19),

f̃_{ℓ,+} = ∑_{k=0}^{ℓ} c̃_{ℓ−k,+}f̂_{k,+}, g̃_{ℓ,+} = ∑_{k=0}^{ℓ} c̃_{ℓ−k,+}ĝ_{k,+}.  (6.12)

Comparing coefficients in (6.10) then yields ^{4}

f̂_{ℓ,+,t\underline{r}} = if̂_{ℓ,+}(g̃_{r_{+},+} − g̃_{r_{−},−}) + 2i ∑_{k=0}^{r_{+}−1} (g̃_{k,+}f̂_{r_{+}+ℓ−k,+} − f̃_{k,+}ĝ_{r_{+}+ℓ−k,+}) (6.13)

− 2i ∑_{k=(ℓ+1−r_{−})∨0}^{ℓ} ĝ_{k,+}f̃_{r_{−}−1−ℓ+k,−} + 2i ∑_{k=(ℓ+2−r_{−})∨0}^{ℓ} f̂_{k,+}g̃_{r_{−}−ℓ+k,−}, ℓ ∈ ℕ_{0}.

By (6.11), the last sum in (6.13) can be rewritten as

∑_{j=0}^{r_{+}−1} (g̃_{j,+}f̂_{r_{+}+ℓ−j,+} − f̃_{j,+}ĝ_{r_{+}+ℓ−j,+})

= ( ∑_{j=0}^{r_{+}+ℓ} − ∑_{j=r_{+}}^{r_{+}+ℓ} )(g̃_{j,+}f̂_{r_{+}+ℓ−j,+} − f̃_{j,+}ĝ_{r_{+}+ℓ−j,+})

= − ∑_{j=r_{+}}^{r_{+}+ℓ} (g̃_{j,+}f̂_{r_{+}+ℓ−j,+} − f̃_{j,+}ĝ_{r_{+}+ℓ−j,+})

= ∑_{j=0}^{ℓ} (ĝ_{j,+}f̃_{r_{+}+ℓ−j,+} − f̂_{j,+}g̃_{r_{+}+ℓ−j,+}).  (6.14)

One performs a similar computation for f̂_{ℓ,−,t\underline{r}} using Theorem C.1 at P → P_{0,+}. In summary, since f_{k,±} = ∑_{ℓ=0}^{k} c_{k−ℓ,±}f̂_{ℓ,±}, (6.13) and (6.14) yield the following

4m ∨ n = max{m, n}.

THE ABLOWITZ–LADIK HIERARCHY INITIAL VALUE PROBLEM 33

autonomous first-order system (for fixed n = n_{0})

f_{ℓ,±,t\underline{r}} = F_{ℓ,±}(f_{j,−}, f_{j,+}, g_{j,−}, g_{j,+}), ℓ = 0, …, p_{±} − 1,  (6.15)

with initial conditions

f_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±} − 1,  (6.16)
g_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±},

where F_{ℓ,±}, ℓ = 0, …, p_{±} − 1, are polynomials in 2p + 3 variables,

F_{ℓ,±} = if_{ℓ,±}(g̃_{r_{±},±} − g̃_{r_{∓},∓})

+ 2i ∑_{k=0}^{ℓ} (f_{k,±}(g̃_{r_{∓}−ℓ+k,∓} − g̃_{r_{±}+ℓ−k,±}) + g_{k,±}(f̃_{r_{±}+ℓ−k,±} − f̃_{r_{∓}−1−ℓ+k,∓}))

+ 2i ∑_{k=0}^{ℓ} c_{ℓ−k,±} × \begin{cases}0, & 0 ≤ k < r_{∓} − 1, \\ ∑ & {k−r_{∓} \atop j=0} & ĝ_{j,±} & f̃_{r_{∓}−1−k+j,∓} \\ − ∑_{j=0}^{k+1−r_{∓}} f̂_{j,±}g̃_{r_{∓}−k+j,∓}, & k ≥ r_{∓} − 1. \end{cases} (6.17)

Explicitly, one obtains (for simplicity, r_{±} > 1)

F_{0,±} = if_{0,±}(g̃_{r_{∓},∓} − g̃_{r_{±},±}) + 2ig_{0,±}(f̃_{r_{±},±} − f̃_{r_{∓}−1,∓}),  (6.18)
F_{1,±} = 2if_{0,±}(g̃_{r_{∓}−1,∓} − g̃_{r_{±}+1,±}) + if_{1,±}(g̃_{r_{∓},∓} − g̃_{r_{±},±})
+ 2ig_{0,±}(f̃_{r_{±}+1,±} − f̃_{r_{∓}−2,∓}) + 2ig_{1,±}(f̃_{r_{±},±} − f̃_{r_{∓}−1,∓}), etc.

By (6.6)–(6.8), the initial conditions (6.16) are uniquely determined by the initial divisor D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} in (6.3) and by the data in (6.5).

Similarly, one transforms

G_{\underline{p},t\underline{r}} = iF\underline{p}H̃\underline{r} − iH\underline{p}F̃\underline{r},  (6.19)
H_{\underline{p},t\underline{r}} = 2iG\underline{p}H̃\underline{r} − i(G̃\underline{r} + K̃\underline{r})H\underline{p}  (6.20)

into (for fixed n = n_{0}) ^{5}

ĝ_{0,±,t\underline{r}} = 0,
ĝ_{ℓ,±,t\underline{r}} = i ∑_{k=0}^{r_{±}−1} (h̃_{k,±}f̂_{r_{±}−1+ℓ−k,±} − f̃_{k,±}ĥ_{r_{±}−1+ℓ−k,±})

+ i ∑_{k=(ℓ−r_{∓})∨0}^{ℓ−1} (f̂_{k,±}h̃_{r_{∓}−ℓ+k,∓} − ĥ_{k,±}f̃_{r_{∓}−ℓ+k,∓})

= i ∑_{k=0}^{ℓ−1} (ĥ_{k,±}f̃_{r_{±}−1+ℓ−k,±} − f̂_{k,±}h̃_{r_{±}−1+ℓ−k,±})

+ i ∑_{k=(ℓ−r_{∓})∨0}^{ℓ−1} (f̂_{k,±}h̃_{r_{∓}−ℓ+k,∓} − ĥ_{k,±}f̃_{r_{∓}−ℓ+k,∓}), ℓ ∈ ℕ,  (6.21)
ĥ_{ℓ,±,t\underline{r}} = iĥ_{ℓ,±}(g̃_{r_{∓},∓} − g̃_{r_{±},±}) + 2i ∑_{k=0}^{r_{±}−1} (h̃_{k,±}ĝ_{r_{±}+ℓ−k,±} − g̃_{k,±}ĥ_{r_{±}+ℓ−k,±})

5m ∨ n = max{m, n}.

34 F. GESZTESY, H. HOLDEN, J. MICHOR, AND G. TESCHL

+ 2i ∑_{k=(ℓ−r_{∓}+1)∨0}^{ℓ} ĝ_{k,±}h̃_{r_{∓}−1−ℓ+k,∓} − 2i ∑_{k=(ℓ−r_{∓})∨0}^{ℓ} ĥ_{k,±}g̃_{r_{∓}−ℓ+k,∓}

= iĥ_{ℓ,±}(g̃_{r_{∓},∓} − g̃_{r_{±},±}) + 2i ∑_{k=0}^{ℓ} (ĥ_{k,±}g̃_{r_{±}+ℓ−k,±} − ĝ_{k,±}h̃_{r_{±}+ℓ−k,±})

+ 2i ∑_{k=(ℓ−r_{∓}+1)∨0}^{ℓ} ĝ_{k,±}h̃_{r_{∓}−1−ℓ+k,∓} − 2i ∑_{k=(ℓ−r_{∓})∨0}^{ℓ} ĥ_{k,±}g̃_{r_{∓}−ℓ+k,∓},

ℓ ∈ ℕ_{0}. (6.22)

Summing over ℓ in (6.21), (6.22) then yields the following first-order system

g_{ℓ,±,t\underline{r}} = G_{ℓ,±}(f_{k,−}, f_{k,+}, h_{k,−}, h_{k,+}), ℓ = 0, …, p_{±},  (6.23)
h_{ℓ,±,t\underline{r}} = H_{ℓ,±}(g_{k,−}, g_{k,+}, h_{k,−}, h_{k,+}), ℓ = 0, …, p_{±} − 1,  (6.24)

with initial conditions

f_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±} − 1,
g_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±},  (6.25)
h_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±} − 1,

where G_{ℓ,±}, H_{ℓ,±}, are polynomials in 2p + 2, 2p + 3 variables

G_{ℓ,±} = i ∑_{k=0}^{ℓ−1} (f_{k,±}(h̃_{r_{∓}−ℓ+k,∓} − h̃_{r_{±}−1+ℓ−k,±}) + h_{k,±}(f̃_{r_{±}−1+ℓ−k,±} − f̃_{r_{∓}−ℓ+k,∓}))

− i ∑_{k=0}^{ℓ−1} c_{ℓ−1−k,±} × \begin{cases}0, & 0 ≤ k ≤ r_{∓}, \\ ∑_{j=0}^{k−r_{∓}−1}(f̂_{j,±}h̃_{r_{∓}−k+j,∓} − ĥ_{j,±}f̃_{r_{∓}−k+j,∓}), & k > r_{∓}, \end{cases} (6.26)

H_{ℓ,±} = ih_{ℓ,±}(g̃_{r_{∓},∓} − g̃_{r_{±},±})

+ 2i ∑_{k=0}^{ℓ} (g_{k,±}(h̃_{r_{∓}−1−ℓ+k,∓} − h̃_{r_{±}+ℓ−k,±}) + h_{k,±}(g̃_{r_{±}+ℓ−k,±} − g̃_{r_{∓}−ℓ+k,∓}))

+ 2i ∑_{k=0}^{ℓ} c_{ℓ−k,±} × \begin{cases}0, & 0 ≤ k < r_{∓}, \\ − ∑_{j=0}^{k−r_{∓}} ĝ_{j,±}h̃_{r_{∓}−1−k+j,∓} \\ + ∑_{j=0}^{k−r_{∓}−1} ĥ_{j,±}g̃_{r_{∓}−k+j,∓}, & k ≥ r_{∓}. \end{cases}  (6.27)

Explicitly (assuming r_{±} > 2),

G_{0,±} = 0,
G_{1,±} = if_{0,±}(h̃_{r_{∓}−1,∓} − h̃_{r_{±},±}) + ih_{0,±}(f̃_{r_{±},±} − f̃_{r_{∓}−1,∓}),  (6.28)
G_{2,±} = if_{0,±}(h̃_{r_{∓}−2,∓} − h̃_{r_{±}+1,±}) + if_{1,±}(h̃_{r_{∓}−1,∓} − h̃_{r_{±},±})
+ ih_{0,±}(f̃_{r_{±}+1,±} − f̃_{r_{∓}−2,∓}) + ih_{1,±}(f̃_{r_{±},±} − f̃_{r_{∓}−1,∓}), etc.,
H_{0,±} = 2ig_{0,±}(h̃_{r_{∓}−1,∓} − h̃_{r_{±},±}) + ih_{0,±}(g̃_{r_{±},±} − g̃_{r_{∓},∓}),  (6.29)
H_{1,±} = 2ig_{0,±}(h̃_{r_{∓}−2,∓} − h̃_{r_{±}+1,±}) + 2ig_{1,±}(h̃_{r_{∓}−1,∓} − h̃_{r_{±},±})
+ 2ih_{0,±}(g̃_{r_{±}+1,±} − g̃_{r_{∓}−1,∓}) + ih_{1,±}(g̃_{r_{±},±} − g̃_{r_{∓},∓}), etc.

Again by (6.6)–(6.8), the initial conditions (6.25) are uniquely determined by the initial divisor D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} in (6.3) and by the data in (6.5).

Being autonomous with polynomial right-hand sides, there exists a T_{0} > 0, such that the first-order initial value problem (6.15), (6.23), (6.24) with initial conditions (6.25) has a unique solution

f_{ℓ,±} = f_{ℓ,±}(n_{0}, t\underline{r}), ℓ = 0, …, p_{±} − 1,
g_{ℓ,±} = g_{ℓ,±}(n_{0}, t\underline{r}), ℓ = 0, …, p_{±},  (6.30)
h_{ℓ,±} = h_{ℓ,±}(n_{0}, t\underline{r}), ℓ = 0, …, p_{±} − 1,
for all t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0})

(cf., e.g., [54, Sect. III.10]). Given the solution (6.30), we proceed as in Section 4 and introduce the following quantities (where t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0})):

α^{+}(n_{0}, t\underline{r}) = α(n_{0}, t\underline{r})( ∏_{m=0}^{2p+1} E_{m})^{1/2} ∏_{k=1}^{q(n_{0},t\underline{r})} μ_{k}(n_{0}, t\underline{r})^{−p_{k}(n_{0},t\underline{r})},  (6.31)

F\underline{p}(z, n_{0}, t\underline{r}) = ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}(n_{0}, t\underline{r})z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}(n_{0}, t\underline{r})z^{ℓ}

= −c_{0,+}α^{+}(n_{0}, t\underline{r})z^{−p_{−}} ∏_{k=1}^{q(n_{0},t\underline{r})}(z − μ_{k}(n_{0}, t\underline{r}))^{p_{k}(n_{0},t\underline{r})},  (6.32)

G\underline{p}(z, n_{0}, t\underline{r}) = \frac{1}{2}(\frac{1}{α(n_{0}, t\underline{r})} − \frac{z}{α^{+}(n_{0}, t\underline{r})})F\underline{p}(z, n_{0}, t\underline{r})  (6.33)

− \frac{z}{2α^{+}(n_{0}, t\underline{r})}F\underline{p}(z, n_{0}, t\underline{r}) ∑_{k=1}^{q(n_{0},t\underline{r})} ∑_{ℓ=0}^{p_{k}(n_{0},t\underline{r})−1} \frac{(d^{ℓ}(ζ^{−1}y(P))/dζ^{ℓ})|_{P =(ζ,η)=μ̂_{k}(n_{0},t\underline{r})}}{ℓ!(p_{k}(n_{0}, t\underline{r}) − ℓ − 1)!}

× (\frac{d^{p_{k}(n_{0},t\underline{r})−ℓ−1}}{dζ^{p_{k}(n_{0},t\underline{r})−ℓ−1}} ((z − ζ)^{−1} ∏_{k′=1, k′≠k}^{q(n_{0},t\underline{r})}(ζ − μ_{k′}(n_{0}, t\underline{r}))^{−p_{k′} (n_{0},t\underline{r})}))|_{ζ=μ_{k}(n_{0},t\underline{r})}.

In particular, this leads to the divisor

D_{\underline{μ̂}(n_{0},t\underline{r})} ∈ Sym^{p}(K_{p})  (6.34)

and the sign of y in (6.32) is chosen as usual by

\begin{aligned}μ̂_{k}(n_{0}, t\underline{r}) & = (μ_{k}(n_{0}, t\underline{r}), (2/c_{0,+})μ_{j}(n_{0}, t\underline{r})^{p_{−}}G\underline{p}(μ_{k}(n_{0}, t\underline{r}), n_{0}, t\underline{r})), \\ & k = 1, …, q(n_{0}, t\underline{r}), \end{aligned}  (6.35)

and

\underline{μ̂}(n_{0}, t\underline{r}) = \{\underbrace{μ_{1}(n_{0}, t\underline{r}), …, μ_{1}(n_{0}, t\underline{r})}_{p_{1}(n_{0},t\underline{r}) times}, …, \underbrace{μ_{q(n_{0},t\underline{r})}(n_{0}, t\underline{r}), …, μ_{q(n_{0},t\underline{r})}(n_{0}, t\underline{r})}_{p_{q(n_{0},t\underline{r})}(n_{0},t\underline{r}) times}\}

(6.36)

with

μ_{k}(n_{0}, t\underline{r}) ≠ μ_{k′}(n_{0}, t\underline{r}) for k ≠ k′, k, k′ = 1, …, q(n_{0}, t\underline{r}),  (6.37)

and

p_{k}(n_{0}, t\underline{r}) ∈ ℕ, k = 1, …, q(n_{0}, t\underline{r}), ∑_{k=1}^{q(n_{0},t\underline{r})} p_{k}(n_{0}, t\underline{r}) = p.  (6.38)

By construction (cf. (6.35)), the divisor D_{\underline{μ̂}(n_{0},t\underline{r})} is nonspecial for all t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}).

In exactly the same manner as in (4.19)–(4.21) one then infers that F\underline{p}( · , n_{0}, t\underline{r}) divides R\underline{p} − G2 ; \underline{p} (since t\underline{r} is just a fixed parameter).

As in Section 4, the assumption that the Laurent polynomial F\underline{p}( · , n_{0} − 1, t\underline{r}) is of full order is implied by the hypothesis that

D_{\underline{μ̂}(n_{0},t\underline{r})} ∈ M_{0} for all t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}).  (6.39)

The explicit formula for β(n_{0}, t\underline{r}) then reads (for t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}))

α^{+}(n_{0}, t\underline{r})β(n_{0}, t\underline{r})  (6.40)

= −\frac{1}{2} ∑_{k=1}^{q(n_{0},t\underline{r})} \frac{(d^{p_{k}(n_{0},t\underline{r})−1}(ζ^{−1}y(P))/dζ^{p_{k}(n_{0},t\underline{r})−1})|_{P =(ζ,η)=μ̂_{k}(n_{0},t\underline{r})}}{(p_{k}(n_{0}, t\underline{r}) − 1)!}

× ∏_{k′=1, k′≠k}^{q(n_{0},t\underline{r})}(μ_{k}(n_{0}, t\underline{r}) − μ_{k′}(n_{0}, t\underline{r}))^{−p_{k}(n_{0},t\underline{r})}

+ \frac{1}{2}(\frac{α^{+}(n_{0}, t\underline{r})}{α(n_{0}, t\underline{r})} + ∑_{k=1}^{q(n_{0},t\underline{r})} p_{k}(n_{0}, t\underline{r})μ_{k}(n_{0}, t\underline{r}) − \frac{1}{2} ∑_{m=0}^{2p+1} E_{m}).  (6.41)

With (6.21)–(6.41) in place, we can now apply the stationary formalism as summarized in Theorem 4.4, subject to the additional hypothesis (6.39), for each fixed t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}). This yields, in particular, the quantities

F\underline{p}, G\underline{p}, H\underline{p}, α, β, and \underline{μ̂}, \underline{ν̂} for (n, t\underline{r}) ∈ ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}),  (6.42)

which are of the form (6.32)–(6.41), replacing the fixed n_{0} ∈ ℤ by an arbitrary n ∈ ℤ. In addition, one has the following result.

Lemma 6.2. Assume Hypothesis 6.1 and condition (6.39). Then the following relations are valid on ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}),

G2 ; \underline{p} − F\underline{p}H\underline{p} = R\underline{p},  (6.43)
z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p} = 0,  (6.44)
zβF − ; \underline{p} + αH\underline{p} − G\underline{p} + G− ; \underline{p} = 0,  (6.45)
−F\underline{p} + zF − ; \underline{p} + α(G\underline{p} + G− ; \underline{p}) = 0,  (6.46)
zβ(G\underline{p} + G− ; \underline{p}) − zH\underline{p} + H− ; \underline{p} = 0,  (6.47)

and hence the stationary part, (5.8), of the algebro-geometric initial value problem holds,

UV\underline{p} − V + ; \underline{p}U = 0 on ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}).  (6.48)

In particular, Lemmas 3.2–3.5 apply.

Lemma 6.2 now raises the following important consistency issue: On the one hand, one can solve the initial value problem (6.57), (6.58) at n = n_{0} in some interval t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}), and then extend the quantities F\underline{p}, G\underline{p}, H\underline{p} to all ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}) using the stationary algorithm summarized in Theorem 4.4 as just recorded in Lemma 6.2. On the other hand, one can solve the initial value problem (6.57), (6.58) at n = n_{1}, n_{1} ≠ n_{0}, in some interval t\underline{r} ∈ (t_{0,\underline{r}} −T_{1}, t_{0,\underline{r}} +T_{1}) with the initial condition obtained by applying the discrete algorithm to the quantities F\underline{p}, G\underline{p}, H\underline{p} starting at (n_{0}, t_{0,\underline{r}}) and ending at (n_{1}, t_{0,\underline{r}}). Consistency then requires that the two approaches yield the same result at n = n_{1} for t\underline{r} in some open neighborhood of t_{0,\underline{r}}.

Equivalently, and pictorially speaking, envisage a vertical t\underline{r}-axis and a horizontal n-axis. Then, consistency demands that first solving the initial value problem (6.57), (6.58) at n = n_{0} in some t\underline{r}-interval around t_{0,\underline{r}} and using the stationary algorithm to extend F\underline{p}, G\underline{p}, H\underline{p} horizontally to n = n_{1} and the same t\underline{r}-interval around t_{0,\underline{r}}, or first applying the stationary algorithm starting at (n_{0}, t_{0,\underline{r}}) to extend F\underline{p}, G\underline{p}, H\underline{p} horizontally to (n_{1}, t_{0,\underline{r}}) and then solving the initial value problem (6.57), (6.58) at n = n_{1} in some t\underline{r}-interval around t_{0,\underline{r}} should produce the same result at n = n_{1} in a sufficiently small open t\underline{r} interval around t_{0,\underline{r}}.

To settle this consistency issue, we will prove the following result. To this end we find it convenient to replace the initial value problem (6.57), (6.58) by the original t\underline{r}-dependent zero-curvature equation (5.7), U_{t\underline{r}} + UṼ\underline{r} − Ṽ + ; \underline{r}U = 0 on ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}).

Lemma 6.3. Assume Hypothesis 6.1 and condition (6.39). Moreover, suppose that (5.47)–(5.49) hold on ℂ × \{n_{0}\} × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}). Then (5.47)–(5.49) hold on ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}), that is,

F_{\underline{p},t\underline{r}}(z, n, t\underline{r}) = −2iG\underline{p}(z, n, t\underline{r})F̃\underline{r}(z, n, t\underline{r})
+ i(G̃\underline{r}(z, n, t\underline{r}) + K̃\underline{r}(z, n, t\underline{r}))F\underline{p}(z, n, t\underline{r}),  (6.49)
G_{\underline{p},t\underline{r}}(z, n, t\underline{r}) = iF\underline{p}(z, n, t\underline{r})H̃\underline{r}(z, n, t\underline{r}) − iH\underline{p}(z, n, t\underline{r})F̃\underline{r}(z, n, t\underline{r}),  (6.50)
H_{\underline{p},t\underline{r}}(z, n, t\underline{r}) = 2iG\underline{p}(z, n, t\underline{r})H̃\underline{r}(z, n, t\underline{r})
− i(G̃\underline{r}(z, n, t\underline{r}) + K̃\underline{r}(z, n, t\underline{r}))H\underline{p}(z, n, t\underline{r}),  (6.51)
(z, n, t\underline{r}) ∈ ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}).

Moreover,

ϕ_{t\underline{r}}(P, n, t\underline{r}) = iF̃\underline{r}(z, n, t\underline{r})ϕ^{2}(P, n, t\underline{r})
− i(G̃\underline{r}(z, n, t\underline{r}) + K̃\underline{r}(z, n, t\underline{r}))ϕ(P, n, t\underline{r}) + iH̃\underline{r}(z, n, t\underline{r}), (6.52)
α_{t\underline{r}}(n, t\underline{r}) = izF̃ − ; \underline{r}(z, n, t\underline{r})
+ iα(n, t\underline{r})(G̃\underline{r}(z, n, t\underline{r}) + K̃− ; \underline{r}(z, n, t\underline{r})) − iF̃\underline{r}(z, n, t\underline{r}),  (6.53)
β_{t\underline{r}}(n, t\underline{r}) = −iβ(n, t\underline{r})(G̃− ; \underline{r}(z, n, t\underline{r}) + K̃\underline{r}(z, n, t\underline{r}))
+ iH̃\underline{r}(z, n, t\underline{r}) − iz^{−1}H̃− ; \underline{r}(z, n, t\underline{r}),  (6.54)
(z, n, t\underline{r}) ∈ ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}).

Proof. By Lemma 6.2 we have (5.22), (5.23), (5.29), (5.31)–(5.33), and (6.43)–(6.47) for (n, t\underline{r}) ∈ ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}) at our disposal.

Differentiating (5.22) at n = n_{0} with respect to t\underline{r} and inserting (6.49) and (6.50) at n = n_{0} then yields (6.52) at n = n_{0}.

We note that the sequences f̃_{ℓ,±}, g̃_{ℓ,±}, h̃_{ℓ,±} satisfy the recursion relations (2.6)– (2.12) (since the homogeneous sequences satisfy these relations). Hence, to prove (6.53) and (6.54) at n = n_{0} it remains to show

\begin{aligned}α_{t\underline{r}} & = iα(g̃_{r_{+},+} + g̃_{r_{−},−}^{−}) + i(f̃_{r_{−}−1,−}^{−} − f̃_{r_{+}−1,+}), \\ β_{t\underline{r}} & = −iβ(g̃_{r_{+},+}^{−} + g̃_{r_{−},−}) − i(h̃_{r_{+}−1,+}^{−} − h̃_{r_{−}−1,−}). \end{aligned}  (6.55)

But this follows from (6.49), (6.51) at n = n_{0} (cf. (6.18), (6.29))

α_{t\underline{r}} = iα(g̃_{r_{+},+} − g̃_{r_{−},−}) + i(f̃_{r_{−},−} − f̃_{r_{+}−1,+}),
β_{t\underline{r}} = iβ(g̃_{r_{+},+} − g̃_{r_{−},−}) + i(h̃_{r_{−}−1,−} − h̃_{r_{+},+}).

Inserting now (2.11) at ℓ = r_{−} − 1 and (2.8) at ℓ = r_{+} − 1 then yields (6.55).

For the step n = n_{0} ∓ 1 we differentiate (4.36)–(4.41) (which are equivalent to (6.43)–(6.47)) and insert (6.49)–(6.51), (5.11)–(5.18) at n = n_{0}. For the case n > n_{0} we obtain α_{t\underline{r}}^{+} and β_{t\underline{r}}^{+} from (6.49), (6.51) at n = n_{0} as before using the other two signs in (6.18), (6.29). Iterating these arguments proves (6.49)–(6.54) for (z, n, t\underline{r}) ∈ ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}). □

We summarize Lemmas 6.2 and 6.3 next.

Theorem 6.4. Assume Hypothesis 6.1 and condition (6.39). Moreover, suppose that

f_{ℓ,±} = f_{ℓ,±}(n_{0}, t\underline{r}), ℓ = 0, …, p_{±} − 1,
g_{ℓ,±} = g_{ℓ,±}(n_{0}, t\underline{r}), ℓ = 0, …, p_{±},  (6.56)
h_{ℓ,±} = h_{ℓ,±}(n_{0}, t\underline{r}), ℓ = 0, …, p_{±} − 1
for all t\underline{r} ∈ (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}),

satisfy the autonomous first-order system of ordinary differential equations (for fixed n = n_{0})

f_{ℓ,±,t\underline{r}} = F_{ℓ,±}(f_{k,−}, f_{k,+}, g_{k,−}, g_{k,+}), ℓ = 0, …, p_{±} − 1,
g_{ℓ,±,t\underline{r}} = G_{ℓ,±}(f_{k,−}, f_{k,+}, h_{k,−}, h_{k,+}), ℓ = 0, …, p_{±},  (6.57)
h_{ℓ,±,t\underline{r}} = H_{ℓ,±}(g_{k,−}, g_{k,+}, h_{k,−}, h_{k,+}), ℓ = 0, …, p_{±} − 1,

with F_{ℓ,±}, G_{ℓ,±}, H_{ℓ,±} given by (6.17), (6.26), (6.27), and with initial conditions

f_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±} − 1,
g_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±},  (6.58)
h_{ℓ,±}(n_{0}, t_{0,\underline{r}}), ℓ = 0, …, p_{±} − 1.

Then F\underline{p}, G\underline{p}, and H\underline{p} as constructed in (6.32)–(6.42) on ℂ×ℤ×(t_{0,\underline{r}} −T_{0}, t_{0,\underline{r}} +T_{0}) satisfy the zero-curvature equations (5.7), (5.8), and (5.50) on ℂ × ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}),

U_{t\underline{r}} + UṼ\underline{r} − Ṽ + ; \underline{r}U = 0,  (6.59)
UV\underline{p} − V + ; \underline{p}U = 0,  (6.60)
V_{\underline{p},t\underline{r}} − [Ṽ\underline{r}, V\underline{p}] = 0  (6.61)

with U, V\underline{p}, and Ṽ\underline{r} given by (5.9). In particular, α, β satisfy (5.1) and the algebrogeometric initial value problem (5.2), (5.3) on ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}),

\begin{matrix}ÃL\underline{r}(α, β) = (\begin{matrix}−iα_{t\underline{r}} − α(g̃_{r_{+},+} + g̃_{r_{−},−}^{−}) + f̃_{r_{+}−1,+} − f̃_{r_{−}−1,−}^{−} \\ −iβ_{t\underline{r}} + β(g̃_{r_{+},+}^{−} + g̃_{r_{−},−}) − h̃_{r_{−}−1,−} + h̃_{r_{+}−1,+}^{−} \end{matrix}) = 0, \\ (α, β)|_{t=t_{0,\underline{r}}} = (α^{(0)}, β^{(0)}), \end{matrix}  (6.62)

s-AL\underline{p} (α^{(0)}, β^{(0)}) = (\begin{matrix}−α^{(0)}(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{−}−1,+} − f_{p_{−}−1,−}^{−} \\ β^{(0)}(g_{p_{+},+}^{−} + g_{p_{−},−}) − h_{p_{−}−1,−} + h_{p_{+}−1,+}^{−} \end{matrix}) = 0. (6.63)

In addition, α, β are given by

α^{+}(n, t\underline{r}) = α(n, t\underline{r})( ∏_{m=0}^{2p+1} E_{m})^{1/2} ∏_{k=1}^{q(n,t\underline{r})} μ_{k}(n, t\underline{r})^{−p_{k}(n,t\underline{r})},  (6.64)

α^{+}(n, t\underline{r})β(n, t\underline{r}) = −\frac{1}{2} ∑_{k=1}^{q(n,t\underline{r})} \frac{(d^{p_{k}(n,t\underline{r})−1}(ζ^{−1}y(P))/dζ^{p_{k}(n,t\underline{r})−1})|_{P =(ζ,η)=μ̂_{k}(n,t\underline{r})}}{(p_{k}(n, t\underline{r}) − 1)!}

× ∏_{k′=1, k′≠k}^{q(n,t\underline{r})}(μ_{k}(n, t\underline{r}) − μ_{k′}(n, t\underline{r}))^{−p_{k}(n,t\underline{r})}  (6.65)

+ \frac{1}{2}(( ∏_{m=0}^{2p+1} E_{m})^{1/2} ∏_{k=1}^{q(n,t\underline{r})} μ_{k}(n, t\underline{r})^{−p_{k}(n,t\underline{r})} + ∑_{k=1}^{q(n,t\underline{r})} p_{k}(n, t\underline{r})μ_{k}(n, t\underline{r})

− \frac{1}{2} ∑_{m=0}^{2p+1} E_{m}), (z, n, t\underline{r}) ∈ ℤ × (t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}).

Moreover, Lemmas 3.2–3.5 and 5.2–5.4 apply.

As in Lemma 4.3 we now show that also in the time-dependent case, most initial divisors are well-behaved in the sense that the corresponding divisor trajectory stays away from P_{∞_{±}}, P_{0,±} for all (n, t\underline{r}) ∈ ℤ × ℝ.

Lemma 6.5. The set M_{1} of initial divisors D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} for which D_{\underline{μ̂}(n,t\underline{r})} and D_{\underline{ν̂}(n,t\underline{r})}, defined via (5.58) and (5.59), are admissible (i.e., do not contain P_{∞_{±}}, P_{0,±}) and hence are nonspecial for all (n, t\underline{r}) ∈ ℤ × ℝ, forms a dense set of full measure in the set Sym^{p}(K_{p}) of nonnegative divisors of degree p.

Proof. Let M_{sing} be as introduced in the proof of Lemma 4.3. Then

⋃_{t\underline{r}∈ℝ} (\underline{α}_{Q_{0}}(M_{sing}) + t\underline{r} ˜\underline{U} (2) ; \underline{r})

⊆ ⋃_{P ∈\{P_{∞_{±}} ,P_{0,±}\}} ⋃_{t\underline{r}∈ℝ} (\underline{A}_{Q_{0}}(P) + \underline{α}_{Q_{0}}(Sym^{p−1}(K_{p})) + t\underline{r} ˜\underline{U} (2) ; \underline{r})  (6.66)

is of measure zero as well, since it is contained in the image of ℝ × Sym^{p−1}(K_{p}) which misses one real dimension in comparison to the 2p real dimensions of J(K_{p}). But then

⋃_{(n,t\underline{r})∈ℤ×ℝ} (\underline{α}_{Q_{0}}(M_{sing}) + n\underline{A}_{P_{0,−}}(P_{∞_{+}}) + t\underline{r} ˜\underline{U} (2) ; \underline{r})

∪ ( ⋃_{(n,t\underline{r})∈ℤ×ℝ} (\underline{α}_{Q_{0}}(M_{sing}) + n\underline{A}_{P_{0,−}}(P_{∞_{+}}) + t\underline{r} ˜\underline{U} (2) ; \underline{r}) + \underline{A}_{P_{0,−}}(P_{∞_{−}})) (6.67)

is also of measure zero. Applying \underline{α}_{Q_{0}}^{−1} to the complement of the set in (6.67) then yields a set M_{1} of full measure in Sym^{p}(K_{p}). In particular, M_{1} is necessarily dense in Sym^{p}(K_{p}). □

Theorem 6.6. Let D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} ∈ M_{1} be an initial divisor as in Lemma 6.5. Then the sequences α, β constructed from \underline{μ̂}(n_{0}, t_{0,\underline{r}}) as described in Theorem 6.4 satisfy Hypothesis 5.1. In particular, the solution α, β of the algebro-geometric initial value problem (6.64), (6.65) is global in (n, t\underline{r}) ∈ ℤ × ℝ.

Proof. Starting with D_{\underline{μ̂}(n_{0},t_{0,\underline{r}})} ∈ M_{1}, the procedure outlined in this section and summarized in Theorem 6.4 leads to D_{\underline{μ̂}(n,t\underline{r})} and D_{\underline{ν̂}(n,t\underline{r})} for all (n, t\underline{r}) ∈ ℤ×(t_{0,\underline{r}} − T_{0}, t_{0,\underline{r}} + T_{0}) such that (5.58) and (5.59) hold. But if α, β should blow up, then D_{\underline{μ̂}(n,t\underline{r})} or D_{\underline{ν̂}(n,t\underline{r})} must hit one of P_{∞_{±}} or P_{0,±}, which is excluded by our choice of initial condition. □

We note, however, that in general (i.e., unless one is, e.g., in the special periodic case), D_{\underline{μ̂}(n,t\underline{r})} will get arbitrarily close to P_{∞_{±}}, P_{0,±} since straight motions on the torus are generically dense (see e.g. [12, Sect. 51] or [35, Sects. 1.4, 1.5]) and hence no uniform bound (and no uniform bound away from zero) on the sequences α(n, t\underline{r}), β(n, t\underline{r}) exists as (n, t\underline{r}) varies in ℤ×ℝ. In particular, these complex-valued algebro-geometric solutions of the Ablowitz–Ladik hierarchy initial value problem, in general, will not be quasi-periodic with respect to n or t\underline{r} (cf. the usual definition of quasi-periodic functions, e.g., in [46, p. 31]).

Appendix A. Hyperelliptic Curves in a Nutshell

We provide a very brief summary of some of the fundamental properties and notations needed from the theory of hyperelliptic curves. More details can be found in some of the standard textbooks [20], [21], and [42], as well as monographs dedicated to integrable systems such as [13, Ch. 2], [29, App. A, B], [51, App. A].

Fix p ∈ ℕ. The hyperelliptic curve K_{p} of genus p used in Sections 3–6 is defined by

K_{p}: F_{p}(z, y) = y^{2} − R_{2p+2}(z) = 0, R_{2p+2}(z) = ∏_{m=0}^{2p+1}(z − E_{m}),  (A.1)
{E_{m}}_{m=0,...,2p+1} ⊂ ℂ, E_{m} ≠ E_{m′} for m ≠ m′, m, m′ = 0, …, 2p + 1. (A.2)

The curve (A.1) is compactified by adding the points P_{∞_{+}} and P_{∞_{−}}, P_{∞_{+}} ≠ P_{∞_{−}}, at infinity. One then introduces an appropriate set of p + 1 nonintersecting cuts C_{j} joining E_{m(j)} and E_{m′(j)} and denotes

C = ⋃_{j∈{1,...,p+1}} C_{j}, C_{j} ∩ C_{k} = ∅, j ≠ k.  (A.3)

Defining the cut plane

Π = ℂ \ C,  (A.4)

and introducing the holomorphic function

R_{2p+2}( ·)^{1/2} : Π → ℂ, z ↦ ( ∏_{m=0}^{2p+1}(z − E_{m}))^{1/2}  (A.5)

on Π with an appropriate choice of the square root branch in (A.5), one considers

M_{p} = {(z, σR_{2p+2}(z)^{1/2}) | z ∈ ℂ, σ ∈ {±1}} ∪ {P_{∞_{+}}, P_{∞_{−}}}  (A.6)

by extending R_{2p+2}( ·)^{1/2} to C. The hyperelliptic curve K_{p} is then the set M_{p} with its natural complex structure obtained upon gluing the two sheets of M_{p} crosswise along the cuts. The set of branch points B(K_{p}) of K_{p} is given by

B(K_{p}) = {(E_{m}, 0)}_{m=0,...,2p+1}  (A.7)

and finite points P on K_{p} are denoted by P = (z, y), where y(P) denotes the meromorphic function on K_{p} satisfying F_{p}(z, y) = y^{2} − R_{2p+2}(z) = 0. Local coordinates near P_{0} = (z_{0}, y_{0}) ∈ K_{p} \ (B(K_{p}) ∪ {P_{∞_{+}}, P_{∞_{−}}}) are given by ζ_{P_{0}} = z − z_{0}, near P_{∞_{±}} by ζ_{P_{∞_{±}}} = 1/z, and near branch points (E_{m_{0}}, 0) ∈ B(K_{p}) by ζ_{(E_{m_{0}} ,0)} = (z−E_{m_{0}})^{1/2}. The Riemann surface K_{p} defined in this manner has topological genus p. Moreover, we introduce the holomorphic sheet exchange map (involution)

∗: K_{p} → K_{p}, P = (z, y) ↦ P^{∗} = (z, −y), P_{∞_{±}} ↦ P_{∞_{±}}^{∗} = P_{∞_{∓}}.  (A.8)

One verifies that dz/y is a holomorphic differential on K_{p} with zeros of order p − 1 at P_{∞_{±}} and hence

η_{j} = \frac{z^{j−1}dz}{y} , j = 1, …, p,  (A.9)

form a basis for the space of holomorphic differentials on K_{p}. Introducing the invertible matrix C in ℂ^{p},

\begin{matrix}C = (C_{j,k})_{j,k=1,...,p}, C_{j,k} = ∫_{a_{k}} η_{j}, \\ \underline{c}(k) = (c_{1}(k), …, c_{p}(k)), c_{j}(k) = C_{j,k}^{−1}, j, k = 1, …, p, \end{matrix}  (A.10)

the corresponding basis of normalized holomorphic differentials ω_{j}, j = 1, …, p, on K_{p} is given by

ω_{j} = ∑_{ℓ=1}^{p} c_{j}(ℓ)η_{ℓ}, ∫_{a_{k}} ω_{j} = δ_{j,k}, j, k = 1, …, p.  (A.11)

Here {a_{j}, b_{j}}_{j=1,...,p} is a homology basis for K_{p} with intersection matrix of the cycles satisfying

a_{j} ∘ b_{k} = δ_{j,k}, a_{j} ∘ a_{k} = 0, b_{j} ∘ b_{k} = 0, j, k = 1, …, p.  (A.12)

Associated with the homology basis {a_{j}, b_{j}}_{j=1,...,p} we also recall the canonical dissection of K_{p} along its cycles yielding the simply connected interior K̂_{p} of the fundamental polygon ∂K̂_{p} given by

∂K̂_{p} = a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}a_{2}b_{2}a_{2}^{−1}b_{2}^{−1} ⋯ a_{p}^{−1}b_{p}^{−1}.  (A.13)

Let M(K_{p}) and M^{1}(K_{p}) denote the set of meromorphic functions (0-forms) and meromorphic differentials (1-forms) on K_{p}. Holomorphic differentials are also called Abelian differentials of the first kind. Abelian differentials of the second kind, ω^{(2)} ∈ M^{1}(K_{p}), are characterized by the property that all their residues vanish. They will usually be normalized by demanding that all their a-periods vanish, that is, ∫_{a_{j}} ω^{(2)} = 0, j = 1, …, p. Any meromorphic differential ω^{(3)} on K_{p} not of the first or second kind is said to be of the third kind. A differential of the third kind ω^{(3)} ∈ M^{1}(K_{p}) is usually normalized by the vanishing of its a-periods, that is, ∫_{a_{j}} ω^{(3)} = 0, j = 1, …, p. A normal differential of the third kind ω_{P_{1},P_{2}}^{(3)} associated

with two points P_{1}, P_{2} ∈ K̂_{p}, P_{1} ≠ P_{2}, by definition, has simple poles at P_{j} with residues (−1)^{j+1}, j = 1, 2 and vanishing a-periods.

Next, define the matrix τ = (τ_{j,ℓ})_{j,ℓ=1,...,p} by

τ_{j,ℓ} = ∫_{b_{ℓ}} ω_{j}, j, ℓ = 1, …, p.  (A.14)

Then

Im(τ) > 0 and τ_{j,ℓ} = τ_{ℓ,j}, j, ℓ = 1, …, p.  (A.15)

Associated with τ one introduces the period lattice

L_{p} = \{\underline{z} ∈ ℂ^{p} | \underline{z} = \underline{m} + \underline{n}τ, \underline{m}, \underline{n} ∈ ℤ^{p}\}.  (A.16)

Next, fix a base point Q_{0} ∈ K_{p} \ {P_{0,±}, P_{∞_{±}}}, denote by J(K_{p}) = ℂ^{p}/L_{p} the Jacobi variety of K_{p}, and define the Abel map \underline{A}_{Q_{0}} by

\underline{A}_{Q_{0}} : K_{p} → J(K_{p}), \underline{A}_{Q_{0}}(P) = ( ∫_{Q_{0}}^{P} ω_{1}, …, ∫_{Q_{0}}^{P} ω_{p}) (mod L_{p}), P ∈ K_{p}.

(A.17)

Similarly, we introduce

\underline{α}_{Q_{0}} : Div(K_{p}) → J(K_{p}), D ↦ \underline{α}_{Q_{0}}(D) = ∑_{P ∈K_{p}} D(P)\underline{A}_{Q_{0}}(P),  (A.18)

where Div(K_{p}) denotes the set of divisors on K_{p}. Here D: K_{p} → ℤ is called a divisor on K_{p} if D(P) ≠ 0 for only finitely many P ∈ K_{p}. (In the main body of this paper we will choose Q_{0} to be one of the branch points, i.e., Q_{0} ∈ B(K_{p}), and for simplicity we will always choose the same path of integration from Q_{0} to P in all Abelian integrals.)

In connection with divisors on K_{p} we shall employ the following (additive) notation,

D_{Q_{0}\underline{Q}} = D_{Q_{0}} + D\underline{Q}, D\underline{Q} = D_{Q_{1}} + ⋯ + D_{Q_{m}},  (A.19)
\underline{Q} = \{Q_{1}, …, Q_{m}\} ∈ Sym^{m} K_{p}, Q_{0} ∈ K_{p}, m ∈ ℕ,

where for any Q ∈ K_{p},

D_{Q}: K_{p} → ℕ_{0}, P ↦ D_{Q}(P) = \begin{cases}1 & for P = Q, \\ 0 & for P ∈ K_{p} \ \{Q\}, \end{cases}  (A.20)

and Sym^{n} K_{p} denotes the nth symmetric product of K_{p}. In particular, Sym^{m} K_{p} can be identified with the set of nonnegative divisors 0 ≤ D ∈ Div(K_{p}) of degree m.

For f ∈ M(K_{p}) \ {0}, ω ∈ M^{1}(K_{p}) \ {0} the divisors of f and ω are denoted by (f) and (ω), respectively. Two divisors D, E ∈ Div(K_{p}) are called equivalent, denoted by D ∼ E, if and only if D − E = (f) for some f ∈ M(K_{p}) \ {0}. The divisor class [D] of D is then given by [D] = {E ∈ Div(K_{p}) | E ∼ D}. We recall that

deg((f)) = 0, deg((ω)) = 2(p − 1), f ∈ M(K_{p}) \ {0}, ω ∈ M^{1}(K_{p}) \ {0}, (A.21)

where the degree deg(D) of D is given by deg(D) = ∑_{P ∈K_{p}} D(P). It is customary to call (f) (respectively, (ω)) a principal (respectively, canonical) divisor.

Introducing the complex linear spaces

L(D) = {f ∈ M(K_{p}) | f = 0 or (f) ≥ D}, r(D) = dim L(D),  (A.22)
L^{1}(D) = {ω ∈ M^{1}(K_{p}) | ω = 0 or (ω) ≥ D}, i(D) = dim L^{1}(D),  (A.23)

with i(D) the index of speciality of D, one infers that deg(D), r(D), and i(D) only depend on the divisor class [D] of D. Moreover, we recall the following fundamental fact.

Theorem A.1. Let D\underline{Q} ∈ Sym^{p} K_{p}, \underline{Q} = \{Q_{1}, …, Q_{p}\}. Then,

1 ≤ i(D\underline{Q}) = s  (A.24)

if and only if {Q_{1}, …, Q_{p}} contains s pairings of the type {P, P^{∗}}. (This includes, of course, branch points for which P = P^{∗}.) One has s ≤ p/2.

Appendix B. Some Interpolation Formulas

In this appendix we recall a useful interpolation formula which goes beyond the standard Lagrange interpolation formula for polynomials in the sense that the zeros of the interpolating polynomial need not be distinct.

Lemma B.1 ([30]). Let p ∈ ℕ and S_{p−1} be a polynomial of degree p − 1. In addition, let F_{p} be a monic polynomial of degree p of the form

F_{p}(z) = ∏_{k=1}^{q}(z − μ_{k})^{p_{k}}, p_{j} ∈ ℕ, μ_{j} ∈ ℂ, j = 1, …, q, ∑_{k=1}^{q} p_{k} = p.  (B.1)

Then,

S_{p−1}(z) = F_{p}(z) ∑_{k=1}^{q} ∑_{ℓ=0}^{p_{k}−1} \frac{S_{p−1}^{(ℓ)}(μ_{k})}{ℓ!(p_{k} − ℓ − 1)!}  (B.2)

× (\frac{d^{p_{k}−ℓ−1}}{dζ^{p_{k}−ℓ−1}} ((z − ζ)^{−1} ∏_{k′=1, k′≠k}^{q}(ζ − μ_{k′})^{−p_{k′}} ))|_{ζ=μ_{k}}, z ∈ ℂ.

In particular, S_{p−1} is uniquely determined by prescribing the p values

S_{p−1}(μ_{k}), S′_{p−1}(μ_{k}), …, S_{p−1}^{(p_{k}−1)}(μ_{k}), k = 1, …, q,  (B.3)

at the given points μ_{1}…. , μ_{q}. Conversely, prescribing the p complex numbers

α_{k}^{(0)}, α_{k}^{(1)}, …, α_{k}^{(p_{k}−1)}, k = 1, …, q,  (B.4)

there exists a unique polynomial T_{p−1} of degree p − 1,

T_{p−1}(z) = F_{p}(z) ∑_{k=1}^{q} ∑_{ℓ=0}^{p_{k}−1} \frac{α_{k}^{(ℓ)}}{ℓ!(p_{k} − ℓ − 1)!}  (B.5)

× (\frac{d^{p_{k}−ℓ−1}}{dζ^{p_{k}−ℓ−1}} ((z − ζ)^{−1} ∏_{k′=1, k′≠k}^{q}(ζ − μ_{k′})^{−p_{k′}} ))|_{ζ=μ_{k}}, z ∈ ℂ,

such that

T_{p−1}(μ_{k}) = α_{k}^{(0)}, T′_{p−1}(μ_{k}) = α_{k}^{(1)}, …, T_{p−1}^{(p_{k}−1)}(μ_{k}) = α_{k}^{(p_{k}−1)}, k = 1, …, q. (B.6)

We briefly mention two special cases of (B.2). First, assume the generic case where all zeros of F_{p} are distinct, that is,

q = p, p_{k} = 1, μ_{k} ≠ μ_{k′} for k ≠ k′, k, k′ = 1, …, p.  (B.7)

In this case (B.2) reduces to the classical Lagrange interpolation formula

S_{p−1}(z) = F_{p}(z) ∑_{k=1}^{p} \frac{S_{p−1}(μ_{k})}{((dF_{p}(ζ)/dζ)|_{ζ=μ_{k}})(z − μ_{k})}, z ∈ ℂ.  (B.8)

Second, we consider the other extreme case where all zeros of F_{p} coincide, that is,

q = 1, p_{1} = p, F_{p}(z) = (z − μ_{1})^{p}, z ∈ ℂ.  (B.9)

In this case (B.2) reduces of course to the Taylor expansion of S_{p−1} around z = μ_{1},

S_{p−1}(z) = ∑_{ℓ=0}^{p−1} \frac{S_{p−1}^{(ℓ)}(μ_{1})}{ℓ!}(z − μ_{1})^{ℓ}, z ∈ ℂ.  (B.10)

Appendix C. Asymptotic Spectral Parameter Expansions

In this appendix we consider asymptotic spectral parameter expansions of F\underline{p}/y, G\underline{p}/y, and H\underline{p}/y, the resulting recursion relations for the homogeneous coefficients f̂_{ℓ}, ĝ_{ℓ}, and ĥ_{ℓ}, their connection with the nonhomogeneous coefficients f_{ℓ}, g_{ℓ}, and h_{ℓ}, and the connection between c_{ℓ,±} and c_{ℓ}(\underline{E}^{±1}) (cf. (C.3)). For detailed proofs of the material in this section we refer to [29], [32]. We will employ the notation

\underline{E}^{±1} = (E_{0}^{±1}, …, E_{2p+1}^{±1}).  (C.1)

We start with the following elementary result (a consequence of the binomial expansion) assuming η ∈ ℂ such that |η| < min{|E_{0}|^{−1}, …, |E_{2p+1}|^{−1}}:

( ∏_{m=0}^{2p+1} (1 − E_{m}η))^{1/2} = ∑_{k=0}^{∞} c_{k}(\underline{E})η^{k},  (C.2)

where

c_{0}(\underline{E}) = 1,
c_{k}(\underline{E}) = ∑_{\begin{matrix}j_{0},...,j_{2p+1}=0 \\ j_{0}+···+j_{2p+1}=k \end{matrix}}^{k}\frac{(2j_{0})! ⋯ (2j_{2p+1})! E_{0}^{j_{0}} ⋯ E_{2p+1}^{j_{2p+1}}}{2^{2k}(j_{0}!)^{2} ⋯ (j_{2p+1}!)^{2}(2j_{0} − 1) ⋯ (2j_{2p+1} − 1)}, k ∈ ℕ. (C.3)

The first few coefficients explicitly are given by

c_{0}(\underline{E}) = 1, c_{1}(\underline{E}) = −\frac{1}{2} ∑_{m=0}^{2p+1} E_{m}, c_{2}(\underline{E}) = \frac{1}{4} ∑_{\begin{matrix}m_{1},m_{2}=0 \\ m_{1}<m_{2} \end{matrix}}^{2p+1} E_{m_{1}}E_{m_{2}} − \frac{1}{8} ∑_{m=0}^{2p+1} E_{m}^{2}, etc. (C.4)

Next we turn to asymptotic expansions. We recall the convention y(P) = ∓ζ^{−p−1} + O(ζ^{−p}) near P_{∞_{±}} (where ζ = 1/z) and y(P) = ±(c_{0,−}/c_{0,+}) + O(ζ) near P_{0,±} (where ζ = z).

Theorem C.1 ([32]). Assume (3.2), s-AL\underline{p}(α, β) = 0, and suppose P = (z, y) ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}\}. Then z^{p_{−}}F\underline{p}/y, z^{p_{−}}G\underline{p}/y, and z^{p_{−}}H\underline{p}/y have the following convergent expansions as P → P_{∞_{±}}, respectively, P → P_{0,±},

\frac{z^{p_{−}}}{c_{0,+}} \frac{F\underline{p}(z)}{y} = \begin{cases}∓ ∑_{ℓ=0}^{∞} f̂_{ℓ,+}ζ^{ℓ+1}, & P → P_{∞_{±}},  ζ = 1/z, \\ ± ∑_{ℓ=0}^{∞} f̂_{ℓ,−}ζ^{ℓ}, & P → P_{0,±},  ζ = z, \end{cases}  (C.5)

\frac{z^{p_{−}}}{c_{0,+}} \frac{G\underline{p}(z)}{y} = \begin{cases}∓ ∑_{ℓ=0}^{∞} ĝ_{ℓ,+}ζ^{ℓ}, & P → P_{∞_{±}},  ζ = 1/z, \\ ± ∑_{ℓ=0}^{∞} ĝ_{ℓ,−}ζ^{ℓ}, & P → P_{0,±},  ζ = z, \end{cases}  (C.6)

\frac{z^{p_{−}}}{c_{0,+}} \frac{H\underline{p}(z)}{y} = \begin{cases}∓ ∑_{ℓ=0}^{∞} ĥ_{ℓ,+}ζ^{ℓ}, & P → P_{∞_{±}},  ζ = 1/z, \\ ± ∑_{ℓ=0}^{∞} ĥ_{ℓ,−}ζ^{ℓ+1}, & P → P_{0,±},  ζ = z, \end{cases}  (C.7)

where ζ = 1/z (resp., ζ = z) is the local coordinate near P_{∞_{±}} (resp., P_{0,±}) and f̂_{ℓ,±}, ĝ_{ℓ,±}, and ĥ_{ℓ,±} are the homogeneous versions of the coefficients f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±} introduced in (2.16)–(2.18).

Moreover, the E_{m}-dependent summation constants c_{ℓ,±}, ℓ = 0, …, p_{±}, in F\underline{p}, G\underline{p}, and H\underline{p} are given by

c_{ℓ,±} = c_{0,±}c_{ℓ}(\underline{E}^{±1}), ℓ = 0, …, p_{±}.  (C.8)

Acknowledgments. F.G., J.M., and G.T. gratefully acknowledge the extraordinary hospitality of the Department of Mathematical Sciences of the Norwegian University of Science and Technology, Trondheim, during extended stays in the summer of 2004–2006, where parts of this paper were written.

References

[1] M. J. Ablowitz. Nonlinear evolution equations – continuous and discrete. SIAM Rev. 19:663– 684, 1977.

[2] M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, 1991.

[3] M. J. Ablowitz and J. F. Ladik. Nonlinear differential-difference equations. J. Math. Phys. 16:598–603, 1975.

[4] M. J. Ablowitz and J. F. Ladik. Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17:1011–1018, 1976.

[5] M. J. Ablowitz and J. F. Ladik. A nonlinear difference scheme and inverse scattering. Studies Appl. Math. 55:213–229, 1976.

[6] M. J. Ablowitz and J. F. Ladik. On the solution of a class of nonlinear partial difference equations. Studies Appl. Math. 57:1–12, 1977.

[7] M. J. Ablowitz, B. Prinari, and A. D. Trubatch. Discrete and Continuous Nonlinear Schrödinger Systems. London Mathematical Society Lecture Note Series, Vol. 302, Cambridge University Press, Cambridge, 2004.

[8] R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications. 2nd ed., Springer, New York, 1988.

[9] S. Ahmad and A. Roy Chowdhury. On the quasi-periodic solutions to the discrete non-linear Schrödinger equation. J. Phys. A 20:293–303, 1987.

[10] S. Ahmad and A. Roy Chowdhury. The quasi-periodic solutions to the discrete non-linear Schrödinger equation. J. Math. Phys. 28:134–137, 1987.

[11] G. S. Ammar and W. B. Gragg. Schur flows for orthogonal Hessenberg matrices. Hamiltonian and Gradient Flows, Algorithms and Control, A. Bloch (ed.), Fields Inst. Commun., Vol. 3, Amer. Math. Soc., Providence, RI, 1994, pp. 27–34.

[12] V. I. Arnold. Mathematical Methods of Classical Mechanics. 2nd ed., Springer, New York, 1989.

[13] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev. Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin, 1994.

[14] N. N. Bogolyubov, A. K. Prikarpatskii, and V. G. Samoilenko. Discrete periodic problem for the modified nonlinear Korteweg–de Vries equation. Sov. Phys. Dokl. 26:490–492, 1981.

[15] N. N. Bogolyubov and A. K. Prikarpatskii. The inverse periodic problem for a discrete approximation of a nonlinear Schrödinger equation. Sov. Phys. Dokl. 27:113–116, 1982.

[16] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl. Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchy. Mem. Amer. Math. Soc. no. 641, 135:1–79, 1998.

[17] A. K. Common. A solution of the initial value problem for half-infinite integrable lattice systems. Inverse Probl. 8:393–408, 1992.

[18] A. K. Common and S. T. Hafez. Continued-fraction solutions to the Riccati equation and integrable lattice systems. J. Phys. A 23:455–466, 1990.

[19] P. Deift. Riemann–Hilbert methods in the theory of orthogonal polynomials. Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Ergodic Schrödinger Operators, Singular Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory, F. Gesztesy, P. Deift, C. Galvez, P. Perry, and W. Schlag (eds.), Proceedings of Symposia in Pure Mathematics, Vol. 76/2, Amer. Math. Soc., Providence, RI, 2007, pp. 715–740.

[20] H. M. Farkas and I. Kra. Riemann Surfaces. Springer, New York, 2nd ed., 1992.

[21] J. Fay. Theta Functions on Riemann Surfaces. Lecture Notes in Mathematics, Vol. 352, Springer, New York, 1973.

[22] L. Faybusovich and M. Gekhtman. On Schur flows. J. Phys. A 32:4671–4680, 1999.

[23] L. Faybusovich and M. Gekhtman. Elementary Toda orbits and integrable lattices. Inverse Probl. 41:2905–2921, 2000.

[24] M. Gekhtman. Non-abelian nonlinear lattice equations on finite interval. J. Phys. A 26:6303– 6317, 1993.

[25] X. Geng, H. H. Dai, and C. Cao. Algebro-geometric constructions of the discrete Ablowitz– Ladik flows and applications. J. Math. Phys. 44:4573–4588, 2003.

[26] X. Geng, H. H. Dai, and J. Zhu. Decompositions of the discrete Ablowitz–Ladik hierarchy. Stud. Appl. Math. 118:281–312, 2007.

[27] J. S. Geronimo, F. Gesztesy, and H. Holden. Algebro-geometric solutions of the Baxter–Szegő difference equation. Comm. Math. Phys. 258:149–177, 2005.

[28] F. Gesztesy and H. Holden. Soliton Equations and Their Algebro-Geometric Solutions. Volume I: (1 + 1)-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics, Vol. 79, Cambridge University Press, Cambridge, 2003.

[29] F. Gesztesy, H. Holden, J. Michor, and G. Teschl. Soliton Equations and Their Algebro-Geometric Solutions. Volume II: (1 + 1)-Dimensional Discrete Models. Cambridge Studies in Advanced Mathematics, Vol. 114, Cambridge University Press, Cambridge, 2008.

[30] F. Gesztesy, H. Holden, and G. Teschl. The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data. Rev. Mat. Iberoamericana, 24, 117–182 (2008).

[31] F. Gesztesy, H. Holden, J. Michor, and G. Teschl. The Ablowitz–Ladik hierarchy revisited. In Methods of Spectral Analysis in Mathematical Physics, Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2006, Lund, Sweden, J. Janas, P. Kurasov, A. Laptev, S. Naboko, and G. Stolz (eds.), Operator Theory: Advances and Applications, Vol. 186, Birkhäuser, Basel, 2008, pp. 139–190.

[32] F. Gesztesy, H. Holden, J. Michor, and G. Teschl. Algebro-geometric finite-band solutions of the Ablowitz–Ladik hierarchy. Int. Math. Res. Notices, 2007:1–55, rnm082.

[33] F. Gesztesy, H. Holden, J. Michor, and G. Teschl. Local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik hierarchy, Stud. Appl. Math. 120, 361–423 (2008).

[34] L. Golinskii. Schur flows and orthogonal polynomials on the unit circle. Sbornik Math. 197:1145–1165, 2006.

[35] A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge Univ. Press, Cambridge, 1995.

[36] R. Killip and I. Nenciu. CMV: The unitary analogue of Jacobi matrices. Comm. Pure Appl. Math. 60:1148–1188, 2007.

[37] L.-C. Li. Some remarks on CMV matrices and dressing orbits. Int. Math. Res. Notices 2005, No. 40, 2437–2446.

[38] A. I. Markushevich. Theory of Functions of a Complex Variable. 2nd. ed., Chelsea, New York, 1985.

[39] P. D. Miller. Macroscopic behavior in the Ablowitz–Ladik equations. Nonlinear Evolution Equations & Dynamical Systems, V. G. Makhankov, A. R. Bishop, and D. D. Holm (eds.), World Scientific, Singapore, 1995, pp. 158–167.

[40] P. D. Miller, N. M. Ercolani, I. M. Krichever, and C. D. Levermore. Finite genus solutions to the Ablowitz–Ladik equations. Comm. Pure Appl. Math. 48:1369–1440, 1995.

[41] A. Mukaihira and Y. Nakamura. Schur flow for orthogonal polynomials on the unit circle and its integrable discretization. J. Comput. Appl. Math. 139:75–94, 2002.

[42] D. Mumford. Tata Lectures on Theta II. Birkhäuser, Boston, 1984.

[43] I. Nenciu. Lax pairs for the Ablowitz–Ladik system via orthogonal polynomials on the unit circle. Int. Math. Res. Notices 11:647–686, 2005.

[44] I. Nenciu. CMV matrices in random matrix theory and integrable systems: a survey. J. Phys. A 39:8811–8822, 2006.

[45] R. J. Schilling. A systematic approach to the soliton equations of a discrete eigenvalue problem. J. Math. Phys. 30:1487–1501, 1989.

[46] L. Pastur and A. Figotin. Spectra of Random and Almost-Periodic Operators. Springer, Berlin, 1992.

[47] B. Simon. Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Part 2: Spectral Theory, AMS Colloquium Publication Series, Vol. 54, Amer. Math. Soc., Providence, Providence, R.I., 2005.

[48] B. Simon. CMV matrices: Five years after. J. Comp. Appl. Math. 208:120–154, 2007.

[49] B. Simon. Zeros of OPUC and long time asymptotics of Schur and related flows. Inverse Probl. Imaging 1:189–215, 2007.

[50] Yu. B. Suris. The Problem of Integrable Discretization: Hamiltonian Approach. Birkhäuser, Basel, 2003.

[51] G. Teschl. Jacobi Operators and Completely Integrable Nonlinear Lattices. Math. Surveys Monographs, Vol. 72, Amer. Math. Soc., Providence, R.I., 2000.

[52] K. L. Vaninsky. An additional Gibbs’ state for the cubic Schrödinger equation on the circle. Comm. Pure Appl. Math. 54:537–582, 2001.

[53] V. E. Vekslerchik. Finite genus solutions for the Ablowitz–Ladik hierarchy. J. Phys. A 32:4983–4994, 1999.

[54] W. Walter. Ordinary Differential Equations. Springer, New York, 1998.

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail address: gesztesyf@missouri.edu
URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO–7491 Trondheim, Norway

E-mail address: holden@math.ntnu.no
URL: http://www.math.ntnu.no/~holden/

Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Johanna.Michor@univie.ac.at
URL: http://www.mat.univie.ac.at/~jmichor/

Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Gerald.Teschl@univie.ac.at
URL: http://www.mat.univie.ac.at/~gerald/