ALGEBRO-GEOMETRIC FINITE-BAND SOLUTIONS OF THE ABLOWITZ–LADIK HIERARCHY

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

To Walter Thirring, on the occasion of his 80th birthday, theoretical and mathematical physicist extraordinaire.

Abstract. We provide a detailed derivation of all complex-valued algebrogeometric finite-band solutions of the Ablowitz–Ladik hierarchy. In addition, we survey a recursive construction of the Ablowitz–Ladik hierarchy and its zero-curvature and Lax formalism.

1. Introduction

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], [22]), 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 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)

We will refer to (1.1) as the Ablowitz–Ladik system. The principal theme of this paper will be to derive the algebro-geometric finite-band solutions of the Ablowitz– Ladik (AL) hierarchy, a completely integrable sequence of systems of nonlinear evolution equations on the lattice ℤ whose first nonlinear member is the Ablowitz– Ladik system (1.1).

Since the mid-seventies there has been an enormous amount of activity in the area of integrable differential-difference equations. Two principal directions of research are responsible for this development: Originally, the development was driven by the theory of completely integrable systems and its applications to fields such as nonlinear optics, and more recently, it gained additional momentum due to its intimate connections with the theory of orthogonal polynomials. In the following we first briefly recall the development in connection with integrable systems and subsequently turn to the one influenced by research on orthogonal polynomials.

2000 Mathematics Subject Classification. Primary 37K15, 37K10; Secondary 39A12, 35Q55.

Key words and phrases. Ablowitz–Ladik hierarchy, discrete NLS, algebro-geometric solutions.

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 Grant No. Y330.

Int. Math. Res. Notices 2007, rnm082, 1–55.

The first systematic discussion of the Ablowitz–Ladik hierarchy appears to be due to Schilling [51] (cf. also [58], [62], [65]); infinitely many conservation laws are derived, for instance, by Ding, Sun, and Xu [26]; the bi-Hamiltonian structure of the AL hierarchy is considered by Ercolani and Lozano [28]; connections between the AL hierarchy and the motion of a piecewise linear curve have been established by Doliwa and Santini [27]; Bäcklund and Darboux transformations were studied by Geng [31] and Vekslerchik [63]; the Hirota bilinear formalism, AL τ-functions, etc., were considered by Vekslerchik [62]. The initial value problem for half-infinite AL systems was discussed by Common [24], for an application of the inverse scattering method to (1.2) we refer to Vekslerchik and Konotop [64]. This just scratches the surface of these developments and the interested reader will find much more material in the references cited in these papers and the ones discussed below.

Algebro-geometric (and periodic) solutions of the AL system (1.1) have briefly been studied by Ahmad and Chowdhury [8], [9], Bogolyubov, Prikarpatskii, and Samoilenko [17], Bogolyubov and Prikarpatskii [18], Chow, Conte, and Xu [23], Geng, Dai, and Cao [32], and Vaninsky [60]. In an effort to analyze models describing oscillations in nonlinear dispersive wave systems, Miller, Ercolani, Krichever, and Levermore [47] (see also [46]) gave a detailed analysis of algebro-geometric solutions of the AL system (1.1). Introducing

U(z) = (\begin{matrix}z & α \\ βz & 1 \end{matrix}) , V (z) = i (\begin{matrix}z − 1 − αβ^{−} & α − α^{−}z^{−1} \\ β^{−}z − β & 1 + α^{−}β − z^{−1} \end{matrix})  (1.3)

with z ∈ ℂ \ {0} a spectral parameter, the authors in [47] relied on the fact that the Ablowitz–Ladik system (1.1) is equivalent to the zero-curvature equations

U_{t} + UV − V^{+}U = 0,  (1.4)

the latter being the compatibility relation for the spatial and temporal linear problems

Φ = UΦ^{−}, Φ_{t}^{−} = V Φ^{−}.  (1.5)

Here we extended the notation f^{±}(n) = f(n ± 1), n ∈ ℤ, to ℂ^{2}-valued and 2 × 2matrix-valued sequences with complex-valued entries.

Miller, Ercolani, Krichever, and Levermore [47] then performed a thorough analysis of the solutions Φ = Φ(z, n, t) associated with the pair (U, V) and derived the theta function representations of α, β satisfying the AL system (1.1). In the particular focusing and defocusing cases they also discussed periodic and quasi-periodic solutions α with respect to n ∈ ℤ and t ∈ ℝ. Vekslerchik [61] also studied finitegenus solutions for the AL hierarchy by establishing connections with Fay’s identity for theta functions.

The connection between the Ablowitz–Ladik system (1.1) and orthogonal polynomials comes about as follows: Let {α(n)}_{n∈ℕ} ⊂ ℂ be a sequence of complex numbers subject to the condition |α(n)| < 1, n ∈ ℕ, and define the transfer matrix

T (z) = (\begin{matrix}z & α \\ \overline{α}z & 1 \end{matrix}) , z ∈ 𝕋,  (1.6)

with spectral parameter z on the unit circle 𝕋 = {z ∈ ℂ | |z| = 1}. Consider the system of difference equations

Φ(z, n) = T (z, n)Φ(z, n − 1), (z, n) ∈ 𝕋 × ℕ,  (1.7)

with initial condition Φ(z, 0) = (\begin{matrix}1 \\ 1 \end{matrix}), where

Φ(z, n) = (\begin{matrix}φ(z, n) \\ z^{n}\overline{φ}(1/z, n) \end{matrix}) , (z, n) ∈ 𝕋 × ℕ_{0}.  (1.8)

(Here ℕ_{0} = ℕ ∪ {0}.) Then φ( · , n) are monic polynomials of degree n first introduced by Szegő in the 1920’s in his seminal work on the asymptotic distribution of eigenvalues of sections of Toeplitz forms [55], [56] (see also [57, Ch. XI]). Szegő’s point of departure was the trigonometric moment problem and hence the theory of orthogonal polynomials on the unit circle. Indeed, given a probability measure dσ supported on an infinite set on the unit circle, one is interested in finding monic polynomials χ( · , n) of degree n ∈ ℕ_{0} in z = e^{iθ}, θ ∈ [0, 2π], such that

∫_{0}^{2π} dσ(e^{iθ}) \overline{χ(e^{iθ}, m)}χ(e^{iθ}, n) = w(n)^{−2}δ_{m,n}, m, n ∈ ℕ_{0},  (1.9)

where w(0)^{2} = 1, w(n)^{2} = ∏_{j=1}^{n} (1 − |α(j)|^{2})^{−1}, n ∈ ℕ. Szegő showed that the corresponding polynomials (1.8) with φ replaced by χ satisfy the recurrence formula (1.7). Early work in this area includes important contributions by Akhiezer, Geronimus, Krein, Tomčuk, Verblunsky, Widom, and others, and is summarized in the books by Akhiezer [10], Geronimus [37], Szegő [57], and especially in the recent two-volume treatise by Simon [52].

Unaware of the paper [47], Geronimo and Johnson [35] studied (1.7) in the case where the coefficients α are random variables. Under appropriate ergodicity assumptions on α and the hypothesis of a vanishing Lyapunov exponent on prescribed spectral arcs on the unit circle 𝕋, Geronimo and Johnson [35] (cf. also [34], [36]) developed the corresponding spectral theory associated with (1.7) and the unitary operator it generates in ℓ^{2}(ℤ). In this sense the discussion in [35] is a purely stationary one and connections with a zero-curvature formalism, theta function representations, and integrable hierarchies are not made in [35] (but in this context we refer to the discussion concerning [33] in the next paragraph). More recently, the defocusing case with periodic and quasi-periodic coefficients was studied in great detail by Deift [25], Golinskii and Nevai [43], Killip and Nenciu [44], Li [45], Nenciu [49], [50], and Simon [52, Ch. 11], [53], [54].

An important extension of (1.7) was developed by Baxter in a series of papers on Toeplitz forms [11]–[14] in 1960–63. In these papers the transfer matrix T in (1.6) is replaced by the more general (complexified) transfer matrix U(z) = ( \begin{matrix}z & α \\ βz & 1 \end{matrix} ) in (1.5), that is, precisely the matrix U responsible for the spatial part in the Ablowitz– Ladik system in its zero-curvature formulation (1.3)–(1.5). Here α = {α(n)}_{n∈ℕ}, β = {β(n)}_{n∈ℕ} are subject to the condition α(n)β(n) ≠ 1, n ∈ ℕ. Studying the following extension of (1.7),

Ψ(z, n) = U(z, n)Ψ(z, n − 1), (z, n) ∈ 𝕋 × ℕ,  (1.10)

Baxter was led to biorthogonal polynomials on the unit circle with respect to a complex-valued measure on 𝕋. In this context of biorthogonal Laurent polynomials we refer to [16] and [20]. Reference [16], in particular, deals with isomonodromic tau functions and is applicable to generalized integrable lattices of the Toda-type. Baxter’s U matrix in (1.5) led to a new hierarchy of nonlinear difference equations, called the Szegő–Baxter (SB) hierarchy in [33], in honor of these two pioneers of orthogonal polynomials on the unit circle. The latter reference also contains an in depth study of algebro-geometric solutions of (1.7).

In addition to these recent developments on the AL system and the AL hierarchy, we offer a variety of results in this paper apparently not covered before. These include:

• An elementary, yet effective recursive construction of the AL hierarchy using Laurent polynomials.

• Explicit formulas for Lax pairs for the AL hierarchy.

• The detailed connection between the AL hierarchy and a “complexified” version of transfer matrices first introduced by Baxter.

• A unified treatment of stationary algebro-geometric finite-band solutions and their theta function representations of the entire AL hierarchy.

• A unified treatment of algebro-geometric solutions and their theta function representations of the time-dependent AL hierarchy by solving the \underline{r}th AL flow with initial data given by stationary algebro-geometric finite-band solutions.

The structure of this paper is as follows: In Section 2 we describe our zerocurvature formalism for the Ablowitz–Ladik (AL) hierarchy. Extending a recursive polynomial approach discussed in great detail in [38] in the continuous case and in [19], [39, Ch. 4], [59, Chs. 6, 12] in the discrete context to the case of Laurent polynomials with respect to the spectral parameter, we derive the AL hierarchy of systems of nonlinear evolution equations whose first nonlinear member is the Ablowitz–Ladik system (1.1). Section 3 is devoted to a detailed study of the stationary AL hierarchy. We employ the recursive Laurent polynomial formalism of Section 2 to describe nonnegative divisors of degree p on a hyperelliptic curve K_{p} of genus p associated with the \underline{p}th system in the stationary AL hierarchy. By means of a fundamental meromorphic function ϕ on K_{p} (an analog of the Weyl– Titchmarsh function for the system (1.7)) we then proceed to derive the theta function representations of the associated Baker–Akhiezer vector and all stationary algebro-geometric finite-band solutions of the AL hierarchy. The corresponding time-dependent results for the AL hierarchy are presented in detail in Section 4. Appendix A collects relevant material on hyperelliptic curves and their theta functions and introduces the terminology freely used in Sections 3 and 4. Appendix B is of a technical nature and summarizes expansions of various key quantities related to the Laurent polynomial recursion formalism as the spectral parameter tends to zero and to infinity.

2. The Ablowitz–Ladik Hierarchy, Recursion Relations, Zero-Curvature Pairs, and Hyperelliptic Curves

In this section we summarize the construction of the Ablowitz–Ladik hierarchy employing a Laurent polynomial recursion formalism and derive the associated sequence of Ablowitz–Ladik zero-curvature pairs (we also hint at Lax pairs). Moreover, we discuss the Burchnall–Chaundy Laurent polynomial in connection with the stationary Ablowitz–Ladik hierarchy and the underlying hyperelliptic curve. For a detailed treatment of this material we refer to [39], [40].

We denote by ℂ^{ℤ} the set of complex-valued sequences indexed by ℤ.

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)

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)

To construct the Ablowitz–Ladik hierarchy one typically introduces appropriate zero-curvature pairs of 2×2 matrices, denoted by U(z) and V\underline{p}(z), \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2} (with z a certain spectral parameter to be discussed later), defined recursively in the following. We take the shortest route to the construction of V\underline{p} and hence to that of the Ablowitz–Ladik hierarchy by starting from the recursion relation (2.5)–(2.12) below.

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)

In a slight abuse of notation we will occasionally stress the dependence of f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±} on α, β by writing f_{ℓ,±}(α, β), g_{ℓ,±}(α, β), and h_{ℓ,±}(α, β).

One can show (cf. [40]) that all homogeneous elements f̂_{ℓ,±}, ĝ_{ℓ,±}, and ĥ_{ℓ,±}, ℓ ∈ ℕ_{0}, are polynomials in α, β, and some of their shifts.

Remark 2.3. As an efficient tool to later distinguish between nonhomogeneous and homogeneous quantities f_{ℓ,±}, g_{ℓ,±}, h_{ℓ,±}, and f̂_{ℓ,±}, ĝ_{ℓ,±}, ĥ_{ℓ,±}, respectively, we now introduce the notion of degree as follows. Denote

f^{(r)} = S^{(r)}f, f = \{f(n)\}_{n∈ℤ} ∈ ℂ^{ℤ}, S^{(r)} = \begin{cases}(S^{+})^{r}, & r ≥ 0, \\ (S^{−})^{−r}, & r < 0, \end{cases} r ∈ ℤ, (2.20)

and define

deg (α^{(r)}) = r, deg (β^{(r)}) = −r, r ∈ ℤ.  (2.21)

This then results in

deg (f̂_{ℓ,+}^{(r)}) = ℓ + 1 + r, deg (f̂_{ℓ,−}^{(r)}) = −ℓ + r, deg (ĝ_{ℓ,±}^{(r)}) = ±ℓ,
deg (ĥ_{ℓ,+}^{(r)}) = ℓ − r, deg (ĥ_{ℓ,−}^{(r)}) = −ℓ − 1 − r, ℓ ∈ ℕ_{0}, r ∈ ℤ.

(2.22)

We also note the following useful result (cf. [40]): Assume (2.1), then,

\begin{aligned}g_{ℓ,+} − g_{ℓ,+}^{−} & = αh_{ℓ,+} + βf_{ℓ,+}^{−}, ℓ ∈ ℕ_{0}, \\ g_{ℓ,−} − g_{ℓ,−}^{−} & = αh_{ℓ,−}^{−} + βf_{ℓ,−}, ℓ ∈ ℕ_{0}. \end{aligned}  (2.23)

Moreover, we note the following symmetries,

f̂_{ℓ,±}(c_{0,±}, α, β) = ĥ_{ℓ,∓}(c_{0,∓}, β, α), ĝ_{ℓ,±}(c_{0,±}, α, β) = ĝ_{ℓ,∓}(c_{0,∓}, β, α), ℓ ∈ ℕ_{0}. (2.24)

Next we relate the homogeneous coefficients f̂_{ℓ,±}, ĝ_{ℓ,±}, and ĥ_{ℓ,±} to certain matrix elements of L, where L will later be identified as the Lax difference expression associated with the Ablowitz–Ladik hierarchy. For this purpose it is useful to introduce the standard basis {δ_{m}}_{m∈ℤ} in ℓ^{2}(ℤ) by

δ_{m} = \{δ_{m,n}\}_{n∈ℤ}, m ∈ ℤ, δ_{m,n} = \begin{cases}1, & m = n, \\ 0, & m ≠ n. \end{cases}  (2.25)

The scalar product in ℓ^{2}(ℤ), denoted by ( · , ·), is defined by

(f, g) = ∑_{n∈ℤ} \overline{f(n)}g(n), f, g ∈ ℓ^{2}(ℤ).  (2.26)

In the standard basis just defined, we introduce the difference expression L 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} ) (2.27)

= 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^{++},

where

δ_{even} = χ_{2ℤ}, δ_{odd} = 1 − δ_{even} = χ_{2ℤ+1}.  (2.28)

In particular, terms of the form −β(n)α(n+1) represent the diagonal (n, n)-entries, n ∈ ℤ, in the infinite matrix (2.27). In addition, we used the abbreviation

ρ = γ^{1/2} = (1 − αβ)^{1/2}.  (2.29)

Next, we introduce the unitary operator Uε̃ in ℓ^{2}(ℤ) by

Uε̃ = (ε̃(n)δ_{m,n})_{(m,n)∈ℤ^{2}}, ε̃(n) ∈ {1, −1}, n ∈ ℤ,  (2.30)

and the sequence ε = {ε(n)}_{n∈ℤ} ∈ ℂ^{ℤ} by

ε(n) = ε̃(n − 1)ε̃(n), n ∈ ℤ.  (2.31)

A straightforward computation then shows that

Uε̃LU −1 ; ε̃ = L̃_{ε},  (2.32)

where L̃_{ε} is associated with the sequences α̃ = α, β̃ = β, and ρ̃ = ερ. Moreover, the recursion formalism in (2.5)–(2.12) yields coefficients which are polynomials in α, β and some of their shifts and hence depends only quadratically on ρ. As a result, the choice of square root of ρ(n), n ∈ ℤ, in (2.29) is immaterial when introducing the AL hierarchy via the Lax equations (2.72).

The half-lattice (i.e., semi-infinite) version of L was rediscovered by Cantero, Moral, and Velázquez [21] in 2003 in the special case where β = \overline{α} (see also Simon [52]–[54] who coined the term CMV matrix in this context).

The next result details the connections between L and the recursion coefficients f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±}:

Lemma 2.4. Let n ∈ ℤ. Then the homogeneous coefficients {f̂_{ℓ,±}}_{ℓ∈ℕ_{0}}, {ĝ_{ℓ,±}}_{ℓ∈ℕ_{0}}, and {ĥ_{ℓ,±}}_{ℓ∈ℕ_{0}} satisfy the following relations:

f̂_{ℓ,+}(n) = α(n)(δ_{n}, L^{ℓ+1}δ_{n}) + ρ(n) \begin{cases}(δ_{n−1}, L^{ℓ+1}δ_{n}), & n even, \\ (δ_{n}, L^{ℓ+1}δ_{n−1}), & n odd, \end{cases} ℓ ∈ ℕ_{0},

f̂_{ℓ,−}(n) = α(n)(δ_{n}, L^{−ℓ}δ_{n}) + ρ(n) \begin{cases}(δ_{n−1}, L^{−ℓ}δ_{n}), & n even, \\ (δ_{n}, L^{−ℓ}δ_{n−1}), & n odd, \end{cases} ℓ ∈ ℕ_{0},
ĝ_{0,±} = 1/2, ĝ_{ℓ,±}(n) = (δ_{n}, L^{±ℓ}δ_{n}), ℓ ∈ ℕ,  (2.33)
ĥ_{ℓ,+}(n) = β(n)(δ_{n}, L^{ℓ}δ_{n}) + ρ(n) \begin{cases}(δ_{n}, L^{ℓ}δ_{n−1}), & n even, \\ (δ_{n−1}, L^{ℓ}δ_{n}), & n odd, \end{cases} ℓ ∈ ℕ_{0},

ĥ_{ℓ,−}(n) = β(n)(δ_{n}, L^{−ℓ−1}δ_{n}) + ρ(n) \begin{cases}(δ_{n}, L^{−ℓ−1}δ_{n−1}), & n even, \\ (δ_{n−1}, L^{−ℓ−1}δ_{n}), & n odd, \end{cases} ℓ ∈ ℕ_{0}.

For the proof of Lemma 2.4 and some of its applications in connection with conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik hierarchy we refer to [42].

Next we define the 2 × 2 zero-curvature matrices

U(z) = (\begin{matrix}z & α \\ zβ & 1 \end{matrix})  (2.34)

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},  (2.35)

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,  (2.36)

one concludes that (2.36) is equivalent with the following relations

z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p} = 0,  (2.37)
zβF − ; \underline{p} + αH\underline{p} − K\underline{p} + K− ; \underline{p} = 0,  (2.38)
−F\underline{p} + zF − ; \underline{p} + α(G\underline{p} + K− ; \underline{p}) = 0,  (2.39)
zβ(G− ; \underline{p} + K\underline{p}) − zH\underline{p} + H− ; \underline{p} = 0.  (2.40)

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^{1}

F\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} f_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}−1} f_{p_{+}−1−ℓ,+}z^{ℓ},  (2.41)

G\underline{p}(z) = ∑_{ℓ=1}^{p_{−}} g_{p_{−}−ℓ,−}z^{−ℓ} + ∑_{ℓ=0}^{p_{+}} g_{p_{+}−ℓ,+}z^{ℓ},  (2.42)

1In this paper, a sum is interpreted as zero whenever the upper limit in the sum is strictly less than its lower limit.

H\underline{p}(z) = ∑_{ℓ=0}^{p_{−}−1} h_{p_{−}−1−ℓ,−}z^{−ℓ} + ∑_{ℓ=1}^{p_{+}} h_{p_{+}−ℓ,+}z^{ℓ},  (2.43)

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_{+},+}.  (2.44)

The corresponding homogeneous quantities are defined by (ℓ ∈ ℕ_{0})

\begin{aligned}F̂_{0,∓}(z) & = 0, F̂_{ℓ,−}(z) = ∑_{k=1}^{ℓ} f̂_{ℓ−k,−}z^{−k}, F̂_{ℓ,+}(z) = ∑_{k=0}^{ℓ−1} f̂_{ℓ−1−k,+}z^{k}, \\ Ĝ_{0,−}(z) & = 0, Ĝ_{ℓ,−}(z) = ∑_{k=1}^{ℓ} ĝ_{ℓ−k,−}z^{−k}, \\ Ĝ_{0,+}(z) & = \frac{1}{2}, Ĝ_{ℓ,+}(z) = ∑_{k=0}^{ℓ} ĝ_{ℓ−k,+}z^{k}, \\ Ĥ_{0,∓}(z) & = 0, Ĥ_{ℓ,−}(z) = ∑_{k=0}^{ℓ−1} ĥ_{ℓ−1−k,−}z^{−k}, Ĥ_{ℓ,+}(z) = ∑_{k=1}^{ℓ} ĥ_{ℓ−k,+}z^{k}, \\ K̂_{0,−}(z) & = \frac{1}{2}, K̂_{ℓ,−}(z) = ∑_{k=0}^{ℓ} ĝ_{ℓ−k,−}z^{−k} = Ĝ_{ℓ,−}(z) + ĝ_{ℓ,−}, \\ K̂_{0,+}(z) & = 0, K̂_{ℓ,+}(z) = ∑_{k=1}^{ℓ} ĝ_{ℓ−k,+}z^{k} = Ĝ_{ℓ,+}(z) − ĝ_{ℓ,+}. \end{aligned} (2.45)

The stationary zero-curvature relation (2.36), 0 = UV\underline{p}−V + ; \underline{p}U, is then equivalent to

−α(g_{p_{+},+} + g_{p_{−},−}^{−}) + f_{p_{+}−1,+} − f_{p_{−}−1,−}^{−} = 0,  (2.46)
β(g_{p_{+},+}^{−} + g_{p_{−},−}) + h_{p_{+}−1,+}^{−} − h_{p_{−}−1,−} = 0.  (2.47)

Thus, varying p_{±} ∈ ℕ_{0}, equations (2.46) and (2.47) 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}  (2.48)

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.,  (2.49)

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}.  (2.50)

By definition, the set of solutions of (2.48), with p_{±} ranging in ℕ_{0} 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 AL system s-AL\underline{p}(α, β) = 0, 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) and (2.45), 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}. (2.51)

In addition, one can show (cf. [40, 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_{−},−},  (2.52)

which in turn implies that

K\underline{p} = G\underline{p}  (2.53)

and hence renders V\underline{p} in (2.35) traceless in the stationary context. (We note that equations (2.52) and (2.53) cease to be valid in the time-dependent context, though.)

Remark 2.5. (i) The particular choice c_{0,+} = c_{0,−} = 1 yields the stationary Ablowitz–Ladik equation. Scaling c_{0,±} with the same constant then amounts to scaling V\underline{p} with this constant which drops out in the stationary zero-curvature equation (2.36). (ii) Different ratios between c_{0,+} and c_{0,−} will lead to different stationary hierarchies. In particular, the choice c_{0,+} = 2, c_{0,−} = ⋯ = c_{p_{−}−1,−} = 0, c_{p_{−},−} ≠ 0, yields the stationary Baxter–Szegő hierarchy considered in detail in [33]. However, in this case some parts from the recursion relation for the negative coefficients still remain. In fact, (2.12) reduces to g_{p_{−},−} − g_{p_{−},−}^{−} = αh_{p_{−}−1,−}, h_{p_{−}−1,−} = 0 and thus requires g_{p_{−},−} to be a constant in (2.48) and (2.64). Moreover, f_{p_{−}−1,−} = 0 in (2.48) in this case.

Next, taking into account (2.53), one infers that the expression R\underline{p}, defined as

R\underline{p} = G2 ; \underline{p} − F\underline{p}H\underline{p},  (2.54)

is a lattice constant, that is, R\underline{p} −R− ; \underline{p} = 0, by taking determinants in the stationary zero-curvature equation (2.36). Hence, R\underline{p}(z) only depends on z, and assuming in addition to (2.1) that

c_{0,±} ∈ ℂ \ \{0\}, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2} \ \{(0, 0)\},  (2.55)

one may write R\underline{p} as^{2}

\begin{matrix}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\}, \\ p = p_{−} + p_{+} − 1 ∈ ℕ_{0}. \end{matrix}  (2.56)

Moreover, multiplying (2.54) by z^{2p_{−}} and taking z → 0 yields

∏_{m=0}^{2p+1} E_{m} = \frac{c_{0,−}^{2}}{c_{0,+}^{2}} .  (2.57)

Relation (2.54) allows one to introduce a hyperelliptic curve K_{p} of (arithmetic) genus p = p_{−} + p_{+} − 1 (possibly with a singular affine part), where

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, p = p_{−} + p_{+} − 1.

(2.58)

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 p_{±}, 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 (2.34), (2.35), and (2.41)–(2.44) 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},  (2.59)

then results in the equations

α_{t\underline{p}} = i(zF − ; \underline{p} + α(G\underline{p} + K− ; \underline{p}) − F\underline{p}),  (2.60)
β_{t\underline{p}} = −i(β(G− ; \underline{p} + K\underline{p}) − H\underline{p} + z^{−1}H− ; \underline{p}),  (2.61)
0 = z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p},  (2.62)
0 = zβF − ; \underline{p} + αH\underline{p} + K− ; \underline{p} − K\underline{p}.  (2.63)

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}^{2}, \end{matrix} (2.64)

then defines the time-dependent Ablowitz–Ladik hierarchy. Explicitly, taking p_{−} = p_{+} for simplicity,

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,

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)  (2.65)

2We 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.

= (\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 (2.50).

By (2.64), (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_{−},−}^{−})).  (2.66)

Remark 2.6. From (2.37)–(2.40) and the explicit computations of the coefficients f_{ℓ,±}, g_{ℓ,±}, and h_{ℓ,±}, one concludes that the zero-curvature equation (2.59) 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}.  (2.67)

Moreover, R\underline{p} = G2 ; \underline{p} − H\underline{p}F\underline{p} and hence \{E_{m}\}_{m=0}^{2p+1} are invariant under this transformation. Furthermore, choosing c = e^{ic\underline{p}t}, one verifies that it is no restriction to assume c\underline{p} = 0. This also indicates that stationary solutions α, β can only be constructed up to a multiplicative constant (compare Theorem 3.7, in particular, (3.64), (3.65)).

Finally, we briefly hint at explicit expressions of the Lax pair for the Ablowitz– Ladik hierarchy. More details will be presented in [42]. First we need some notation. Let T be a bounded operator in ℓ^{2}(ℤ). Given the standard basis (2.25) 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}.  (2.68)

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} (2.69)

Next, consider the finite difference expression P\underline{p} defined by

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},  (2.70)

with L given by (2.27), c\underline{p} = (c_{p_{−},−} +c_{p_{+},+})/2, and Q_{d} denoting the doubly infinite diagonal matrix

Q_{d} = ((−1)^{k}δ_{k,ℓ})_{k,ℓ∈ℤ}.  (2.71)

Then one can show that (L, P\underline{p}) represents the Lax pair for the Ablowitz–Ladik equations (2.64) for \underline{p} ∈ ℕ_{0}^{2} \ \{(0, 0\}. In particular, the hierarchy of nonlinear Ablowitz–Ladik evolution equations (2.64) then can alternatively be derived by imposing the Lax commutator equations

L_{t\underline{p}}(t\underline{p}) − [P\underline{p}(t\underline{p}), L(t\underline{p})] = 0, t\underline{p} ∈ ℝ, \underline{p} ∈ ℕ_{0}^{2}.  (2.72)

For additional representations of P\underline{p} in terms of L and a particular factorization of L we refer to [42]. The Ablowitz–Ladik Lax pair in the special defocusing case, where β = \overline{α}, in the finite-dimensional context, was recently derived by Nenciu [50].

In the special stationary case, where P\underline{p} and L commute, [P\underline{p}, L] = 0, they satisfy an algebraic relationship of the type

\begin{matrix}P 2 ; \underline{p} + R\underline{p}(L) = P 2 ; \underline{p} + (c_{0,+}^{2}/4)L^{−2p_{−}} ∏_{m=0}^{2p+1}(L − E_{m}) = 0, \\ R\underline{p}(z) = (c_{0,+}^{2}/4)z^{−2p_{−}} ∏_{m=0}^{2p+1}(z − E_{m}), p = p_{−} + p_{+} − 1. \end{matrix}  (2.73)

Thus, the expression P 2 ; \underline{p} + R\underline{p}(L) in (2.73) represents the Burchnall–Chaundy Laurent polynomial of the Lax pair (L, P\underline{p}).

3. The Stationary Ablowitz–Ladik Formalism

This section is devoted to a detailed study of the stationary Ablowitz–Ladik hierarchy and its algebro-geometric solutions. Our principal tools are derived from combining the Laurent polynomial recursion formalism introduced in Section 2 and a fundamental meromorphic function ϕ on a hyperelliptic curve K_{p}. With the help of ϕ we study the Baker–Akhiezer vector Ψ, trace formulas, and theta function representations of ϕ, Ψ, α, and β. For proofs of the elementary results of the stationary formalism we refer to [39], [40].

Unless explicitly stated otherwise, we suppose throughout this section that

α, β ∈ ℂ^{ℤ}, α(n)β(n) ∉ {0, 1}, n ∈ ℤ,  (3.1)

and assume (2.5)–(2.13), (2.34)–(2.36), (2.41)–(2.44), (2.48), (2.53), (2.54), (2.56), keeping \underline{p} ∈ ℕ_{0}^{2} \ \{(0, 0\} fixed.

We recall the hyperelliptic curve

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,  (3.2)

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,

as introduced in (2.58). Throughout this section we assume the affine part of K_{p} to be nonsingular, that is, we suppose that

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

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 (B.8) 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 and hence \underline{p} ∈ ℕ_{0}^{2} \ \{(0, 0), (0, 1), (1, 0)\}.

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. (2.54))

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} of ( ·)^{p_{−}}F\underline{p} and ( ·)^{p_{−}−1}H\underline{p} 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}. 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)}  (3.11)

= \frac{−H\underline{p}(z, n)}{(c_{0,+}/2)z^{−p_{−}}y − G\underline{p}(z, n)},  (3.12)
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_{∞_{−}} ˆ \underline{μ}(n)},  (3.13)

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

ˆ\underline{μ} = \{μ̂_{1}, …, μ̂_{p}\}, \underline{ν̂} = \{ν̂_{1}, …, ν̂_{p}\} ∈ Sym^{p}(K_{p}).  (3.14)

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}  (3.15)

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} (3.16)

Basic properties of ϕ and Ψ are summarized in the following result. Lemma 3.1 ([40]). Suppose that α, β satisfy (3.1) and the \underline{p}th stationary Ablowitz– Ladik system (2.48). Moreover, assume (3.2) 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,  (3.17)

as well as

ϕ(P)ϕ(P^{∗}) = \frac{H\underline{p}(z)}{F\underline{p}(z)},  (3.18)
ϕ(P) + ϕ(P^{∗}) = 2\frac{G\underline{p}(z)}{F\underline{p}(z)},  (3.19)
ϕ(P) − ϕ(P^{∗}) = c_{0,+}z^{−p_{−}} \frac{y(P)}{F\underline{p}(z)}.  (3.20)

The vector Ψ satisfies

U(z)Ψ^{−}(P) = Ψ(P),  (3.21)
V\underline{p}(z)Ψ^{−}(P) = −(i/2)c_{0,+}z^{−p_{−}}yΨ^{−}(P),  (3.22)
ψ_{2}(P, n, n_{0}) = ϕ(P, n)ψ_{1}(P, n, n_{0}),  (3.23)
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}),  (3.24)
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}),  (3.25)
ψ_{1}(P, n, n_{0})ψ_{2}(P^{∗}, n, n_{0}) + ψ_{1}(P^{∗}, n, n_{0})ψ_{2}(P, n, n_{0})  (3.26)

= 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})  (3.27)

= −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}  (3.28)

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 3.2 ([40]). Suppose that α, β satisfy (3.1) and the \underline{p}th stationary Ablowitz– Ladik system (2.48). Then,

\frac{α}{α^{+}} = (−1)^{p+1} \frac{c_{0,+}}{c_{0,−}} ∏_{j=1}^{p} μ_{j},  (3.29)

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

∑_{j=1}^{p} μ_{j} = α^{+}β − γ^{+} \frac{α^{++}}{α^{+}} − \frac{c_{1,+}}{c_{0,+}} ,  (3.31)

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

Next we turn to asymptotic properties of ϕ and Ψ in a neighborhood of P_{∞_{±}} and P_{0,±}.

Lemma 3.3 ([40]). Suppose that α, β satisfy (3.1) and the \underline{p}th stationary Ablowitz– Ladik system (2.48). Moreover, let P = (z, y) ∈ K_{p} \ {P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}}, (n, n_{0}) ∈ ℤ^{2}. Then ϕ has the asymptotic behavior

ϕ(P) =_{ζ→0} \begin{cases}β + β^{−}γζ + O(ζ^{2}), & P → P_{∞_{+}}, \\ −(α^{+})^{−1}ζ^{−1} + (α^{+})^{−2}α^{++}γ^{+} + O(ζ), & P → P_{∞_{−}}, \end{cases} ζ = 1/z, (3.33)

ϕ(P) =_{ζ→0} \begin{cases}α^{−1} − α^{−2}α^{−}γζ + O(ζ^{2}), & P → P_{0,+}, \\ −β^{+}ζ − β^{++}γ^{+}ζ^{2} + O(ζ^{3}), & P → P_{0,−}, \end{cases} ζ = z.  (3.34)

The components of the Baker–Akhiezer vector Ψ have the asymptotic behavior

ψ_{1}(P, n, n_{0}) =_{ζ→0} \begin{cases}ζ^{n_{0}−n}(1 + O(ζ)), & P → P_{∞_{+}}, \\ \frac{α^{+}(n)}{α^{+}(n_{0})}Γ(n, n_{0}) + O(ζ), & P → P_{∞_{−}}, \end{cases} ζ = 1/z,  (3.35)

ψ_{1}(P, n, n_{0}) =_{ζ→0} \begin{cases}\frac{α(n)}{α(n_{0})} + O(ζ), & P → P_{0,+}, \\ ζ^{n−n_{0}}Γ(n, n_{0})(1 + O(ζ)), & P → P_{0,−}, \end{cases} ζ = z,  (3.36)

ψ_{2}(P, n, n_{0}) =_{ζ→0} \begin{cases}β(n)ζ^{n_{0}−n}(1 + O(ζ)), & P → P_{∞_{+}}, \\ −\frac{1}{α^{+}(n_{0})}Γ(n, n_{0})ζ^{−1}(1 + O(ζ)), & P → P_{∞_{−}}, \end{cases} ζ = 1/z, (3.37)

ψ_{2}(P, n, n_{0}) =_{ζ→0} \begin{cases}\frac{1}{α(n_{0})} + O(ζ), & P → P_{0,+}, \\ −β^{+}(n)Γ(n, n_{0})ζ^{n+1−n_{0}}(1 + O(ζ)), & P → P_{0,−}, \end{cases} ζ = z.(3.38)

The divisors (ψ_{j}) of ψ_{j}, j = 1, 2, are given by

(ψ_{1}( · , n, n_{0})) = D_{ˆ\underline{μ}(n)} − D_{ˆ\underline{μ}(n_{0})} + (n − n_{0})(D_{P_{0,−}} − D_{P_{∞_{+}}} ),  (3.39)

(ψ_{2}( · , n, n_{0})) = D_{\underline{ν̂}(n)} − D_{ˆ\underline{μ}(n_{0})} + (n − n_{0})(D_{P_{0,−}} − D_{P_{∞_{+}}} ) + D_{P_{0,−}} − D_{P_{∞_{−}}} . (3.40)

Since nonspecial divisors play a fundamental role in this section and the next, we now take a closer look at them.

Lemma 3.4 ([40]). Suppose that α, β satisfy (3.1) and the \underline{p}th stationary Ablowitz– Ladik system (2.48). Moreover, assume (3.2) and (3.3) and let n ∈ ℤ. Let Dˆ\underline{μ}, ˆ\underline{μ} = \{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}  (3.41)

Then D_{ˆ\underline{μ}(n)} and D_{\underline{ν̂}(n)} are nonspecial for all n ∈ ℤ.

Next, we shall provide an explicit representation of ϕ, Ψ, α, and β in terms of the Riemann theta function associated with K_{p}. We freely employ the notation established in Appendix A. (We recall our tacit assumption p ∈ ℕ to avoid the trivial case p = 0.)

Let θ denote the Riemann theta function associated with K_{p} and introduce a fixed homology basis {a_{j}, b_{j}}_{j=1,...,p} on K_{p}. Choosing as a convenient fixed base point one of the branch points, Q_{0} = (E_{m_{0}}, 0), the Abel maps \underline{A}_{Q_{0}} and \underline{α}_{Q_{0}} are defined by (A.20) and (A.21) and the Riemann vector \underline{Ξ}_{Q_{0}} is given by (A.31). Let ω_{P_{+},P_{−}}^{(3)} be the normal differential of the third kind holomorphic on K_{p} \ {P_{+}, P_{−}} with simple poles at P_{±} and residues ±1, respectively. In particular, one obtains for ω_{P_{0,−},P_{∞_{±}}}^{(3)} ,

ω_{P_{0,−},P_{∞_{±}}}^{(3)} = (\frac{y + y_{0,−}}{z} ∓ ∏_{j=1}^{p}(z − λ_{±,j}))\frac{dz}{2y} , P_{0,−} = (0, y_{0,−}),  (3.42)

where the constants {λ_{±,j}}_{j=1}^{p} ⊂ ℂ are uniquely determined by employing the normalization

∫_{a_{j}} ω_{P_{0,−},P_{∞_{±}}}^{(3)} = 0, j = 1, …, p.  (3.43)

The explicit formula (3.42) then implies the following asymptotic expansions (using the local coordinate ζ = z near P_{0,±} and ζ = 1/z near P_{∞_{±}}),

∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} =_{ζ→0} \begin{cases}0 \\ ln(ζ) \end{cases}\} + ω_{0}^{0,±}(P_{0,−}, P_{∞_{−}}) + O(ζ) as P → P_{0,±},  (3.44)

∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} =_{ζ→0} \begin{cases}0 \\ − ln(ζ) \end{cases}\} + ω_{0}^{∞_{±}}(P_{0,−}, P_{∞_{−}}) + O(ζ) as P → P_{∞_{±}}, (3.45)

∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{+}}}^{(3)} =_{ζ→0} \begin{cases}0 \\ ln(ζ) \end{cases}\} + ω_{0}^{0,±}(P_{0,−}, P_{∞_{+}}) + O(ζ) as P → P_{0,±},  (3.46)

∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{+}}}^{(3)} =_{ζ→0} \begin{cases}− ln(ζ) \\ 0 \end{cases}\} + ω_{0}^{∞_{±}}(P_{0,−}, P_{∞_{+}}) + O(ζ) as P → P_{∞_{±}}. (3.47)

Lemma 3.5. With ω_{0}^{∞_{σ}}(P_{0,−}, P_{∞_{±}}) and ω_{0}^{0,σ′}(P_{0,−}, P_{∞_{±}}), σ, σ′ ∈ {+, −}, defined as in (3.44)–(3.47) one has

\begin{matrix}exp (ω_{0}^{0,−}(P_{0,−}, P_{∞_{±}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{±}}) \\ − ω_{0}^{∞_{−}}(P_{0,−}, P_{∞_{±}}) + ω_{0}^{0,+}(P_{0,−}, P_{∞_{±}})) = 1. \end{matrix}  (3.48)

Proof. Pick Q_{1,±} = (z_{1}, ±y_{1}) ∈ K_{p} \ {P_{∞_{±}}} in a neighborhood of P_{∞_{±}} and Q_{2,±} = (z_{2}, ±y_{2}) ∈ K_{p} \ {P_{0,±}} in a neighborhood of P_{0,±}. Without loss of generality one may assume that P_{∞_{+}} and P_{0,+} lie on the same sheet. Then by (3.42),

∫_{Q_{0}}^{Q_{2,−}} ω_{P_{0,−},P_{∞_{−}}}^{(3)} − ∫_{Q_{0}}^{Q_{1,+}} ω_{P_{0,−},P_{∞_{−}}}^{(3)} − ∫_{Q_{0}}^{Q_{1,−}} ω_{P_{0,−},P_{∞_{−}}}^{(3)} + ∫_{Q_{0}}^{Q_{2,+}} ω_{P_{0,−},P_{∞_{−}}}^{(3)}

= ∫_{Q_{0}}^{Q_{2,+}} \frac{dz}{z} − ∫_{Q_{0}}^{Q_{1,+}} \frac{dz}{z} = ln(z_{2}) − ln(z_{1}) + 2πik,  (3.49)

for some k ∈ ℤ. On the other hand, by (3.44)–(3.47) one obtains

∫_{Q_{0}}^{Q_{2,−}} ω_{P_{0,−},P_{∞_{−}}}^{(3)} − ∫_{Q_{0}}^{Q_{1,+}} ω_{P_{0,−},P_{∞_{−}}}^{(3)} − ∫_{Q_{0}}^{Q_{1,−}} ω_{P_{0,−},P_{∞_{−}}}^{(3)} + ∫_{Q_{0}}^{Q_{2,+}} ω_{P_{0,−},P_{∞_{−}}}^{(3)}

= ln(z_{2}) + ln(1/z_{1}) + ω_{0}^{0,−}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}})

− ω_{0}^{∞_{−}}(P_{0,−}, P_{∞_{−}}) + ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}}) + O(z_{2}) + O(1/z_{1}),  (3.50)

and hence the part of (3.48) concerning ω_{P_{0,−},P_{∞_{−}}}^{(3)} follows. The corresponding

result for ω_{P_{0,−},P_{∞_{+}}}^{(3)} is proved analogously. □

In the following it will be convenient to use the abbreviation

\begin{matrix}\underline{z}(P, \underline{Q}) = \underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P) + \underline{α}_{Q_{0}}(D\underline{Q}), \\ P ∈ K_{p}, \underline{Q} = \{Q_{1}, …, Q_{p}\} ∈ Sym^{p}(K_{p}). \end{matrix}  (3.51)

We note that \underline{z}( · , \underline{Q}) is independent of the choice of base point Q_{0}.

For later use we state the following result.

Lemma 3.6. The following relations hold:

\underline{z}(P_{∞_{+}}, ˆ \underline{μ}^{+}) = \underline{z}(P_{∞_{−}}, \underline{ν̂}) = \underline{z}(P_{0,−}, ˆ \underline{μ}) = \underline{z}(P_{0,+}, \underline{ν̂}^{+}),  (3.52)
\underline{z}(P_{∞_{+}}, \underline{ν̂}^{+}) = \underline{z}(P_{0,−}, \underline{ν̂}), \underline{z}(P_{0,+}, ˆ \underline{μ}^{+}) = \underline{z}(P_{∞_{−}}, ˆ \underline{μ}).  (3.53)

Proof. We indicate the proof of some of the relations to be used in (3.69) and (3.70). Let ˆ \underline{λ} denote either ˆ \underline{μ} or \underline{ν̂}. Then,

\underline{z}(P_{0,+}, ˆ \underline{λ}^{+}) = \underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P_{0,+}) + \underline{α}_{Q_{0}}(Dˆ\underline{λ}^{+})
= \underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P_{0,+}) + \underline{α}_{Q_{0}}(Dˆ\underline{λ}) + \underline{A}_{P_{0,−}}(P_{∞_{+}})
= \underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P_{∞_{−}}) + \underline{α}_{Q_{0}}(Dˆ\underline{λ})
= \underline{z}(P_{∞_{−}}, ˆ \underline{λ}),  (3.54)
\underline{z}(P_{∞_{+}}, ˆ \underline{λ}^{+}) = \underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P_{∞_{+}}) + \underline{α}_{Q_{0}}(Dˆ\underline{λ}^{+})
= \underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P_{∞_{+}}) + \underline{α}_{Q_{0}}(Dˆ\underline{λ}) + \underline{A}_{P_{0,−}}(P_{∞_{+}})
= \underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P_{0,−}) + \underline{α}_{Q_{0}}(Dˆ\underline{λ})

= \underline{z}(P_{0,−}, ˆ \underline{λ}), etc.  (3.55)

Here we used \underline{A}_{Q_{0}}(P^{∗}) = −\underline{A}_{Q_{0}}(P), P ∈ K_{p}, since Q_{0} is a branch point of K_{p}, and \underline{α}_{Q_{0}}(Dˆ\underline{λ}^{+}) = \underline{α}_{Q_{0}}(Dˆ\underline{λ}) + \underline{A}_{P_{0,−}}(P_{∞_{+}}). The latter equality immediately follows

from (3.39) in the case ˆ \underline{λ} = ˆ \underline{μ} and from combining (3.13) and (3.40) in the case ˆ\underline{λ} = \underline{ν̂}. □

Given these preparations, the theta function representations of ϕ, ψ_{1}, ψ_{2}, α, and β then read as follows.

Theorem 3.7. Suppose that α, β satisfy (3.1) and the \underline{p}th stationary Ablowitz– Ladik system (2.48). Moreover, assume hypothesis (3.2) and (3.3), and let P ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}\} and (n, n_{0}) ∈ ℤ^{2}. Then for each n ∈ ℤ, D_{ˆ\underline{μ}(n)} and D_{\underline{ν̂}(n)} are nonspecial. Moreover,

ϕ(P, n) = C(n)\frac{θ(\underline{z}(P, \underline{ν̂}(n)))}{θ(\underline{z}(P, ˆ \underline{μ}(n)))} exp ( ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} ),  (3.56)

ψ_{1}(P, n, n_{0}) = C(n, n_{0})\frac{θ(\underline{z}(P, ˆ \underline{μ}(n)))}{θ(\underline{z}(P, ˆ \underline{μ}(n_{0})))} exp ((n − n_{0}) ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{+}}}^{(3)} ), (3.57)
ψ_{2}(P, n, n_{0}) = C(n)C(n, n_{0})\frac{θ(\underline{z}(P, \underline{ν̂}(n)))}{θ(\underline{z}(P, ˆ \underline{μ}(n_{0})))}

× exp ( ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} + (n − n_{0}) ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{+}}}^{(3)} ),  (3.58)

where

C(n) = (−1)^{n−n_{0}} exp ((n − n_{0})(ω_{0}^{0,−}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}})))
× \frac{1}{α(n_{0})} exp ( − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}}))\frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n_{0})))}{θ(\underline{z}(P_{0,+}, \underline{ν̂}(n_{0})))},  (3.59)
C(n, n_{0}) = exp ( − (n − n_{0})ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}}))\frac{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0})))}{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n)))}.  (3.60)

The Abel map linearizes the auxiliary divisors D_{ˆ\underline{μ}(n)} and D_{\underline{ν̂}(n)} in the sense that

\underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n)}) = \underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n_{0})}) + (n − n_{0})\underline{A}_{P_{0,−}}(P_{∞_{+}}),  (3.61)
\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_{∞_{+}}),  (3.62)

in addition,

\underline{α}_{Q_{0}}(D_{\underline{ν̂}(n)}) = \underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n)}) − \underline{A}_{Q_{0}}(P_{0,−}) + \underline{A}_{Q_{0}}(P_{∞_{−}})
= \underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n)}) + \underline{A}_{P_{0,−}}(P_{∞_{−}}).  (3.63)

Finally, α, β are of the form

α(n) = α(n_{0})(−1)^{n−n_{0}} exp ( − (n − n_{0})(ω_{0}^{0,−}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}})))

× \frac{θ(\underline{z}(P_{0,+}, \underline{ν̂}(n_{0})))θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n)))}{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n_{0})))θ(\underline{z}(P_{0,+}, \underline{ν̂}(n)))},  (3.64)

β(n) = β(n_{0})(−1)^{n−n_{0}} exp ((n − n_{0})(ω_{0}^{0,−}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}})))

× \frac{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0})))θ(\underline{z}(P_{∞_{+}}, \underline{ν̂}(n)))}{θ(\underline{z}(P_{∞_{+}}, \underline{ν̂}(n_{0})))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n)))},  (3.65)

α(n)β(n) = exp (ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}}))

× \frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n)))θ(\underline{z}(P_{∞_{+}}, \underline{ν̂}(n)))}{θ(\underline{z}(P_{0,+}, \underline{ν̂}(n)))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n)))},  (3.66)

Γ(n, n_{0}) = exp ((n − n_{0})(ω_{0}^{0,−}(P_{0,−}, P_{∞_{+}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}})))

× \frac{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n)))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0})))}{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n_{0})))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n)))}.  (3.67)

Proof. Applying Abel’s theorem (cf. Theorem A.1, (A.29)) to (3.13) proves (3.63), and applying it to (3.39), (3.40) results in (3.61) and (3.62). By Lemma 3.4, Dˆ\underline{μ} and D\underline{ν̂} are nonspecial. By equation (3.13) and Theorem A.3, ϕ(P, n) exp ( −

∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} ) must be of the type

ϕ(P, n) exp ( − ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} ) = C(n)\frac{θ(\underline{z}(P, \underline{ν̂}(n)))}{θ(\underline{z}(P, ˆ \underline{μ}(n)))}  (3.68)

for some constant C(n). A comparison of (3.68) and the asymptotic relations (3.33) then yields, with the help of (3.44), (3.45) and (3.52), (3.53), the following expressions for α and β:

(α^{+})^{−1} = C^{+}e^{ω_{0}^{0,+}(P_{0,−},P_{∞_{−}})} \frac{θ(\underline{z}(P_{0,+}, \underline{ν̂}^{+}))}{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}^{+}))}

= C^{+}e^{ω_{0}^{0,+}(P_{0,−},P_{∞_{−}})} \frac{θ(\underline{z}(P_{∞_{−}}, \underline{ν̂}))}{θ(\underline{z}(P_{∞_{−}}, ˆ \underline{μ}))}

= −Ce^{ω_{0}^{∞_{−}} (P_{0,−},P_{∞_{−}})} \frac{θ(\underline{z}(P_{∞_{−}}, \underline{ν̂}))}{θ(\underline{z}(P_{∞_{−}}, ˆ \underline{μ}))}.  (3.69)

Similarly one obtains

β^{+} = C^{+}e^{ω_{0}^{∞_{+}} (P_{0,−},P_{∞_{−}})} \frac{θ(\underline{z}(P_{∞_{+}}, \underline{ν̂}^{+}))}{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}^{+}))}

= C^{+}e^{ω_{0}^{∞_{+}} (P_{0,−},P_{∞_{−}})} \frac{θ(\underline{z}(P_{0,−}, \underline{ν̂}))}{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}))}

= −Ce^{ω_{0}^{0,−}(P_{0,−},P_{∞_{−}})} \frac{θ(\underline{z}(P_{0,−}, \underline{ν̂}))}{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}))}.  (3.70)

Here we used (3.61) and (3.62), more precisely,

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

(3.71)

Thus, one concludes

C(n + 1) = − exp [ω_{0}^{0,−}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}})]C(n), n ∈ ℤ, (3.72)

and

C(n + 1) = − exp [ω_{0}^{∞_{−}}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}})]C(n), n ∈ ℤ, (3.73)

which is consistent with (3.48). The first-order difference equation (3.72) then implies

C(n) = (−1)^{(n−n_{0})} exp [(n − n_{0})(ω_{0}^{0,−}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}}))]C(n_{0}),
n, n_{0} ∈ ℤ. (3.74)

Thus one infers (3.64) and (3.65). Moreover, (3.74) and taking n = n_{0} in the first line in (3.69) yield (3.59). Dividing the first line in (3.70) by the first line in (3.69) then proves (3.66).

By (3.39) and Theorem A.3, ψ_{1}(P, n, n_{0}) must be of the type (3.57). A comparison of (3.15), (3.33), and (3.57) as P → P_{∞_{+}} (with local coordinate ζ = 1/z) then yields

ψ_{1}(P, n, n_{0}) =_{ζ→0} ζ^{n_{0}−n}(1 + O(ζ))  (3.75)

and

ψ_{1}(P, n, n_{0}) =_{ζ→0} C(n, n_{0})\frac{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n)))}{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0})))}
× exp [(n − n_{0})ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}})]ζ^{n_{0}−n}(1 + O(ζ))  (3.76)

proving (3.60). Equation (3.58) is clear from (3.23), (3.56), and (3.57).

Finally, a comparison of (3.36) and (3.57) as P → P_{0,−} (with local coordinate ζ = z) yields

ψ_{1}(P, n, n_{0}) =_{ζ→0} Γ(n, n_{0})ζ^{n−n_{0}}(1 + O(ζ))  (3.77)

=_{ζ→0} C(n, n_{0})\frac{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n)))}{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n_{0})))} exp ((n − n_{0})ω_{0}^{0,−}(P_{0,−}, P_{∞_{+}}))

× ζ^{n−n_{0}}(1 + O(ζ))  (3.78)

and hence

Γ(n, n_{0}) = C(n, n_{0})\frac{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n)))}{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n_{0})))} exp ((n − n_{0})ω_{0}^{0,−}(P_{0,−}, P_{∞_{+}}))
= exp ((n − n_{0})(ω_{0}^{0,−}(P_{0,−}, P_{∞_{+}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}})))
× \frac{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n)))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0})))}{θ(\underline{z}(P_{0,−}, ˆ \underline{μ}(n_{0})))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n)))}  (3.79)

using (3.60). □

We note that the apparent n_{0}-dependence of C(n) in the right-hand side of (3.59) actually drops out to ensure the n_{0}-independence of ϕ in (3.56).

The theta function representations (3.64), (3.65) for α, β and that for Γ in (3.67) also show that γ(n) ∉ {0, 1} for all n ∈ ℤ, and hence condition (3.1) is satisfied for the stationary algebro-geometric AL solutions discussed in this section, provided the associated divisors Dˆ\underline{μ}(n) and D\underline{ν̂}(n) stay away from P_{∞_{±}}, P_{0,±} for all n ∈ ℤ.

The stationary algebro-geometric initial value problem for the Ablowitz–Ladik hierarchy with complex-valued initial data, that is, the construction of α and β by starting from a set of initial data (nonspecial divisors) of full measure, will be presented in [41].

4. The Time-Dependent Ablowitz–Ladik Formalism

In this section we extend the algebro-geometric analysis of Section 3 to the timedependent Ablowitz–Ladik hierarchy. For proofs of the elementary results of the time-dependent formalism we refer to [39], [40].

For most of this section we assume the following hypothesis.

Hypothesis 4.1. (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}  (4.1)

(ii) Assume that the hyperelliptic curve K_{p} satisfies (3.2) and (3.3).

The basic problem in the analysis of algebro-geometric solutions of the Ablowitz– Ladik hierarchy consists of solving the time-dependent \underline{r}th Ablowitz–Ladik flow with initial data a stationary solution of the \underline{p}th system in the hierarchy. More precisely, given \underline{p} ∈ ℕ_{0}^{2}\\{(0, 0\} we consider a solution α^{(0)}, β^{(0)} of the \underline{p}th stationary Ablowitz–Ladik system s-AL\underline{p}(α^{(0)}, β^{(0)}) = 0, \underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2}\\{(0, 0\}, associated with the hyperelliptic curve K_{p} and a corresponding set of summation constants \{c_{ℓ,±}\}_{ℓ=1,...,p_{±}} ⊂ ℂ. Next, let \underline{r} = (r_{−}, r_{+}) ∈ ℕ_{0}^{2}; we intend to construct a solution α, β of the Ablowitz–Ladik 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.

Summing up, we are looking for solutions α, β of the time-dependent algebrogeometric 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=t_{0,\underline{r}}} = (α^{(0)}, β^{(0)}), \end{matrix}  (4.2)

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 (4.3)

for some t_{0,\underline{r}} ∈ ℝ, where α = α(n, t\underline{r}), β = β(n, t\underline{r}) satisfy (4.1) and a fixed curve K_{p} is associated with the stationary solutions α^{(0)}, β^{(0)} in (4.3). Here,

\underline{p} = (p_{−}, p_{+}) ∈ ℕ_{0}^{2} \ \{(0, 0)\}, \underline{r} = (r_{−}, r_{+}) ∈ ℕ_{0}^{2}, p = p_{−} + p_{+} − 1.  (4.4)

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,  (4.5)
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.  (4.6)

One can show (cf. [41]) that the stationary Ablowitz–Ladik system (4.6) is actually satisfied for all times t\underline{r} ∈ ℝ. Thus, we impose

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,  (4.7)
U(z, t\underline{r})V\underline{p}(z, t\underline{r}) − V + ; \underline{p}(z, t\underline{r})U(z, t\underline{r}) = 0,  (4.8)

instead of (4.5) and (4.6). For further reference, we recall the relevant quantities here (cf. (2.34), (2.35), (2.41)–(2.43), (2.53)):

\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}  (4.9)

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},  (4.10)

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, (4.7) and (4.8) are equivalent to (cf. (2.37)–(2.40), (2.60)–(2.63)),

α_{t\underline{r}} = i(zF̃ − ; \underline{r} + α(G̃\underline{r} + K̃− ; \underline{r}) − F̃\underline{r}),  (4.11)
β_{t\underline{r}} = −i(β(G̃− ; \underline{r} + K̃\underline{r}) − H̃\underline{r} + z^{−1}H̃− ; \underline{r}),  (4.12)
0 = z(G̃− ; \underline{r} − G̃\underline{r}) + zβF̃\underline{r} + αH̃− ; \underline{r},  (4.13)
0 = zβF̃ − ; \underline{r} + αH̃\underline{r} + K̃− ; \underline{r} − K̃\underline{r},  (4.14)
0 = z(G− ; \underline{p} − G\underline{p}) + zβF\underline{p} + αH− ; \underline{p},  (4.15)
0 = zβF − ; \underline{p} + αH\underline{p} − G\underline{p} + G− ; \underline{p},  (4.16)
0 = −F\underline{p} + zF − ; \underline{p} + α(G\underline{p} + G− ; \underline{p}),  (4.17)
0 = zβ(G\underline{p} + G− ; \underline{p}) − zH\underline{p} + H− ; \underline{p},  (4.18)

respectively. In particular, (2.54) holds in the present t\underline{r}-dependent setting, that is,

G2 ; \underline{p} − F\underline{p}H\underline{p} = R\underline{p}.  (4.19)

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}  (4.20)

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}  (4.21)

and note that the regularity assumptions (4.1) 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 (3.11), (3.12), 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})}  (4.22)

= \frac{−H\underline{p}(z, n, t\underline{r})}{(c_{0,+}/2)z^{−p_{−}}y − G\underline{p}(z, n, t\underline{r})},  (4.23)
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_{∞_{−}} ˆ \underline{μ}(n,t\underline{r})}.  (4.24)

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}) ,  (4.25)

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)))

× \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}  (4.26)

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)))

× 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} (4.27)

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}  (4.28)

The following lemma records basic properties of ϕ and Ψ in analogy to the stationary case discussed in Lemma 3.1.

Lemma 4.2 ([40]). Assume Hypothesis 4.1 and suppose that (4.7), (4.8) 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,  (4.29)
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),  (4.30)
ϕ(P)ϕ(P^{∗}) = \frac{H\underline{p}(z)}{F\underline{p}(z)},  (4.31)

ϕ(P) + ϕ(P^{∗}) = 2\frac{G\underline{p}(z)}{F\underline{p}(z)},  (4.32)
ϕ(P) − ϕ(P^{∗}) = c_{0,+}z^{−p_{−}} \frac{y(P)}{F\underline{p}(z)}.  (4.33)

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}}),  (4.34)
U(z)Ψ^{−}(P) = Ψ(P),  (4.35)
V\underline{p}(z)Ψ^{−}(P) = −(i/2)c_{0,+}z^{−p_{−}}yΨ^{−}(P),  (4.36)
Ψ_{t\underline{r}}(P) = Ṽ + ; \underline{r}(z)Ψ(P),  (4.37)
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}), (4.38)

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}), (4.39)

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}),  (4.40)

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}),  (4.41)

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}  (4.42)

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}.

Next we consider the t\underline{r}-dependence of F\underline{p}, G\underline{p}, and H\underline{p}.

Lemma 4.3 ([40]). Assume Hypothesis 4.1 and suppose that (4.7), (4.8) 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},  (4.43)
G_{\underline{p},t\underline{r}} = iF\underline{p}H̃\underline{r} − iH\underline{p}F̃\underline{r},  (4.44)

H_{\underline{p},t\underline{r}} = 2iG\underline{p}H̃\underline{r} − i(G̃\underline{r} + K̃\underline{r})H\underline{p}.  (4.45)

In particular, (4.43)–(4.45) are equivalent to

V_{\underline{p},t\underline{r}} = [Ṽ\underline{r}, V\underline{p}].  (4.46)

Next we turn to the Dubrovin 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 4.4 ([40]). Assume Hypothesis 4.1 and suppose that (4.7), (4.8) 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 (4.20), 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,  (4.47)

with

μ̂_{j}(n, ·) ∈ C^{∞}(I_{μ}, K_{p}), j = 1, …, p, n ∈ ℤ.  (4.48)

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 (4.21), 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,  (4.49)

with

ν̂_{j}(n, ·) ∈ C^{∞}(I_{ν}, K_{p}), j = 1, …, p, n ∈ ℤ.  (4.50)

When attempting to solve the Dubrovin systems (4.47) and (4.49), they must be augmented with appropriate divisors D_{ˆ\underline{μ}(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. The algebro-geometric initial value problem for the AL hierarchy with appropriate initial divisors will be discussed in detail in [41].

Next, we turn to the asymptotic expansions of ϕ and Ψ in a neighborhood of P_{∞_{±}} and P_{0,±}. Since this is a bit more involved we provide some details.

Lemma 4.5. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold. Moreover, let P = (z, y) ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}\}, (n, n_{0}, t\underline{r}, t_{0,\underline{r}}) ∈ ℤ^{2} × ℝ^{2}. Then ϕ has the asymptotic behavior

ϕ(P) =_{ζ→0} \begin{cases}β + β^{−}γζ + O(ζ^{2}), & P → P_{∞_{+}}, \\ −(α^{+})^{−1}ζ^{−1} + (α^{+})^{−2}α^{++}γ^{+} + O(ζ), & P → P_{∞_{−}}, \end{cases} ζ = 1/z, (4.51)

ϕ(P) =_{ζ→0} \begin{cases}α^{−1} − α^{−2}α^{−}γζ + O(ζ^{2}), & P → P_{0,+}, \\ −β^{+}ζ − β^{++}γ^{+}ζ^{2} + O(ζ^{3}), & P → P_{0,−}, \end{cases} ζ = z.  (4.52)

The component ψ_{1} of the Baker–Akhiezer vector Ψ has the asymptotic behavior

ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) =_{ζ→0} exp ( ± \frac{i}{2}(t\underline{r} − t_{0,\underline{r}}) ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}ζ^{−s})(1 + O(ζ))

× \begin{cases}ζ^{n_{0}−n}, & P → P_{∞_{+}}, \\ \begin{matrix}Γ(n, n_{0}, t\underline{r})\frac{α^{+}(n,t\underline{r})}{α^{+}(n_{0},t_{0,\underline{r}})} \\ × exp (i ∫_{t_{0,\underline{r}}}^{t\underline{r}} ds(g̃_{r_{+},+}(n_{0}, s) − g̃_{r_{−},−}(n_{0}, s))), \end{matrix} & P → P_{∞_{−}}, \end{cases} ζ = 1/z, (4.53)

ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) =_{ζ→0} exp ( ± \frac{i}{2}(t\underline{r} − t_{0,\underline{r}}) ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}ζ^{−s})(1 + O(ζ))

× \begin{cases}\frac{α(n,t\underline{r})}{α(n_{0},t_{0,\underline{r}})}, & P → P_{0,+}, \\ \begin{matrix}Γ(n, n_{0}, t\underline{r})ζ^{n−n_{0}} \\ × exp (i ∫_{t_{0,\underline{r}}}^{t\underline{r}} ds(g̃_{r_{+},+}(n_{0}, s) − g̃_{r_{−},−}(n_{0}, s))), \end{matrix} & P → P_{0,−}, \end{cases} ζ = z.(4.54)

Proof. Since by the definition of ϕ in (4.22) the time parameter t\underline{r} can be viewed as an additional but fixed parameter, the asymptotic behavior of ϕ remains the same as in Lemma 3.3. Similarly, also the asymptotic behavior of ψ_{1}(P, n, n_{0}, t\underline{r}, t\underline{r}) is derived in an identical fashion to that in Lemma 3.3. This proves (4.53) and (4.54) for t_{0,\underline{r}} = t\underline{r}, that is,

ψ_{1}(P, n, n_{0}, t\underline{r}, t\underline{r}) =_{ζ→0} \begin{cases}ζ^{n_{0}−n}(1 + O(ζ)), & P → P_{∞_{+}}, \\ Γ(n, n_{0}, t\underline{r})\frac{α^{+}(n,t\underline{r})}{α^{+}(n_{0},t\underline{r})} + O(ζ), & P → P_{∞_{−}}, \end{cases}  ζ = 1/z, (4.55)

ψ_{1}(P, n, n_{0}, t\underline{r}, t\underline{r}) =_{ζ→0} \begin{cases}\frac{α(n,t\underline{r})}{α(n_{0},t\underline{r})} + O(ζ), & P → P_{0,+}, \\ Γ(n, n_{0}, t\underline{r})ζ^{n−n_{0}}(1 + O(ζ)), & P → P_{0,−}, \end{cases}  ζ = z. (4.56)

Remaining to be investigated is

ψ_{1}(P, n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}}) = exp (i ∫_{t_{0,\underline{r}}}^{t\underline{r}} dt(G̃\underline{r}(z, n_{0}, t)−F̃\underline{r}(z, n_{0}, t)ϕ(P, n_{0}, t))). (4.57)

The asymptotic expansion of the integrand is derived using Theorem B.1. Focusing on the homogeneous coefficients first, one computes as P → P_{∞_{±}},

Ĝ_{s,+} − F̂_{s,+}ϕ = Ĝ_{s,+} − F̂_{s,+} \frac{G\underline{p} + (c_{0,+}/2)z^{−p_{−}}y}{F\underline{p}}

= Ĝ_{s,+} − F̂_{s,+}(\frac{2z^{p_{−}}}{c_{0,+}} \frac{G\underline{p}}{y} + 1)(\frac{2z^{p_{−}}}{c_{0,+}} \frac{F\underline{p}}{y} )^{−1}

=_{ζ→0} ±\frac{1}{2}ζ^{−s} + \frac{ĝ_{0,+} ∓ \frac{1}{2}}{f̂_{0,+}} f̂_{s,+} + O(ζ), P → P_{∞_{±}}, ζ = 1/z.  (4.58)

Since

F̃\underline{r} =_{ζ→0} ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}F̂_{s,+} + O(ζ), G̃\underline{r} =_{ζ→0} ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}Ĝ_{s,+} + O(ζ),  (4.59)

one infers from (4.51)

G̃\underline{r} − F̃\underline{r}ϕ =_{ζ→0} \frac{1}{2} ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}ζ^{−s} + O(ζ), P → P_{∞_{+}}, ζ = 1/z.  (4.60)

Insertion of (4.60) into (4.57) then proves (4.53) as P → P_{∞_{+}}.

As P → P_{∞_{−}}, we need one additional term in the asymptotic expansion of F̃\underline{r}, that is, we will use

F̃\underline{r} =_{ζ→0} ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}F̂_{s,+} + ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}f̂_{s−1,−}ζ + O(ζ^{2}).  (4.61)

This then yields

G̃\underline{r} − F̃\underline{r}ϕ =_{ζ→0} −\frac{1}{2} ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}ζ^{−s} − (α^{+})^{−1}(f̃_{r_{+},+} − f̃_{r_{−}−1,−}) + O(ζ). (4.62)

Invoking (2.7) and (4.2) one concludes that

f̃_{r_{−}−1,−} − f̃_{r_{+},+} = −iα_{t\underline{r}}^{+} + α^{+}(g̃_{r_{+},+} − g̃_{r_{−},−})  (4.63)

and hence

G̃\underline{r} − F̃\underline{r}ϕ =_{ζ→0} −\frac{1}{2} ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}ζ^{−s} − \frac{iα_{t\underline{r}}^{+}}{α^{+}} + g̃_{r_{+},+} − g̃_{r_{−},−} + O(ζ).  (4.64)

Insertion of (4.64) into (4.57) then proves (4.53) as P → P_{∞_{−}}.

Using Theorem B.1 again, one obtains in the same manner as P → P_{0,±},

Ĝ_{s,−} − F̂_{s,−}ϕ =_{ζ→0} ±\frac{1}{2}ζ^{−s} − ĝ_{s,−} + \frac{ĝ_{0,−} ± \frac{1}{2}}{f̂_{0,−}} f̂_{s,−} + O(ζ).  (4.65)

Since

F̃\underline{r} =_{ζ→0} ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}F̂_{s,−} + f̃_{r_{+}−1,+} + O(ζ), P → P_{0,±}, ζ = z,  (4.66)

G̃\underline{r} =_{ζ→0} ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}Ĝ_{s,−} + g̃_{r_{+},+} + O(ζ), P → P_{0,±}, ζ = z,  (4.67)

(4.65)–(4.67) yield

G̃\underline{r}−F̃\underline{r}ϕ =_{ζ→0} ±\frac{1}{2} ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}ζ^{−s}+g̃_{r_{+},+}−g̃_{r_{−},−}− \frac{ĝ_{0,−} ± \frac{1}{2}}{f̂_{0,−}} (f̃_{r_{+}−1,+}−f̃_{r_{−},−})+O(ζ),

(4.68)

where we again used (4.52), (2.19), and (4.2). As P → P_{0,−}, one thus obtains

G̃\underline{r} − F̃\underline{r}ϕ =_{ζ→0} −\frac{1}{2} ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}ζ^{−s} + g̃_{r_{+},+} − g̃_{r_{−},−}, P → P_{0,−}, ζ = z. (4.69)

Insertion of (4.69) into (4.57) then proves (4.54) as P → P_{0,−}.

As P → P_{0,+}, one obtains

G̃\underline{r} − F̃\underline{r}ϕ =_{ζ→0} \frac{1}{2} ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}ζ^{−s} + g̃_{r_{+},+} − g̃_{r_{−},−} − \frac{1}{α}(f̃_{r_{+}−1,+} − f̃_{r_{−},−}) + O(ζ),

=_{ζ→0} \frac{1}{2} ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}ζ^{−s} − \frac{iα_{t\underline{r}}}{α} + O(ζ), P → P_{0,+}, ζ = z,  (4.70)

using f̃_{r_{−},−} = f̃_{r_{−}−1,−}^{−} + α(g̃_{r_{−},−} − g̃_{r_{−},−}^{−}) (cf. (2.11)) and (4.2). Insertion of (4.70) into (4.57) then proves (4.54) as P → P_{0,+}. □

Next, we note that Lemmas 3.2 and 3.4 on trace formulas and nonspecial divisors in the stationary context immediately extend to the present time-dependent situation since t\underline{r} ∈ ℝ just plays the role of a parameter. We thus omit the details.

Finally, we turn to the principal result of this section, the representation of ϕ, Ψ, α, and β in terms of the Riemann theta function associated with K_{p}, assuming p ∈ ℕ for the remainder of this section.

In addition to (3.42)–(3.48), 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},  (4.71)
ω_{P_{0,±},q}^{(2)} =_{ζ→0} (ζ^{−2−q} + O(1))dζ, P → P_{0,±}, ζ = z, q ∈ ℕ_{0},  (4.72)

with vanishing a-periods,

∫_{a_{j}} ω_{P_{∞_{±}} ,q}^{(2)} = ∫_{a_{j}} ω_{P_{0,±},q}^{(2)} = 0, j = 1, …, p.  (4.73)

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)})),  (4.74)

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.  (4.75)

Finally, we abbreviate

Ω̃∞_{±} ; \underline{r} = lim_{P →P_{∞_{±}}} ( ∫_{Q_{0}}^{P} Ω̃(2) ; \underline{r} ± \frac{i}{2} ∑_{s=0}^{r_{+}} c̃_{r_{+}−s,+}ζ^{−s}),  (4.76)

Ω̃0,± ; \underline{r} = lim_{P →P_{0,±}} ( ∫_{Q_{0}}^{P} Ω̃(2) ; \underline{r} ± \frac{i}{2} ∑_{s=0}^{r_{−}} c̃_{r_{−}−s,−}ζ^{−s}).  (4.77)

Theorem 4.6. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold. In addition, let P ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}\} and (n, n_{0}, t\underline{r}, t_{0,\underline{r}}) ∈ ℤ^{2} × ℝ^{2}. Then for each (n, t\underline{r}) ∈ ℤ × ℝ, D_{ˆ\underline{μ}(n,t\underline{r})} and D_{\underline{ν̂}(n,t\underline{r})} are nonspecial. Moreover,

ϕ(P, n, t\underline{r}) = C(n, t\underline{r})\frac{θ(\underline{z}(P, \underline{ν̂}(n, t\underline{r})))}{θ(\underline{z}(P, ˆ \underline{μ}(n, t\underline{r})))} exp ( ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} ),  (4.78)

ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = C(n, n_{0}, t\underline{r}, t_{0,\underline{r}})\frac{θ(\underline{z}(P, ˆ \underline{μ}(n, t\underline{r})))}{θ(\underline{z}(P, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}  (4.79)

× exp ((n − n_{0}) ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{+}}}^{(3)} − (t\underline{r} − t_{0,\underline{r}}) ∫_{Q_{0}}^{P} Ω̃(2) ; \underline{r}),

ψ_{2}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = C(n, t\underline{r})C(n, n_{0}, t\underline{r}, t_{0,\underline{r}})\frac{θ(\underline{z}(P, \underline{ν̂}(n, t\underline{r})))}{θ(\underline{z}(P, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}  (4.80)

× exp ( ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{−}}}^{(3)} + (n − n_{0}) ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{+}}}^{(3)} − (t\underline{r} − t_{0,\underline{r}}) ∫_{Q_{0}}^{P} Ω̃(2) ; \underline{r}),

where

C(n, t\underline{r}) = \frac{(−1)^{n−n_{0}}}{α(n_{0}, t\underline{r})} exp ((n − n_{0})(ω_{0}^{0,−}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}}))) (4.81)

× exp ( − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}}))\frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n_{0}, t\underline{r})))}{θ(\underline{z}(P_{0,+}, \underline{ν̂}(n_{0}, t\underline{r})))},

C(n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = \frac{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n, t\underline{r})))}  (4.82)

× exp ((t\underline{r} − t_{0,\underline{r}})Ω̃∞_{+} ; \underline{r} − (n − n_{0})ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}})).

The Abel map linearizes the auxiliary divisors D_{ˆ\underline{μ}(n,t\underline{r})} and D_{\underline{ν̂}(n,t\underline{r})} in the sense that

\underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n,t\underline{r})}) = \underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(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}, (4.83)
\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}. (4.84)

Finally, α, β are of the form

α(n, t\underline{r}) = α(n_{0}, t_{0,\underline{r}}) exp ((n − n_{0})(ω_{0}^{0,+}(P_{0,−}, P_{∞_{+}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}}))) (4.85)

× exp ((t\underline{r} − t_{0,\underline{r}})(Ω̃∞_{+} ; \underline{r} − Ω̃0,+ ; \underline{r}))\frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n, t\underline{r})))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n, t\underline{r})))},

β(n, t\underline{r}) = \frac{1}{α(n_{0}, t_{0,\underline{r}})} exp ((n − n_{0})(ω_{0}^{0,+}(P_{0,−}, P_{∞_{+}}) − ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}})))

× exp (ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}}))  (4.86)

× exp ( − (t\underline{r} − t_{0,\underline{r}})(Ω̃∞_{+} ; \underline{r} − Ω̃0,+ ; \underline{r}))\frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))θ(\underline{z}(P_{∞_{+}}, \underline{ν̂}(n, t\underline{r})))}{θ(\underline{z}(P_{0,+}, \underline{ν̂}(n_{0}, t_{0,\underline{r}})))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n, t\underline{r})))},

and

\begin{aligned}α(n, t\underline{r})β(n, t\underline{r}) & = exp (ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}}) − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}})) \\ & × \frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n, t\underline{r})))θ(\underline{z}(P_{∞_{+}}, \underline{ν̂}(n, t\underline{r})))}{θ(\underline{z}(P_{0,+}, \underline{ν̂}(n, t\underline{r})))θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n, t\underline{r})))}. \end{aligned}  (4.87)

Proof. As in Theorem 3.7 one concludes that ϕ(P, n, t\underline{r}) is of the form (4.78) and that for t_{0,\underline{r}} = t\underline{r}, ψ_{1}(P, n, n_{0}, t\underline{r}, t\underline{r}) is of the form

ψ_{1}(P, n, n_{0}, t\underline{r}, t\underline{r}) = C(n, n_{0}, t\underline{r}, t\underline{r})\frac{θ(\underline{z}(P, ˆ \underline{μ}(n, t\underline{r})))}{θ(\underline{z}(P, ˆ \underline{μ}(n_{0}, t\underline{r})))} exp ((n−n_{0}) ∫_{Q_{0}}^{P} ω_{P_{0,−},P_{∞_{+}}}^{(3)} ).

(4.88)

To discuss ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) we recall (4.28), that is,

ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = ψ_{1}(P, n, n_{0}, t\underline{r}, t\underline{r})ψ_{1}(P, n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}}),  (4.89)

and hence remaining to be studied is

ψ_{1}(P, n_{0}, 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))).

(4.90)

Introducing ψ̂_{1}(P) on K_{p} \ {P_{∞_{+}}, P_{∞_{−}}} by

ψ̂_{1}(P, n_{0}, t\underline{r}, t_{0,\underline{r}}) = C(n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}})\frac{θ(\underline{z}(P, ˆ \underline{μ}(n_{0}, t\underline{r})))}{θ(\underline{z}(P, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}

× exp ( − (t\underline{r} − t_{0,\underline{r}}) ∫_{Q_{0}}^{P} Ω̃(2) ; \underline{r}),  (4.91)

we intend to prove that

\begin{matrix}ψ_{1}(P, n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}}) = ψ̂_{1}(P, n_{0}, t\underline{r}, t_{0,\underline{r}}), \\ P ∈ K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}\}, n_{0} ∈ ℤ, t\underline{r}, t_{0,\underline{r}} ∈ ℝ, \end{matrix}  (4.92)

for an appropriate choice of the normalization constant C(n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}}) in (4.91). We start by noting that a comparison of (4.53), (4.54) and (4.71), (4.72), (4.74), (4.79) shows that ψ_{1} and ψ̂_{1} have the same essential singularities at P_{∞_{±}} and P_{0,±}. Thus, we turn to the local behavior of ψ_{1} and ψ̂_{1}. By (4.91), ψ̂_{1} has zeros and poles at ˆ \underline{μ}(n_{0}, t\underline{r}) and ˆ \underline{μ}(n_{0}, t_{0,\underline{r}}), respectively. Similarly, by (4.90), ψ_{1} can have zeros and poles only at poles of ϕ(P, n_{0}, s), s ∈ [t_{0,\underline{r}}, t\underline{r}] (resp., s ∈ [t\underline{r}, t_{0,\underline{r}}]). In the following we temporarily restrict t_{0,\underline{r}} and t\underline{r} to a sufficiently small nonempty interval I ⊆ ℝ and pick n_{0} ∈ ℤ such that for all s ∈ I, μ_{j}(n_{0}, s) ≠ μ_{k}(n_{0}, s) for all j ≠ k, j, k = 1, …, p. One computes

iG̃\underline{r}(z, n_{0}, s) − iF̃\underline{r}(z, n_{0}, s)ϕ(P, n_{0}, s)

= iG̃\underline{r}(z, n_{0}, s) − iF̃\underline{r}(z, n_{0}, s)\frac{(c_{0,+}/2)z^{−p_{−}}y + G\underline{p}(z, n_{0}, s)}{F\underline{p}(z, n_{0}, s)}

=_{P →μ̂_{j}(n_{0},s)} \frac{iF̃\underline{r}(μ_{j}(n_{0}, s), n_{0}, s)y(μ̂_{j}(n_{0}, s))}{α^{+}(n_{0}, s)(z − μ_{j}(n_{0}, s)) ∏p ; \begin{matrix}k=1 \\ k≠j \end{matrix}(μ_{j}(n_{0}, s) − μ_{k}(n_{0}, s))} + O(1)

=_{P →μ̂_{j}(n_{0},s)} \frac{∂}{∂s} ln (μ_{j}(n_{0}, s) − z) + O(1).  (4.93)

Restricting P to a sufficiently small neighborhood U_{j}(n_{0}) of {μ̂_{j}(n_{0}, s) ∈ K_{p} | s ∈ [t_{0,\underline{r}}, t\underline{r}] ⊆ I\} such that μ̂_{k}(n_{0}, s) ∉ U_{j}(n_{0}) for all s ∈ [t_{0,\underline{r}}, t\underline{r}] ⊆ I and all k ∈ {1, …, p} \ {j}, (4.91) and (4.93) imply

ψ_{1}(P, n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}})

= \begin{cases}(μ_{j}(n_{0}, t\underline{r}) − z)O(1) & as P → μ̂_{j}(n_{0}, t\underline{r}) ≠ μ̂_{j}(n_{0}, t_{0,\underline{r}}), \\ O(1) & as P → μ̂_{j}(n_{0}, t\underline{r}) = μ̂_{j}(n_{0}, t_{0,\underline{r}}), \\ (μ_{j}(n_{0}, t_{0,\underline{r}}) − z)^{−1}O(1) & as P → μ̂_{j}(n_{0}, t_{0,\underline{r}}) ≠ μ̂_{j}(n_{0}, t\underline{r}), \end{cases}  (4.94)

P = (z, y) ∈ K_{p},

with O(1) ≠ 0. Thus, ψ_{1} and ψ̂_{1} have the same local behavior and identical essential singularities at P_{∞_{±}} and P_{0,±}. By Lemma A.4, ψ_{1} and ψ̂_{1} coincide up to a multiple constant (which may depend on n_{0}, t\underline{r}, t_{0,\underline{r}}). This proves (4.92) for t_{0,\underline{r}}, t\underline{r} ∈ I and for n_{0} as restricted above. By continuity with respect to divisors this extends to all n_{0} ∈ ℤ since by hypothesis D_{ˆ\underline{μ}(n,s)} remain nonspecial for all (n, s) ∈ ℤ × ℝ. Moreover, since by (4.90), for fixed P and n_{0}, ψ_{1}(P, n_{0}, n_{0}, ., t_{0,\underline{r}}) is entire in t\underline{r} (and this argument is symmetric in t\underline{r} and t_{0,\underline{r}}), (4.92) holds for all t\underline{r}, t_{0,\underline{r}} ∈ ℝ (for an appropriate choice of C(n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}})). Together with (4.89), this proves (4.79) for all (n, t\underline{r}), (n_{0}, t_{0,\underline{r}}) ∈ ℤ × ℝ. The expression (4.80) for ψ_{2} then immediately follows from (4.78) and (4.79).

To determine the constant C(n, n_{0}, t\underline{r}, t_{0,\underline{r}}) one compares the asymptotic expansions of ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) for P → P_{∞_{+}} in (4.53) and (4.79)

C(n, n_{0}, t\underline{r}, t_{0,\underline{r}}) = exp ((t\underline{r} − t_{0,\underline{r}})Ω̃∞_{+} ; \underline{r} − (n − n_{0})ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}}))
× \frac{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n, t\underline{r})))}.  (4.95)

Remaining to be computed are the expressions for α and β. Comparing the asymptotic expansions of ψ_{1}(P, n, n_{0}, t\underline{r}, t_{0,\underline{r}}) for P → P_{0,+} in (4.54) and (4.79) shows

\frac{α(n, t\underline{r})}{α(n_{0}, t_{0,\underline{r}})} = C(n, n_{0}, t\underline{r}, t_{0,\underline{r}}) exp ((n − n_{0})(ω_{0}^{0,+}(P_{0,−}, P_{∞_{+}}) − (t\underline{r} − t_{0,\underline{r}})Ω̃0,+ ; \underline{r})
× \frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n, t\underline{r})))}{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}  (4.96)

and inserting C(n, n_{0}, t\underline{r}, t_{0,\underline{r}}) proves (4.85). Equation (4.81) for C(n, t\underline{r}) follows as in the stationary case since t\underline{r} can be viewed as an additional but fixed parameter. By the first line of (3.69),

α(n, t\underline{r}) = \frac{1}{C(n, t\underline{r})} exp ( − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}}))\frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n, t\underline{r})))}{θ(\underline{z}(P_{0,+}, \underline{ν̂}(n, t\underline{r})))}.  (4.97)

Inserting the result (4.85) for α(n, t\underline{r}) into (4.97) then yields (using Lemma 3.6)

C(n, t\underline{r}) = \frac{1}{α(n_{0}, t_{0,\underline{r}})} \frac{θ(\underline{z}(P_{0,+}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))}{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n_{0}, t_{0,\underline{r}})))} exp ((t\underline{r} − t_{0,\underline{r}})(Ω̃0,+ ; \underline{r} − Ω̃∞_{+} ; \underline{r}))

× exp ((n − n_{0})(ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{+}}) − ω_{0}^{0,+}(P_{0,−}, P_{∞_{+}})) − ω_{0}^{0,+}(P_{0,−}, P_{∞_{−}})). (4.98)

Also, since the first line of (3.70) holds,

β(n, t\underline{r}) = C(n, t\underline{r}) exp (ω_{0}^{∞_{+}}(P_{0,−}, P_{∞_{−}}))\frac{θ(\underline{z}(P_{∞_{+}}, \underline{ν̂}(n, t\underline{r})))}{θ(\underline{z}(P_{∞_{+}}, ˆ \underline{μ}(n, t\underline{r})))},  (4.99)

an insertion of (4.98) into (4.99), observing Lemma 3.6, yields equation (4.86) for β(n, t\underline{r}). Finally, multiplying (4.97) and (4.99) proves (4.87).

Single-valuedness of ψ_{1}( · , n_{0}, n_{0}, t\underline{r}, t_{0,\underline{r}}) on K_{p} implies

\underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n_{0},t\underline{r})}) = \underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n_{0},t_{0,\underline{r}})}) + i(t\underline{r} − t_{0,\underline{r}})\underline{Ũ}(2) ; \underline{r}.  (4.100)

Inserting (4.100) into (3.61),

\underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n,t\underline{r})}) = \underline{α}_{Q_{0}}(D_{ˆ\underline{μ}(n_{0},t\underline{r})}) + \underline{A}_{P_{0,−}}(P_{∞_{+}})(n − n_{0}),  (4.101)

one obtains the result (4.83). □

Again we note that the apparent n_{0}-dependence of C(n, t\underline{r}) in the right-hand side of (4.81) actually drops out to ensure the n_{0}-independence of ϕ in (4.78).

The theta function representations (4.85) and (4.86) for α and β, and the one for Γ( · , · , t\underline{r}) analogous to that in (4.87) also show that γ(n, t\underline{r}) ∉ \{0, 1\} for all (n, t\underline{r}) ∈ ℤ × ℝ. Hence, condition (4.1) is satisfied for the time-dependent algebrogeometric AL solutions discussed in this section, provided the associated divisors Dˆ\underline{μ}(n, t\underline{r}) and D\underline{ν̂}(n, t\underline{r}) stay away from P_{∞_{±}}, P_{0,±} for all (n, t\underline{r}) ∈ ℤ × ℝ.

The time-dependent algebro-geometric initial value problem for the Ablowitz– Ladik hierarchy with complex-valued initial data, that is, the construction of α and β by starting from a set of initial data (nonspecial divisors) of full measure, will be presented in [41].

Appendix A. Hyperelliptic Curves and Their Theta Functions

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 [29], [30], and [48], as well as monographs dedicated to integrable systems such as [15, Ch. 2], [38, App. A, B].

Fix p ∈ ℕ. The hyperelliptic curve K_{p} of genus p used in Sections 2 and 3 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)

and the Riemann theta function associated with K_{p} and the given homology basis {a_{j}, b_{j}}_{j=1,...,p},

θ(\underline{z}) = ∑_{\underline{n}∈ℤ^{p}} exp (2πi(\underline{n}, \underline{z}) + πi(\underline{n}, \underline{n}τ)), \underline{z} ∈ ℂ^{p},  (A.17)

where (\underline{u}, \underline{v}) = \overline{\underline{u}} \underline{v}^{⊤} = ∑_{j=1}^{p} \overline{u_{j}} v_{j} denotes the scalar product in ℂ^{p}. It has the fundamental properties

θ(z_{1}, …, z_{j−1}, −z_{j}, z_{j+1}, …, z_{p}) = θ(\underline{z}),  (A.18)
θ(\underline{z} + \underline{m} + \underline{n}τ) = exp ( − 2πi(\underline{n}, \underline{z}) − πi(\underline{n}, \underline{n}τ))θ(\underline{z}), \underline{m}, \underline{n} ∈ ℤ^{p}.  (A.19)

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.20)

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.21)

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.22)
\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.23)

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.24)

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.25)
L^{1}(D) = {ω ∈ M^{1}(K_{p}) | ω = 0 or (ω) ≥ D}, i(D) = dim L^{1}(D),  (A.26)

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 facts.

Theorem A.1. Let D ∈ Div(K_{p}), ω ∈ M^{1}(K_{p}) \ {0}. Then,

i(D) = r(D − (ω)), p ∈ ℕ_{0}.  (A.27)

The Riemann–Roch theorem reads

r(−D) = deg(D) + i(D) − p + 1, p ∈ ℕ_{0}.  (A.28)

By Abel’s theorem, D ∈ Div(K_{p}), p ∈ ℕ is principal if and only if

deg(D) = 0 and \underline{α}_{Q_{0}}(D) = \underline{0}.  (A.29)

Finally, assume p ∈ ℕ. Then \underline{α}_{Q_{0}} : Div(K_{p}) → J(K_{p}) is surjective (Jacobi’s inversion theorem).

Theorem A.2. Let D\underline{Q} ∈ Sym^{p} K_{p}, \underline{Q} = \{Q_{1}, …, Q_{p}\}. Then,

1 ≤ i(D\underline{Q}) = s  (A.30)

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.

Denote by \underline{Ξ}_{Q_{0}} = (Ξ_{Q_{0,1}}, …, Ξ_{Q_{0,p}}) the vector of Riemann constants,

Ξ_{Q_{0,j}} = \frac{1}{2}(1 + τ_{j,j}) − ∑_{\begin{matrix}ℓ=1 \\ ℓ≠j \end{matrix}}^{p} ∫_{a_{ℓ}} ω_{ℓ}(P) ∫_{Q_{0}}^{P} ω_{j}, j = 1, …, p.  (A.31)

Theorem A.3. Let \underline{Q} = \{Q_{1}, …, Q_{p}\} ∈ Sym^{p} K_{p} and assume D\underline{Q} to be nonspecial, that is, i(D\underline{Q}) = 0. Then

θ(\underline{Ξ}_{Q_{0}} − \underline{A}_{Q_{0}}(P) + α_{Q_{0}}(D\underline{Q})) = 0 if and only if P ∈ \{Q_{1}, …, Q_{p}\}.  (A.32)

Lemma A.4. [19, Lemmas 5.4 and 6.1] Let (n, t\underline{r}), (n_{0}, t_{0,\underline{r}}) ∈ Ω for some Ω ⊆ ℤ × ℝ. Assume ψ( · , n, t\underline{r}) to be meromorphic on K_{p} \ \{P_{∞_{+}}, P_{∞_{−}}, P_{0,+}, P_{0,−}\} with possible essential singularities at P_{∞_{±}}, P_{0,±} such that ψ̃( · , n, t\underline{r}) defined by

ψ̃(P, n, t\underline{r}) = ψ(P, n, t\underline{r}) exp ((t\underline{r} − t_{0,\underline{r}}) ∫_{Q_{0}}^{P} Ω̃(2) ; \underline{r})  (A.33)

is meromorphic on K_{p} and its divisor satisfies

(ψ̃( · , n, t\underline{r})) ≥ −D_{ˆ\underline{μ}(n_{0},t_{0,\underline{r}})} + (n − n_{0})(D_{P_{0,−}} − D_{P_{∞_{+}}} )  (A.34)

for some positive divisor D_{ˆ\underline{μ}(n_{0},t_{0,\underline{r}})} of degree p. Here Ω̃(2) ; \underline{r} is defined in (4.74) and the path of integration is chosen identical to that in the Abel maps^{3} (A.20) and (A.21). Define a divisor D_{0}(n, t\underline{r}) by

(ψ̃( · , n, t\underline{r})) = D_{0}(n, t\underline{r}) − D_{ˆ\underline{μ}(n_{0},t_{0,\underline{r}})} + (n − n_{0})(D_{P_{0,−}} − D_{P_{∞_{+}}} ).  (A.35)

3This is to avoid multi-valued expressions and hence the use of the multiplicative Riemann– Roch theorem in the proof of Lemma A.4.

Then

D_{0}(n, t\underline{r}) ∈ Sym^{p} K_{p}, D_{0}(n, t\underline{r}) > 0, deg(D_{0}(n, t\underline{r})) = p.  (A.36)

Moreover, if D_{0}(n, t\underline{r}) is nonspecial for all (n, t\underline{r}) ∈ Ω, that is, if

i(D_{0}(n, t\underline{r})) = 0, (n, t\underline{r}) ∈ Ω,  (A.37)

then ψ( · , n, t\underline{r}) is unique up to a constant multiple (which may depend on the parameters (n, t\underline{r}), (n_{0}, t_{0,\underline{r}}) ∈ Ω).

Appendix B. 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. (B.3)). For detailed proofs of the material in this section we refer to [39], [40]. We will employ the notation

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

We start with the following elementary results (consequences 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},  (B.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. (B.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. (B.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 B.1 ([40]). Assume (3.1), 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}  (B.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}  (B.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}  (B.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_{±}.  (B.8)

Acknowledgments. We thank P. D. Miller for helpful discussions in connection with the Ablowitz–Ladik hierarchy. 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 summers of 2004–2006, where parts of this paper were written.

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

E-mail address: fritz@math.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@esi.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/