ON THE SPATIAL ASYMPTOTICS OF SOLUTIONS OF THE ABLOWITZ–LADIK HIERARCHY

JOHANNA MICHOR

Dedicated with great pleasure to Peter W. Michor on the occasion of his 60th birthday.

Abstract. We show that for decaying solutions of the Ablowitz–Ladik system, the leading asymptotic term is time independent. In addition, two arbitrary bounded solutions of the Ablowitz–Ladik system which are asymptotically close at the initial time stay close. All results are also derived for the associated hierarchy.

1. Introduction

When solving completely integrable wave equations via the inverse scattering transform, a method developed by Gardner et al. [12] in 1967 for the Korteweg– de Vries (KdV) equation, one intends to prove existence of solutions within the respective class. In particular, short-range perturbations of the background solution should remain short-range during the time evolution. So to what extend are spatial asymptotical properties time independent?

For the KdV equation, this question was answered by Bondareva and Shubin [9], [10], who considered the Cauchy problem for initial conditions which have a prescribed asymptotic expansion in terms of powers of the spatial variable and showed that the leading term of the expansion is time independent. Teschl [18] considered the initial value problem for the Toda lattice in the class of decaying solutions and obtained time independence of the leading term.

In this note we want to address the same question for the Ablowitz–Ladik (AL) system, an integrable discretization of the AKNS-ZS system derived by Ablowitz and Ladik ([3]–[6]) in the mid seventies. The AL system is given by

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

where α = α(n, t), β = β(n, t), (n, t) ∈ ℤ × ℝ, are complex valued sequences and f^{±}(n, t) = f(n ± 1, t). In the defocusing (β = \overline{α}) and focusing case (β = −\overline{α}), (1.1) is a discrete analog of the nonlinear Schrödinger (NLS) equation

iq_{t} + q_{xx} ± 2q|q|^{2} = 0.

We refer to the monographs [2], [7], or [13] for further information.

2000 Mathematics Subject Classification. Primary 37K40, 37K15; Secondary 35Q55, 37K10.

Key words and phrases. Spatial asymptotics, Ablowitz–Ladik hierarchy.

Research supported by the Austrian Science Fund (FWF) under Grant No. V120.

Proc. Amer. Math. Soc. 138, 4249–4258 (2010).

Our main result in Theorem 2.4 yields that the dominant term of suitably decaying solutions α(n, t), β(n, t) of (1.1), for instance weighted ℓ^{2p} sequences whose spatial difference is in ℓ^{p}, 1 ≤ p < ∞, is time independent. For example,

α(n, t) = \frac{a}{n^{δ}} + O(\frac{1}{n^{min(2δ,δ+1)}} ), β(n, t) = \frac{b}{n^{δ}} + O(\frac{1}{n^{min(2δ,δ+1)}} ), n → ∞,

(1.2)

holds for fixed t, provided it holds at the initial time t = t_{0}. Here a, b ∈ ℂ and δ ≥ 0. A similar expression is valid for n → −∞.

The inverse scattering transform for the AL system with vanishing boundary conditions was studied in [4]. Ablowitz, Biondini, and Prinari [1] (compare also Vekslerchik and Konotop [19]) considered nonvanishing steplike boundary conditions α(n) → α_{0}e^{iθ_{±}} as |n| → ∞, α_{0} > 0, in the class

∑_{j=n}^{±∞}(α(j) − α_{0}e^{iθ_{±}}) < ∞  (1.3)

for the defocusing discrete NLS equation. Quasi-periodic boundary conditions for the AL hierarchy will be considered in Michor [17]. As mentioned, a crucial step is to show that short-range perturbations like (1.3) of solutions stay short-range. Here we show in general that arbitrary bounded solutions of the AL system which are asymptotically close at the initial time stay close.

In Section 2 we derive our results for the AL system and extend them in Section 3 to the AL hierarchy, a completely integrable hierarchy of nonlinear evolution equations whose first nonlinear member is (1.1).

2. The initial value problem for the Ablowitz–Ladik system

Let us begin by recalling some basic facts on the system (1.1). We will only consider bounded solutions and hence require

Hypothesis H.2.1. Suppose that α, β: ℤ × ℝ → ℂ satisfy

\begin{matrix}sup_{(n,t)∈ℤ×ℝ} (|α(n, t)| + |β(n, t)|) < ∞, \\ α(n, ·), β(n, ·) ∈ C^{1}(ℝ), n ∈ ℤ, α(n, t)β(n, t) ∉ \{0, 1\}, (n, t) ∈ ℤ × ℝ. \end{matrix} (2.1)

The AL system (1.1) is equivalent to the zero-curvature equation

U_{t} + UV − V^{+}U = 0,  (2.2)

where

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})  (2.3)

for the spectral parameter z ∈ ℂ \ {0}. The AL system can also be formulated in terms of Lax pairs, see [15]. Then (1.1) is equivalent to the Lax equation

\frac{d}{dt}L(t) − [P (t), L(t)] = 0,  t ∈ ℝ,  (2.4)

where L reads in the standard basis of ℓ^{2}(ℤ) (abbreviate ρ = (1 − αβ)^{1/2})

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} ) (2.5)

and P is given by

P = \frac{i}{2}(L_{+} − L_{−} + (L^{−1})_{−} − (L^{−1})_{+} + 2Q_{d}).

Here Q_{d} is the doubly infinite diagonal matrix Q_{d} = ((−1)^{k}δ_{k,ℓ})_{k,ℓ∈ℤ} and L_{±} denote the upper and lower triangular parts of L,

L_{±} = (L_{±}(m, n))_{(m,n)∈ℤ^{2}}, L_{±}(m, n) = \begin{cases}L(m, n), & ±(n − m) > 0, \\ 0, & otherwise. \end{cases}  (2.6)

The Lax equation (2.4) implies existence of a propagator W (s, t) such that the family of operators L(t), t ∈ ℝ, is similar,

L(s) = W (s, t)L(t)W (s, t)^{−1}, s, t ∈ ℝ.

By [13, Sec. 3.8] or [15], existence, uniqueness, and smoothness of local solutions of the AL initial value problem follow from [8, Thm 4.1.5], since the AL flows are autonomous.

Theorem 2.2. Let t_{0} ∈ ℝ and suppose (α_{0}, β_{0}) ∈ M = ℓ^{p}(ℤ) ⊕ ℓ^{p}(ℤ) for some p ∈ [1, ∞) ∪ {∞}. Then there exists T > 0 and a unique local integral curve t ↦ (α(t), β(t)) in C^{∞}((t_{0} − T, t_{0} + T), M) of the Ablowitz–Ladik system (1.1) such that (α, β)|_{t=t_{0}} = (α_{0}, β_{0}).

Our first lemma shows that the leading asymptotics as n → ±∞ are preserved by the AL flow. We only state the result for the AL system, whose proof follows as the one of Lemma 3.2. Define

∥(α, β)∥_{w,p} = \begin{cases}( ∑_{n∈ℤ} w(n)(|α(n)|^{p} + |β(n)|^{p}))^{1/p}, & 1 ≤ p < ∞, \\ sup_{n∈ℤ} w(n)(|α(n)| + |β(n)|), & p = ∞. \end{cases}  (2.7)

Lemma 2.3. Let w(n) ≥ 1 be some weight with sup_{n}(|\frac{w(n+1)}{w(n)}| + |\frac{w(n)}{w(n+1)}|) < ∞. Fix 1 ≤ p ≤ ∞ and suppose (α(n, t), β(n, t)) and (α̃(n, t), β̃(n, t)) are arbitrary bounded solutions of the AL system (1.1). If

∥(α(t) − α̃(t), β(t) − β̃(t))∥_{w,p} < ∞  (2.8)

holds for one t = t_{0} ∈ ℝ, then it holds for all t ∈ (t_{0} − T, t_{0} + T).

But even the leading term is preserved by the time evolution.

Theorem 2.4. Let w(n) ≥ 1 be some weight with sup_{n}(|\frac{w(n+1)}{w(n)}| + |\frac{w(n)}{w(n+1)}|) < ∞. Fix 1 ≤ p ≤ ∞ and suppose α_{0}, β_{0} and α̃_{0}, β̃_{0} are bounded sequences such that

\begin{matrix}∥(α_{0}, β_{0})∥_{w,2p} < ∞, & ∥(α_{0} − α_{0}^{+}, β_{0} − β_{0}^{+})∥_{w,p} < ∞, \\ ∥(α̃_{0}, β̃_{0})∥_{w,p} < ∞, \end{matrix}  if 1 ≤ p < ∞,

\begin{matrix}∥(α_{0}, β_{0})∥_{w,∞} < ∞, & ∥(α_{0} − α_{0}^{+}, β_{0} − β_{0}^{+})∥_{w^{2},∞} < ∞, \\ ∥(α̃_{0}, β̃_{0})∥_{w^{2},∞} < ∞, \end{matrix}  if p = ∞.

Let (α(t), β(t)), t ∈ (−T, T), be the unique solution of the Ablowitz–Ladik system (1.1) corresponding to the initial conditions

α(0) = α_{0} + α̃_{0},  β(0) = β_{0} + β̃_{0}.  (2.9)

Then this solution is of the form

α(t) = α_{0} + α̃(t),  β(t) = β_{0} + β̃(t),  (2.10)

where ∥(α̃(t), β̃(t))∥_{w,p} < ∞, respectively, ∥(α̃(t), β̃(t))∥_{w^{2},∞} < ∞.

Proof. The proof relies on the idea to consider our differential equation in two nested spaces of sequences, the Banach space of all (α(n), β(n)) with sup norm, and the Banach space with norm ∥.∥_{w,p}, as follows. Plugging (α_{0} + α̃(t), β_{0} + β̃(t)) into the AL equations (1.1) yields a differential equation for (α̃(t), β̃(t))

iα̃_{t}(t) = −(1 − (α_{0} + α̃(t))(β_{0} + β̃(t)))(α̃^{+}(t) + α_{0}^{+} + α̃^{−}(t) + α_{0}^{−}) + 2α̃(t) + 2α_{0}
= α_{0} − α_{0}^{−} + α_{0} − α_{0}^{+} + α_{0}β_{0}(α_{0}^{+} + α_{0}^{−})
+ α̃(t)(2 + (β_{0} + β̃(t))(α̃^{+}(t) + α_{0}^{+} + α̃^{−}(t) + α_{0}^{−}))
+ β̃(t)α_{0}(α̃^{+}(t) + α_{0}^{+} + α̃^{−}(t) + α_{0}^{−})
+ α̃^{+}(t)(α_{0}β_{0} − 1) + α̃^{−}(t)(α_{0}β_{0} − 1),
iβ̃_{t}(t) = (1 − (α_{0} + α̃(t))(β_{0} + β̃(t)))(β̃^{+}(t) + β_{0}^{+} + β̃^{−}(t) + β_{0}^{−}) − 2β̃(t) − 2β_{0}
= β_{0}^{+} − β_{0} + β_{0}^{−} − β_{0} − α_{0}β_{0}(β_{0}^{+} + β_{0}^{−})
− β̃(t)(2 + (α_{0} + α̃(t))(β̃^{+}(t) + β_{0}^{+} + β̃^{−}(t) + β_{0}^{−}))
− α̃(t)β_{0}(β̃^{+}(t) + β_{0}^{+} + β̃^{−}(t) + β_{0}^{−}))
− β̃^{+}(t)(α_{0}β_{0} − 1) − β̃^{−}(t)(α_{0}β_{0} − 1).  (2.11)

The requirement on w(n) implies that the shift operators are continuous with respect to the norm ∥.∥_{w,p} and the same is true for the multiplication operator with a bounded sequence. Therefore, using the generalized Hölder inequality yields that (2.11) is a system of inhomogeneous linear differential equations in the Banach space with norm ∥.∥_{w,p} and has a local solution with respect to this norm (see e.g. [11] for the theory of ordinary differential equations in Banach spaces). Since w(n) ≥ 1, this solution is bounded and the corresponding coefficients (α̃, β̃) coincide with the solution (α, β) of the AL system (1.1) from Theorem 2.2.

Moreover, (α̃(t), β̃(t)) are uniformly bounded for t ∈ (−T, T), as writing (2.11) in integral form yields

∥(α̃(t), β̃(t))∥_{w,p} ≤ ∥(α̃(0), β̃(0))∥_{w,p} + tC∥(α_{0}, β_{0})∥_{w,2p} + C ∫_{0}^{t} ∥(α̃(s), β̃(s))∥_{w,p}ds

for some constants C. □

Example (1.2) in the introduction follows if we let α̃_{0} = β̃_{0} ≡ 0 and

α_{0}(n) = \frac{a}{n^{δ}} , β_{0}(n) = \frac{b}{n^{δ}} , a, b ∈ ℂ, δ ≥ 0,

for n > 0, α_{0}(n) = β_{0}(n) = 0 for n ≤ 0. Now choose p = ∞ with

w(n) = \begin{cases}(1 + n)^{min(δ,(δ+1)/2)}, & n > 0, \\ 1, & n ≤ 0, \end{cases}

and apply Theorem 2.4.

Finally, we remark that if a solution (α(n, t), β(n, t)) vanishes at two consecutive points n = n_{0}, n = n_{0} + 1 in an arbitrarily small time intervall t ∈ (t_{1}, t_{2}), then it vanishes identically for all (n, t) in ℤ×ℝ, see [16]. In particular, a compact support of the solution is not preserved. The corresponding result for the AL hierarchy is derived in [16] as well.

3. Extension to the Ablowitz–Ladik hierarchy

In this section we show how our results extend to the AL hierarchy. The hierarchy can be constructed by generalizing the matrix V (z) in the zero-curvature equation (2.2) to a 2 × 2 matrix V\underline{r}(z), \underline{r} = (r_{−}, r_{+}) ∈ ℕ_{0}^{2}, with Laurent polynomial entries, see [13, Sec. 3.2] or [14]. Suppose that U(z) and V\underline{r}(z) satisfy the zero-curvature equation

U_{t} + UV\underline{r} − V + ; \underline{r}U = 0.  (3.1)

Then the coefficients {f_{ℓ,±}}_{ℓ=0,...,r_{±}−1}, {g_{ℓ,±}}_{ℓ=0,...,r_{±}}, and {h_{ℓ,±}}_{ℓ=0,...,r_{±}−1} of the Laurent polynomials in the entries of V\underline{r}(z) are recursively defined by

\begin{aligned}g_{0,+} & = \frac{1}{2}c_{0,+}, f_{0,+} = −c_{0,+}α^{+}, h_{0,+} = c_{0,+}β, \\ g_{ℓ+1,+} − g_{ℓ+1,+}^{−} & = αh_{ℓ,+}^{−} + βf_{ℓ,+}, 0 ≤ ℓ ≤ r_{+} − 1, \\ f_{ℓ+1,+}^{−} & = f_{ℓ,+} − α(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}), 0 ≤ ℓ ≤ r_{+} − 2, \\ h_{ℓ+1,+} & = h_{ℓ,+}^{−} + β(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}), 0 ≤ ℓ ≤ r_{+} − 2, \end{aligned}  (3.2)

and

\begin{aligned}g_{0,−} & = \frac{1}{2}c_{0,−}, f_{0,−} = c_{0,−}α, h_{0,−} = −c_{0,−}β^{+}, \\ g_{ℓ+1,−} − g_{ℓ+1,−}^{−} & = αh_{ℓ,−} + βf_{ℓ,−}^{−}, 0 ≤ ℓ ≤ r_{−} − 1, \\ f_{ℓ+1,−} & = f_{ℓ,−}^{−} + α(g_{ℓ+1,−} + g_{ℓ+1,−}^{−}), 0 ≤ ℓ ≤ r_{−} − 2, \\ h_{ℓ+1,−}^{−} & = h_{ℓ,−} − β(g_{ℓ+1,−} + g_{ℓ+1,−}^{−}), 0 ≤ ℓ ≤ r_{−} − 2. \end{aligned}  (3.3)

Note that g_{ℓ,±} are only defined up to summation constants {c_{ℓ,±}}_{ℓ=0,...,r_{±}} by the difference equations in (3.2), (3.3). In addition, the zero-curvature equation (3.1) is equivalent to

0 = i (\begin{matrix}0 & \begin{matrix}−iα_{t} − α(g_{r_{+},+} + g_{r_{−},−}^{−}) \\ +f_{r_{+}−1,+} − f_{r_{−}−1,−}^{−} \end{matrix} \\ \begin{matrix}z( − iβ_{t} + β(g_{r_{+},+}^{−} + g_{r_{−},−}) \\ −h_{r_{−}−1,−} + h_{r_{+}−1,+}^{−}) \end{matrix} & 0 \end{matrix}) .

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

\begin{matrix}AL\underline{r}(α, β) = (\begin{matrix}−iα_{t} − α(g_{r_{+},+} + g_{r_{−},−}^{−}) + f_{r_{+}−1,+} − f_{r_{−}−1,−}^{−} \\ −iβ_{t} + β(g_{r_{+},+}^{−} + g_{r_{−},−}) − h_{r_{−}−1,−} + h_{r_{+}−1,+}^{−} \end{matrix}) = 0, \\ t ∈ ℝ, \underline{r} = (r_{−}, r_{+}) ∈ ℕ_{0}^{2}, \end{matrix} (3.4)

then defines the time-dependent Ablowitz–Ladik hierarchy. Explicitly, taking r_{−} = r_{+} for simplicity, the first few equations are

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

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

AL_{(2,2)}(α, β) = (\begin{matrix}−iα_{t} − 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} + 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.,  (3.5)

where we abbreviated c\underline{r} = (c_{r,−} + c_{r,+})/2 and γ = 1 − αβ. Different ratios of c_{0,+}/c_{0,−} lead to different hierarchies. The AL system (1.1) corresponds to the case \underline{r} = (1, 1), c_{0,±} = 1, and c_{(1,1)} = −2. The special choices β = ±\overline{α}, c_{0,±} = 1 lead to the discrete NLS hierarchy, the choices β = \overline{α}, c_{0,±} = ∓i yield the hierarchy of Schur flows. The AL hierarchy is invariant under the scaling transform

{(α(n), β(n))}_{n∈ℤ} → {(c α(n), β(n)/c)}_{n∈ℤ}, c ∈ ℂ\{0}.  (3.6)

Hence choosing c = e^{ic\underline{r}t} it is no restriction to assume c\underline{r} = 0.

By [15], the AL hierarchy is equivalent to the Lax equation

\frac{d}{dt}L(t) − [P\underline{r}(t), L(t)] = 0, t ∈ ℝ, \underline{r} ∈ ℕ_{0}^{2},  (3.7)

where L is the doubly infinite five-diagonal matrix (2.5) and (recall (2.6))

P\underline{r} = \frac{i}{2} ∑_{ℓ=1}^{r_{+}} c_{r_{+}−ℓ,+}((L^{ℓ})_{+} − (L^{ℓ})_{−}) − \frac{i}{2} ∑_{ℓ=1}^{r_{−}} c_{r_{−}−ℓ,−}((L^{−ℓ})_{+} − (L^{−ℓ})_{−}) − \frac{i}{2}c\underline{r}Q_{d}.

Since the AL flows are autonomous and f_{r_{±}−1,±}, g_{r_{±},±}, and h_{r_{±}−1,±} depend polynomially on α, β and their shifts, [8, Thm 4.1.5] implies local existence, uniqueness, and smoothness of the solution of the initial value problem of the hierarchy as well (see [13, Sec. 3.8], [15]).

Theorem 3.1. Let t_{0} ∈ ℝ and suppose α_{0}, β_{0} ∈ ℓ^{p}(ℤ) for some p ∈ [1, ∞) ∪ {∞}. Then the \underline{r}th Ablowitz–Ladik initial value problem

AL\underline{r}(α, β) = 0, (α, β)|_{t=t_{0}} = (α_{0}, β_{0})  (3.8)

for some \underline{r} ∈ ℕ_{0}^{2}, has a unique, local, and smooth solution in time, that is, there exists a T > 0 such that α( ·), β( ·) ∈ C^{∞}((t_{0} − T, t_{0} + T), ℓ^{p}(ℤ)).

Next we show that short-range perturbations of bounded solutions remain shortrange. In fact, we will be more general to include perturbations of steplike background solutions as for example (1.3).

Lemma 3.2. Let w(n) ≥ 1 be some weight with sup_{n}(|\frac{w(n+1)}{w(n)}| + |\frac{w(n)}{w(n+1)}|) < ∞ and fix 1 ≤ p ≤ ∞. Suppose (α(t), β(t)) and (α_{ℓ,r}(t), β_{ℓ,r}(t)) are arbitrary bounded solutions of some equation AL\underline{r} in the AL hierarchy and abbreviate

α̃(n, t) = \begin{cases}α_{r}(n, t), & n ≥ 0, \\ α_{ℓ}(n, t), & n < 0, \end{cases}  β̃(n, t) = \begin{cases}β_{r}(n, t), & n ≥ 0, \\ β_{ℓ}(n, t), & n < 0. \end{cases}  (3.9)

If

∥(α(t) − α̃(t), β(t) − β̃(t))∥_{w,p} < ∞  (3.10)

holds for one t = t_{0} ∈ ℝ, then it holds for all t ∈ (t_{0} − T, t_{0} + T).

Proof. Without loss we assume that t_{0} = 0. First we derive the differential equation for the differences δ(n, t) = (α(n, t) − α̃(n, t), β(n, t) − β̃(n, t)) in the Banach space of pairs of bounded sequences δ = (δ_{1}, δ_{2}) for which the norm ∥δ∥_{w,p} is finite.

Let us show by induction on r_{±} that f_{r_{±}−1,±}(t)−f̃_{r_{±}−1,±}(t), g_{r_{±},±}(t)−g̃_{r_{±},±}(t), and h_{r_{±}−1,±}(t) − h̃_{r_{±}−1,±}(t) can be written as a linear combination of shifts of δ with the coefficients depending only on (α(t), β(t)) and (α_{ℓ,r}(t), β_{ℓ,r}(t)). It suffices to consider the homogeneous case where c_{j,±} = 0, 1 ≤ j ≤ r_{±}, since all involved sums are finite. In this case [14, Lemma A.3] yields that f_{j,+}, g_{j,+}, and h_{j,+} can be recursively computed from f_{0,+} = −α^{+}, g_{0,+} = \frac{1}{2}, and h_{0,+} = β via

f_{ℓ+1,+}^{−} = f_{ℓ,+} − α(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}),
h_{ℓ+1,+} = h_{ℓ,+}^{−} + β(g_{ℓ+1,+} + g_{ℓ+1,+}^{−}),
g_{ℓ+1,+} = ∑_{k=0}^{ℓ} f_{ℓ−k,+}h_{k,+} − ∑_{k=1}^{ℓ} g_{ℓ+1−k,+}g_{k,+},

and similarly for the minus sign and f̃_{j,±}, g̃_{j,±}, and h̃_{j,±}. The fact that (α̃, β̃) does not solve AL\underline{r} only affects finitely many terms and gives rise to an inhomogeneous term B\underline{r}(t) which is nonzero only for a finite number of terms.

Hence δ satisfies an inhomogeneous linear differential equation of the form

i\frac{d}{dt}δ(t) = ∑_{|j|≤max(r_{−},r_{+})} A_{\underline{r},j}(t)(S^{+})^{j}δ(t) + B\underline{r}(t)

Here S^{±}(δ_{1}(n, t), δ_{2}(n, t)) = (δ_{1}(n ± 1, t), δ_{2}(n ± 1, t)) are the shift operators,

A_{\underline{r},j}(n, t) = (\begin{matrix}A_{\underline{r},j}^{11}(n, t) & A_{\underline{r},j}^{12}(n, t) \\ A_{\underline{r},j}^{21}(n, t) & A_{\underline{r},j}^{22}(n, t) \end{matrix}) ,

are multiplication operators with bounded 2 × 2 matrix-valued sequences, and B\underline{r}(n, t) = (B_{\underline{r},1}(n, t), B_{\underline{r},2}(n, t)) with B_{r,i}(n, t) = 0 for |n| > max(r_{−}, r_{+}). All entries of A_{\underline{r},j}(t) and B\underline{r}(t) are polynomials with respect to (α(n + j, t), β(n + j, t)), (α_{ℓ,r}(n + j, t), β_{ℓ,r}(n + j, t)), |j| ≤ max(r_{−}, r_{+}). Thus ∥B\underline{r}(t)∥_{w,p} ≤ D\underline{r}, where the constant depends only on the sup norms of (α(t), β(t)) and (α_{ℓ,r}(t), β_{ℓ,r}(t)). Moreover, by our assumption the shift operators are continuous,

∥S^{±}∥ = \begin{cases}sup_{n∈ℤ} |\frac{w(n)}{w(n±1)}|^{1/p}, & p ∈ [1, ∞), \\ sup_{n∈ℤ} |\frac{w(n)}{w(n±1)}|, & p = ∞, \end{cases}

and the same is true for the multiplication operators A_{\underline{r},j}(t) whose norms depend only on the supremum of the entries by Hölder’s inequality, that is, again on the sup norms of (α(t), β(t)) and (α_{ℓ,r}(t), β_{ℓ,r}(t)). Consequently, for t ∈ (−T, T) there is a constant such that ∑_{|j|≤max(r_{−},r_{+})} ∥A_{\underline{r},j}(t)∥∥(S^{+})^{j}∥ ≤ C\underline{r}. Hence

∥δ(t)∥_{w,p} ≤ ∥δ(0)∥_{w,p} + ∫_{0}^{t} (C\underline{r}∥δ(s)∥_{w,p} + D\underline{r})

and Gronwall’s inequality implies

∥δ(t)∥_{w,p} ≤ ∥δ(0)∥_{w,p}e^{C\underline{r}t} + \frac{D\underline{r}}{C\underline{r}} (e^{C\underline{r}t} − 1) .

Since w(n) ≥ 1, this solution is again bounded and hence coincides with the solution of the AL equation from Theorem 3.1. □

For certain equations in the AL hierarchy, i.e. for certain configurations of summation coefficients {c_{j,±}}, our main result remains valid.

Theorem 3.3. Let \underline{r} = (r_{−}, r_{+}) ∈ ℕ_{0}^{2}\(0, 0) and assume that c_{j,±} ∈ ℂ, j = 0, …, r_{±}, satisfy

∑_{j=0}^{r_{+}−1} c_{j,+} + ∑_{j=0}^{r_{−}−1} c_{j,−} = 0.  (3.11)

Let w(n) ≥ 1 be some weight with sup_{n}(|\frac{w(n+1)}{w(n)}| + |\frac{w(n)}{w(n+1)}|) < ∞. Fix 1 ≤ p ≤ ∞ and suppose α_{0}, β_{0} and α̃_{0}, β̃_{0} are bounded sequences such that

\begin{matrix}∥(α_{0}, β_{0})∥_{w,2p} < ∞, & ∥(α_{0} − α_{0}^{+}, β_{0} − β_{0}^{+})∥_{w,p} < ∞, \\ ∥(α̃_{0}, β̃_{0})∥_{w,p} < ∞, \end{matrix}  if 1 ≤ p < ∞,

\begin{matrix}∥(α_{0}, β_{0})∥_{w,∞} < ∞, & ∥(α_{0} − α_{0}^{+}, β_{0} − β_{0}^{+})∥_{w^{2},∞} < ∞, \\ ∥(α̃_{0}, β̃_{0})∥_{w^{2},∞} < ∞, \end{matrix}  if p = ∞.

Let (α(t), β(t)), t ∈ (−T, T), be the unique solution of the equation AL\underline{r}(α, β) = 0 with summation coefficients {c_{j,±}}_{j=0}^{r_{±}}, corresponding to the initial conditions

α(0) = α_{0} + α̃_{0},  β(0) = β_{0} + β̃_{0}.  (3.12)

Then this solution is of the form

α(t) = α_{0} + α̃(t),  β(t) = β_{0} + β̃(t),  (3.13)

where ∥(α̃(t), β̃(t))∥_{w,p} < ∞, respectively, ∥(α̃(t), β̃(t))∥_{w^{2},∞} < ∞.

Proof. The proof is similar to the one of Theorem 2.4. From AL\underline{r}(α, β) = 0 we obtain an inhomogeneous differential equation for (α̃, β̃). The homogeneous part is a finite sum over shifts of (α̃, β̃). The inhomogeneous part consists of products of α_{0}, β_{0} and their shifts, whose ∥.∥_{w,p} norm is finite by Hölder’s inequality, and of sums of the form c_{j,±}α_{0}, c_{j,±}β_{0} and shifts thereof,

−(c_{0,+}S^{+r_{+}} + c_{1,+}S^{+r_{+}−1} + ⋯ + c_{r_{+}−1,+}S^{+1} + c\underline{r}
+ c_{0,−}S^{−r_{−}} + c_{1,−}S^{−r_{−}−1} + ⋯ + c_{r_{−}−1,−}S^{−1})α_{0},

(and analogously for β_{0}) from which restriction (3.11) arises. Again S^{±j} denote the shift operators S^{±j}α_{0}(n) = α_{0}(n±j). The requirement ∥(α_{0}−α_{0}^{+}, β_{0}−β_{0}^{+})∥_{w,p} < ∞ yields the algebraic constraint (3.11) for c_{j,±}. Finally, note that it is no restriction to assume c\underline{r} = 0 by (3.6). □

For example, we obtain such decaying solutions for AL_{(0,1)}(α, β) if c_{0,+} = 0, for AL_{(1,1)}(α, β) if c_{0,+} = −c_{0,−} (or c_{0,+} + c_{0,−} + c\underline{1} = 0 as in Theorem 2.4).

Acknowledgment. The author thanks Armin Rainer and Gerald Teschl for valuable discussions on this topic.

References

[1] M. J. Ablowitz, G. Biondini, and B. Prinari, Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions, Inverse Problems 23, 1711–1758 (2007).

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

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

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

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

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

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

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

[9] I. N. Bondareva, The Korteweg-de Vries equation in classes of increasing functions with prescribed asymptotic behaviour as |x| → ∞, Math. USSR Sb. 50-1, 125–135 (1983).

[10] I. N. Bondareva and M. Shubin, Increasing asymptotic solutions of the Korteweg-de Vries equation and its higher analogues, Sov. Math. Dokl. 26-3, 716–719 (1982).

[11] K. Deimling, Ordinary Differential Equations on Banach Spaces, Lecture Notes in Mathematics 596, Springer, Berlin, 1977.

[12] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, A method for solving the Korteweg-de Vries equation, Phys. Rev. Letters 19, 1095–1097 (1967).

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

[14] F. Gesztesy, H. Holden, J. Michor, and G. Teschl, The Ablowitz–Ladik hierarchy revisited, in Methods of Spectral Analysis in Math. Physics, J. Janas et al. (ed.), 139–190, Oper. Theory Adv. Appl. 186, Birkhäuser, Basel, 2008.

[15] F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik hierarchy, Stud. Appl. Math. 120-4, 361–423 (2008).

[16] H. Krüger and G. Teschl, Unique continuation for discrete nonlinear wave equations, arXiv:0904.0011.

[17] J. Michor, Inverse scattering transform for the Ablowitz–Ladik Hierarchy with quasi-periodic background, in preparation.

[18] G. Teschl, On the spatial asymptotics of solutions of the Toda lattice, Discrete Contin. Dyn. Sys. 27-3, 1233–1239 (2010).

[19] V. E. Vekslerchik and V. V. Konotop, Discrete nonlinear Schrödinger equation under nonvanishing boundary conditions, Inverse Problems 8, 889–909 (1992).

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

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