On the spatial asymptotics of solutions of the Ablowitz-Ladik hierarchy

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.


Introduction
When solving completely integrable wave equations via the inverse scattering transform, a method developed by Gardner et al. [12] in 1967 for the Kortewegde 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.
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, (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 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).

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 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. (2.1) The AL system (1.1) is equivalent to the zero-curvature equation 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 where L reads in the standard basis of ℓ 2 (Z) (abbreviate ρ = ( and P is given by 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, 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, 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.
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 (2.7) and suppose (α(n, t), β(n, t)) and (α(n, t),β(n, t)) are arbitrary bounded solutions of the AL system (1.1). If 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.
Let (α(t), β(t)), t ∈ (−T, T ), be the unique solution of the Ablowitz-Ladik system (1.1) corresponding to the initial conditions Then this solution is of the form 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 +α(t), β 0 +β(t)) into the AL equations (1.1) yields a differential equation 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 (α,β) coincide with the solution (α, β) of the AL system (1.1) from Theorem 2.2. Moreover, (α(t),β(t)) are uniformly bounded for t ∈ (−T, T ), as writing (2.11) in integral form yields (α(t),β(t)) w,p ≤ (α(0),β(0)) w,p + tC (α 0 , β 0 ) w,2p + C t 0 (α(s),β(s)) w,p ds for some constants C.

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 r (z), r = (r − , r + ) ∈ N 2 0 , with Laurent polynomial entries, see [13,Sec. 3.2] or [14]. Suppose that U (z) and V r (z) satisfy the zero-curvature equation 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 r (z) are recursively defined by and Varying r ∈ N 2 0 , the collection of evolution equations then defines the time-dependent Ablowitz-Ladik hierarchy. Explicitly, taking r − = r + for simplicity, the first few equations are Hence choosing c = e icrt it is no restriction to assume c r = 0. By [15], the AL hierarchy is equivalent to the Lax equation where L is the doubly infinite five-diagonal matrix (2.5) and (recall (2.6)) 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]).
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).