\(l^2\)-Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems (Q2677796)

The authors consider a discrete version of the nonlinear Schrödinger equation, the Ablowitz-Ladik system, and investigate its integrability properties for a certain class of solutions by means of the inverse scattering method. The class of solution under consideration is \(l^{2,k}\), which is the \(l^2\) space with a certain weight defined for each \(k\). The authors consider first the direct scattering problem and establish the corresponding Riemann-Hilbert problem. The main result of this investigation is concerned with the reflection coefficient, which is proven to belong to the Sobolev space, and an estimate on its norm is also given. The inverse scattering problem is considered as well, and it is proven that the relation between the classes of the potential and the reflection coefficient is in fact a bijection. Thus, given the reflection coefficient which belongs to the Sobolev space, the authors show that the solution of the Ablowitz-Ladik system can be obtained from the reconstruction formula (given explicitly in the paper), and show that it belongs to the \(l^{2,k}\) space. The time evolution is also considered. The paper is clearly written and the authors provide detailed proofs of the main theorems, together with many relevant intermediate results.