$L^2$ Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems

TL;DR

This paper proves the bijectivity of the inverse scattering transform in $L^2$-Sobolev spaces for the defocusing Ablowitz-Ladik system, establishing a correspondence between potentials and reflection coefficients with implications for initial-value problems.

Contribution

It establishes the $L^2$-Sobolev space bijectivity of the inverse scattering transform for the defocusing Ablowitz-Ladik system, linking potential spaces and reflection coefficient spaces.

Findings

Reflection coefficient in $H^k_ heta(\Sigma)$ implies potential in $l^{2,k}$.

Potential in $l^{2,k}$ leads to reflection coefficient in $H^k_ heta(\Sigma)$.

Solutions maintain $l^{2,k}$ regularity for $t eq 0$ if initial potential is in $l^{2,k}$ and bounded by 1.

Abstract

In this paper, we establish $L^2$-Sobolev space bijectivity of the inverse scattering transform related to the defocusing Ablowitz-Ladik system. On the one hand, in the direct problem, based on the spectral problem, we establish the reflection coefficient and the corespondent Riemann-Hilbert problem. And we also prove that if the potential belongs to $l^{2,k}$ space, then the reflection coefficient belongs to $H^k_\theta(\Sigma)$. On the other hand, in the inverse problem, based on the Riemann-Hilbert problem, we obtain the corespondent reconstructed formula and recover potentials from reflection coefficients. And we also confirm that if reflection coefficients are in $H^k_\theta(\Sigma)$, then we show that potentials also belong to $l^{2,k}$. This study also confirm that for the initial-valued problem of defocusing Ablowitz-Ladik equations, it the initial potential belongs to $l^{2,k}$ and satisfying $\parallel q\parallel_\infty<1$, then the solution for $t\ne0$ also belongs to $l^{2,k}$.