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arXiv · 2204.06897

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

Meisen Chen·Engui Fan·Jingsong He
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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 $l^{2,k}$.

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Meisen Chen, Engui Fan, Jingsong He. 2022-04-14. $L^2$ Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems. https://doi.org/10.1016/j.physd.2022.133565

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