论文标题
$ l^2 $ sobolev散射分散散射转换的空间射击与散落的ablowitz-ladik系统有关
$L^2$ Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems
论文作者
论文摘要
在本文中，我们建立了$ l^2 $ -sobolev空间射击与散射变换有关的偶数散射变换。一方面，在直接问题中，基于频谱问题，我们建立了反射系数和Coreconsent Riemann-Hilbert问题。而且我们还证明，如果电势属于$ l^{2，k} $空间，那么反射系数属于$ h^k_θ（σ）$。另一方面，在逆问题中，基于Riemann-Hilbert问题，我们获得了核心的重建公式，并从反射系数中恢复了电势。而且我们还确认，如果反射系数在$ h^k_θ（σ）$中，那么我们表明电势也属于$ l^{2，k} $。这项研究还证实，对于散热ablowitz-ladik方程的初始值问题，它的初始电位属于$ l^{2，k} $，并且满足$ \ parallel q \ parallel_ \ parallel_ \ infty <1 $，然后是$ t \ ne0 $的解决方案，也属于$ l^{2，K {2，k} $。
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 $H^k_θ(Σ)$. 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 $H^k_θ(Σ)$, 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 $l^{2,k}$.
