Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions
Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions
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DOI：
10.1088/0266-5611/23/4/021
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发表时间：
2007-08
期刊：
影响因子：
2.1
通讯作者：
M. Ablowitz;G. Biondini;B. Prinari
中科院分区：
文献类型：
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作者：
M. Ablowitz;G. Biondini;B. Prinari
构造了在无穷远处具有非零边值的散焦非线性薛定谔方程的可积离散的逆散射变换。这个问题以前已经研究过，并建立了许多关键的结果。在这里，一个合适的变换的散射问题，以解决公开的问题，本征函数和散射数据的解析。此外，反问题被配制为单位圆上的Riemann-Hilbert问题，并且需要修改的标准程序，以处理本征函数的渐近性对势的依赖性。本文还导出了Gel'fand-Levitan-Marchenko方程的离散形式。最后得到了孤子解和小振幅极限解，并讨论了连续极限。
The inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed. This problem had been previously studied, and many key results had been established. Here, a suitable transformation of the scattering problem is introduced in order to address the open issue of analyticity of eigenfunctions and scattering data. Moreover, the inverse problem is formulated as a Riemann–Hilbert problem on the unit circle, and a modification of the standard procedure is required in order to deal with the dependence of asymptotics of the eigenfunctions on the potentials. The discrete analog of Gel'fand–Levitan–Marchenko equations is also derived. Finally, soliton solutions and solutions in the small-amplitude limit are obtained and the continuum limit is discussed.