论文标题
Riemann-Hilbert的方法，用于具有非变化边界条件的通用可变的非线性schrödinger方程
Riemann-Hilbert approach to the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions
论文作者
论文摘要
在这项工作中，我们考虑了具有无限性的非呈现边界条件的广义可变的非线性schrödinger方程，包括散射系数的简单和双极。通过引入适当的Riemann表面和均匀化坐标变量，我们首先将直接散射过程中发生的双值函数转换为单值函数。然后，我们通过分析JOST函数的分析性，对称性和渐近行为以及源自方程的LAX对的散射矩阵来建立直接散射问题。基于这些结果，成功建立了一个广义的Riemann-Hilbert问题。系统地研究了离散的光谱和残留条件，痕量foumulae和theta条件，包括简单的杆子案例和双杆案例。此外，通过Riemann-Hilbert方法解决了反向散射问题。最后，在无反射电势的条件下，孤子和呼吸溶液得到了很好的衍生。通过评估每个参数的影响，以图形方式分析了这些解决方案的一些有趣现象。
In this work, we consider the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions at infinity including the simple and double poles of the scattering coefficients. By introducing an appropriate Riemann surface and uniformization coordinate variable, we first convert the double-valued functions which occur in the process of direct scattering to single-value functions. Then, we establish the direct scattering problem via analyzing the analyticity, symmetries and asymptotic behaviors of Jost functions and scattering matrix derived from Lax pairs of the equation. Based on these results, a generalized Riemann-Hilbert problem is successfully established for the equation. The discrete spectrum and residual conditions, trace foumulae and theta conditions are investigated systematically including the simple poles case and double poles case. Moreover, the inverse scattering problem is solved via the Riemann-Hilbert approach. Finally, under the condition of reflection-less potentials, the soliton and breather solutions are well derived. Via evaluating the impact of each parameters, some interesting phenomena of these solutions are analyzed graphically.
