非零边界条件下广义变系数非线性薛定谔方程的Riemann-Hilbert逼近
Riemann-Hilbert approach to the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions
摘要：
本文研究了在无穷远处具有非零边界条件的广义变系数非线性薛定谔方程，其中包括散射系数的单极点和双极点.通过引入合适的黎曼曲面和一致化坐标变量，将直接散射过程中出现的双值函数转化为单值函数。然后，通过分析由方程的拉克斯对导出的Jost函数和散射矩阵的解析性、对称性和渐近性，建立了直接散射问题。在此基础上，成功地建立了该方程的广义Riemann-Hilbert问题.系统地研究了单极点和双极点情况下的离散谱和残差条件、迹公式和θ条件。此外，通过Riemann-Hilbert方法求解逆散射问题。最后，在无反射势的条件下，得到了孤子解和呼吸子解。通过分析各参数对解的影响，对解的一些有趣现象进行了图解分析。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
