A new Riemann–Hilbert problem in a model of stimulated Raman Scattering

E. Moskovchenko, V. Kotlyarov
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引用次数: 19

Abstract

The Riemann–Hilbert problem proposed in [2] for the integrable stimulated Raman scattering (SRS) model was shown to be solvable under an additional condition: the boundary data have to be chosen in such a way that a corresponding spectral problem has no spectral singularities. In the general case, it can be shown that a spectral singularity occurs at k = 0. On the other hand, the initial boundary value (IBV) problem for the SRS equations is known to be well posed: using PDE techniques, this has been established in [3]. Therefore, it seems natural to try to find a new RH problem that is solvable in the presence of arbitrary spectral singularities. The formulation of such a RH problem is the main aim of the paper. Then the solution of the nonlinear initial boundary value problem for the SRS equations is expressed in terms of the solution of a linear problem which is the Riemann–Hilbert problem for a sectionally analytic matrix function.
受激拉曼散射模型中的一个新的黎曼-希尔伯特问题
[2]中提出的可积受激拉曼散射(SRS)模型的Riemann-Hilbert问题在一个附加条件下是可解的:边界数据的选择必须使相应的光谱问题没有光谱奇点。一般情况下,可以证明在k = 0处出现谱奇点。另一方面,已知SRS方程的初始边值(IBV)问题是定态良好的:使用PDE技术,[3]已经建立了这一点。因此,试图找到一个在任意谱奇点存在下可解的新RH问题似乎是很自然的。这类RH问题的表述是本文的主要目的。然后将SRS方程的非线性初边值问题的解表示为一个线性问题的解,该线性问题是一个截面解析矩阵函数的Riemann-Hilbert问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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