具有特殊初值的散焦Kundu-Eckhaus方程的正散射问题
On the Direct Scattering Problem for Defocusing the Kundu-Eckhaus Equation with Special Initial Value
摘要: 在本文中，我们主要考虑了具有特殊初值条件的散焦Kundu-Eckhaus (KE)方程的正散射问题。正散射问题是根据一个合适的均匀化变量提出的，这使我们能够在标准复平面上发展讨论相关问题，而不是在两片黎曼曲面或沿切口具有不连续性的切面上。我们首先通过KE方程的Lax对求解出该初值条件下的Jost函数解，然后根据Jost函数解之间存在的散射关系得出相应的散射数据，最后重点讨论了散射数据 的零点，利用一些特殊的三角函数来研究其零点规律，然后我们发现一个有趣的现象，即在某些情况下散射数据 的零点不存在或者只存在一个零点。
Abstract:
In this article, we consider the forward scattering problem of the defocused Kundu-Eckhaus (KE) equation with special initial conditions. The forward scattering problem is proposed based on a suitable uniformization variable, which enables us to discuss related problems on the standard complex plane, rather than on two-sheeted Riemannian surface or cut plane with discontinuities along the cuts. We first solved the Jost function solution under this initial condition by using Lax pairs, and then obtained the scattering data based on the scattering relationship between the Jost function solutions. Finally, we focused on discussing the scattering data . The zero point of should be studied using some special trigonometric function values, and an interesting phenomenon was discovered, namely, in certain cases, the zero point of scattering data should not exist or only one zero point should exist.
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