中间长波方程的直接散射变换
The Direct Scattering Transform for the Intermediate Long Wave Equation
- University of Kentucky（肯塔基大学）
- University of Oklahoma（俄克拉荷马大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文严格研究了中间长波方程的直接散射问题，推导了Lax算子并构造了散射数据，为通过逆散射变换求解ILW方程奠定基础。
AI中文摘要：
本文对Kodama、Ablowitz和Satsuma针对中间长波（ILW）方程的完全可积性理论所提出的直接散射问题进行了严谨的研究。我们推导了ILW的Lax算子，并研究了其谱理论。我们证明了离散谱的有限性和简单性，Jost解的存在性和唯一性，并构造了Lax算子的散射数据。我们还证明了Jost解满足一个非局部Riemann-Hilbert问题，其奇点数据由散射数据给出，并且散射势可以从Jost解重构。这项工作为通过逆散射变换求解ILW方程铺平了道路。
英文摘要：
In this paper, we provide a rigorous study of the direct scattering problem proposed by Kodama, Ablowitz and Satsuma for the complete integrability theory of the Intermediate Long Wave (ILW) equation. We derive the Lax operators for ILW, and study their spectral theory. We show finiteness and simplicity of the discrete spectrum, existence and uniqueness of the Jost solutions, and construct the scattering data for the Lax operator. We also show that the Jost solutions satisfy a nonlocal Riemann-Hilbert problem whose singularity data are given by the scattering data, and that the scattering potential can be reconstructed from the Jost solutions. This work paves the way for a solution to ILW via the inverse scattering transform.
