非局域非线性色散方程Cauchy问题的反散射变换

三波共振方程组在水波、内波、等离子物理、非线性光学中具有广泛的应用。耦合非线性薛定谔方程也出现在许多物理环境中，例如：深水波、非线性光学、声学、玻色-爱因斯坦凝聚态。本项目计划应用反散射变换研究一些具有重要物理意义的非线性色散方程在不同边值条件下的初值问题，包括非局域复反时空、非局域实反时空三波共振方程组以及耦合非局域 PT、耦合非局域反时空非线性薛定谔方程，预计应用Riemann-Hilbert方法以及散射数据的对称关系，以最少的数据刻画方程的孤立子解。此外，对于两个非局域三波共振系统，本项目将探索三波在不同时刻的相互作用情况并解释其物理意义。并且，与经典的三波共振方程组所得到的三个波包进行对比，分别从数学和物理两个方面阐述经典的和两个非局域三波共振系统的区别。

The three-wave interaction equations arise widely in applications, including water waves, internal waves, plasma physics and nonlinear optics. The coupled nonlinear Schroedinger equation also appear in many physical contexts, such as deep water waves, nonlinear optics, acoustics and Bose–Einstein condensation. This project plans to employ the inverse scattering transform to study the initial value problems for some physical-important nonlinear dispersive equations with different boundary conditions, including the nonlocal complex and real reverse space-time three-wave equations and the coupled nonlocal PT and reverse space-time nonlinear Schroedinger equations. To derive soliton solutions by minimal data, Riemann-Hilbert methods and symmetries will be applied. In addition, the interaction properties of three envelopes for the classical and two nonlocal three-wave systems will be explained mathematically and physically.