物理中某些耦合可积方程的研究

The integrable equations play a vital role in mathematical Physics. Recently the study of rogue wave has drawn a lot of attentions in the nonlinear science. In the earlier research, we were mainly looking for the exact solutions to describe the dynamics. For the further researches, it is natural to explore the physical mechanism for these phenomena. Since the background of rogue wave is nonzero, to understand the physical mechanism we need to study the related problems on the nonzero background. This project focuses on the study of the following models: multi-component nonlinear Schrödinger equations, multi-component derivative nonlinear Schrödinger equations, multi-component Fokas-Lenells equations and so on. To understand the physical mechanism from distinct aspects, we mainly study the inverse scattering theory and classification of solutions for the integrable models with non-vanishing background in this project. The theories will be established through combining Riemann-Hilbert approach with Darboux transformation. The exact solutions obtained from these systems would be simplified and classified. The present research will help us to understand and design the nonlinear and localized phenomena arising in the nonlinear optics, Bose-Einstein condensates and so on.

可积方程在数学物理的研究中扮演非常重要的角色.近年来,怪波的研究已经成为非线性科学的热点问题. 早期的研究中, 着重于寻求描叙这些动力学行为的精确解. 随着研究的深入,一个自然的问题就是探索这些现象产生的物理机制. 怪波具有非零的边界条件, 因而要理解其物理机理就必须研究非零背景的相关问题. 本项目拟针对以下几类重要的耦合可积方程组开展研究工作: 多分量非线性薛定谔方程组, 多分量导数的非线性薛定谔方程组, 多分量 Fokas-Lenells 方程组等模型. 主要研究这些耦合可积模型在非零背景条件下的反散射理论以及解的分类, 从不同角度理解这些现象的物理机制. 拟采用 Riemann-Hilbert 方法结合Darboux变换方法建立对应系统的反散射理论, 并对系统中具有物理意义的精确解进行分类和化简. 这些研究有助于理解和设计非线性光学, 波色-爱因斯坦凝聚等物理系统的非线性局域现象.

本项目主要致力于耦合可积模型解的构造，分类和分析的研究。研究的模型主要包括，耦合的非线性薛定谔方程，耦合的Fokas-Lenells方程以及半离散的非线性薛定谔方程等。解的构造上，我们主要采用基于Loop群作用理论的Darboux变换方法。通过Darboux变换方法构造基本的局域波解，利用动力学分析方法给出这些基本局域波解的分类。进一步，利用Darboux变换的迭代方法，给出复杂局域波解的动力学行为分析。 值得一提的工作是，我们结合Darboux变换和Riemann-Hilbert问题方法成功给出了无穷阶怪波的理论，并将其用于分析大阶数怪波解在进场区域的渐近行为。我们将这一工作发表在Duke Math杂志l上。