具有非零边界条件可积系统初值问题的Riemann-Hilbert方法

The Riemann-Hilbert (RH) problem was raised by Hilbert at the International Conference of Mathematicians in Paris in 1900. In the late 1970s, RH problem began to be applied to the study of integrable systems. However, the study of long-time asymptotic behavior of solutions to initial value problems of integrable systems has always been a challenging topic. Until the 1990s, Deift proposed a nonlinear steepest descent method for solving oscillating RH problems to the long-time asymptotic behavior of initial value problems. At present, the nonlinear steepest descent method is mainly used to study the long-time behavior of solutions for initial value problems with decaying initial conditions, but the research on initial value problems with general initial conditions is very few. In this paper, based on Inverse Scattering method, operator spectrum theory and Deift-Zhou nonlinear steepest descent method, the asymptotic behavior of solutions with non-zero boundary initial conditions is studied. The following problems are emphatically studied: 1. The long-time behavior of solutions for classical integrable equations with general initial values; 2. The long-time behavior of solutions for nonlocal integrable equations with decaying initial values and general initial values.

Riemann-Hilbert（RH）问题是Hilbert于1900年在巴黎国际数学家大会上提出的。20世纪70年代末，RH问题开始应用于可积系统的研究，但是关于可积系统初值问题解的长时间渐近行为的研究一直是具有挑战性的课题。直到90年代，Deift提出了一套求解振荡RH问题的非线性速降法来解决初值问题解的长时间渐进性。目前非线性速降法主要用于研究具有快速衰减初始条件下初值问题解的长时间行为，对于一般初始条件的初值问题的研究非常少。本课题主要基于反散射方法、算子谱理论以及Deift-Zhou非线性速降法研究具有非零边界初始条件下解的渐近行为。重点研究如下几个问题：1. 经典可积方程具有一般初值的解的长时间行为；2. 非局部可积方程具有快速衰减初值和一般初值条件下解的长时间行为。

Riemann-Hilbert方法应用于可积系统领域的研究可追溯到二十世纪七十年代，但是关于高阶以及负阶族Lax对的可积发展方程的黎曼-希尔伯特问题的研究较少。本项目主要研究了具有3阶矩阵Lax对的推广Sasa-Satsuma方程在半直线和有限区间上的初边值问题，通过分析全局关系得到了Dirichlet边值和Neumann边值之间的映射；同时，讨论了具有负阶族矩阵谱问题的Sawada-Kotera方程在半直线上的黎曼-希尔伯特问题，分析了谱参数在零点处的奇异性。另外，我们分别给出了在无穷远处有零边界条件和非零边界条件下，拓展的修正Korteweg-de Vries方程的黎曼-希尔伯特问题，以及它们的简单极点和高阶极点。