可积方程初边值问题的 Riemann-Hilbert 方法及其应用

可积方程初边值问题无论在理论上还是应用上都是数学物理的重要研究课题。由Riemann-Hilbert问题演变而成的Riemann-Hilbert 方法已经成为可积系统研究的一种强有力的分析工具，本课题将着重研究以下两个方面内容: (1)可积方程的初边值问题（解的唯一性）表达式及解的长时间渐近行为；(2)随机矩阵理论研究。与此同时，对准晶材料领域相关问题开展应用研究，利用Riemann-Hilbert方法求解一维、三维准晶与压头的无摩擦接触等边值问题。将Riemann-Hilbert 方法和统计方法相结合研究随机矩阵理论和在准晶材料领域中统一进行数学物理机理研究，期望形成具有一定特色的随机矩阵理论和准晶材料领域应用研究的新局面。

The initial boundary value problem of integrable equations is an important research topic of mathematical physics both in theory and in application. The Riemann-Hilbert problem (RHP) has evolved into the Riemann-Hilbert Approach(RHA) and becomes a powerful analysis tool, which is being applied to the following two aspects. (1) The expressions of initial boundary value problem of the integrable equation (the uniqueness of the solution) and long-term asymptotic behavior of solutions are being studied; (2) Random matrix theory is being researched. This is two important issues to be studied in our team. In addition, in the field of quasicrystal materials, the Riemann-Hilbert Approach can be used to solve the frictionless contact of one-dimensional and three-dimensional quasicrystals and indenters with boundary value problem. The Riemann-Hilbert method and the statistical method are combined to study the random matrix theory and the mathematical physics mechanism research is considered in the field of quasicrystal materials. It is expected to form a new situation in the field of random matrix theory and new application of quasicrystal materials with certain characteristics.