Riemann-Hilbert方法与可积系统解的长期行为

Riemann-Hilbert(RH)问题是著名数学家Hilbert所提出的23个问题之一， 80年代,RH问题开始应用于可积系统求解，而可积系统解的长期行为一直是具有挑战性的研究问题。90年代，美国科学院院士、纽约大学Deift所提出求解震荡RH问题的非线性速降法是可积系统领域的一项突破性工作，并广泛应用于正交多项式、随机矩阵、组合学等其它研究领域。为此，Deift二次被国际数学家大会邀请做大会报告，2007年，他提出可积系统和随机矩阵有待解决的16个重要公开问题。.本项目将基于算子谱理论、多复变和散射理论，在Deift的若干公开问题方向上，重点研究以下三个问题：1. 具有非自伴算子可积系统初值问题解的长期行为。2.具有高阶算子的可积系统初值问题解的长期行为。3. 可积系统初边值解的长期行为。本项目研究成果将促进学科交叉发展，为揭示可积系统的性质、结构和内在联系提供新的理论和方法。

Hilbert's twenty-three problmes have played an important role in the development of mathematics in twentieth century. His twenty-first problem is the Riemann-Hilbert problem, which started to applly in integrable systems in 1980s.At the beginning of the 90's, Deift and Zhou proposed nonlinear descent method for oscillatory matrix Riemann-Hilbert problems. With this new method came the nice possibility to rewrite known asympotic resultss for different integrable models in a rigorous and transparent form and obtain numerous new significant results in the theory of completely integrable nonlinear equations, random matrix models, orthogonal polynomials and integrable statistical mechanics. . Based on the operator and spectral theory, multi-complex functions and scattering theory, this project concentrate on the following three problems related to Deift's open problems: 1. Long-time asymptotics for integrable systems with non-adjoint Lax operator and initial date. 2. Long-time asymptotics for integrable systems with higher order Lax operator and initial date. 3. Long-time asymptotics for initial boundary problem of integrable systems. The results of this project will contribute the study of properties, structure and inner connnections of integrable systems.