Riemann-Hilbert方法在若干非线性可积系统中的应用

本项目发展Riemann-Hilbert方法研究若干非线性可积系统的精确孤立子解、初值问题和初边值问题解的长时间渐近行为。拟开展以下四个方面的研究：首先，基于Riemann-Hilbert方法寻找五分量Gross-Pitaevskii方程新的精确孤立子解，特别是多暗孤立子解；其次，发展Deift-Zhou的非线性速降法，研究谱问题是三阶的具有二阶色散项的Boussinesq方程初值问题解的长时间渐近行为；再次，将非线性速降法推广到导数非线性薛定谔方程，分析此方程三类step-like初值问题解的长时间渐近行为；最后，将Fokas统一方法与非线性速降法相结合，求得半经典非线性薛定谔方程具有相位调制的方势垒初（边）值问题的极限解。本项目将对探索可积系统初（边）值问题的整体渐近性质、完善和发展Riemann-Hilbert方法以及现实物理应用具有重要的科学意义和理论价值。

This project develops Riemann-Hilbert method to investigate the exact soliton solutions, long-time asymptotics of initial (boundary) value problems of several integrable systems. It plans to carry out four aspects of studies: First of all, based on Riemann-Hilbert method find new exact soliton solutions, especially the multi-dark soliton solutions of the five-component Gross-Pitaevskii equations. Secondly, extend the nonlinear steepest descent method of Deift and Zhou to study the long-time asymptotics of Schwartz class initial problem of the Boussinesq equation with second-order dispersive term, which has three-order Lax pair. Thirdly, extend the nonlinear steepest descent method to the derivative nonlinear Schrödinger equation to analyze the long-time asymptotics of three kinds of step-like initial value problems. Finally, combine the unified transform method of Fokas with nonlinear steepest descent method to find the zero-dispersion limit solutions of semiclassical nonlinear Schrödinger equation square barrier initial (boundary) problem with phase. In a word, this project has important scientific significance and theoretic value for the whole asymptotic behaviors of the initial (boundary) value problems of integrable systems and the development of Riemann-Hilbert method.