Riemann-Hilbert Problem and Integrable Systems

摘要本课题涉及聚焦NLS的微扰理论、多重正交多项式、非解析相的振荡riemann - hilbertproblem、聚焦NLS的零色散、正交有理函数和随机矩阵理论中的Janossy密度。所有这些问题都需要黎曼-希尔伯特问题技术来研究。该项目旨在进一步发展黎曼-希尔伯特问题方法的分析方面和跨学科应用，并在其智力和教育优点之间取得平衡。黎曼-希尔伯特问题的研究在数学和数学物理的几个领域都有应用，包括偏微分方程、统计力学和组合学。

AbstractZhouThis proposal project covers, perturbation theory for focusing NLS,multiple orthogonal polynomials, oscillatory Riemann-Hilbertproblems with nonanalytic phase, zero dispersion of focusing NLS,orthogonal rational functions, and Janossy densities in the randommatrix theory. All these problems require Riemann-Hilbert problemtechniques to study.This project is an effort to develop further the analytic aspectsand the interdisciplinary applications of the fruitful field of theRiemann--Hilbert problem method, with the balance between itsintellectual and educational merits. The study of Riemann--Hilbertproblem has applications in several areas of mathematics andmathematical physics, including partial differential equations,statistical mechanics and combinatorics.