Inverse Scattering Theory

dms -0071398本项目包含五个研究问题：1.；散射/逆散射变换的解析性质。对nls的摄动。Toda格的摄动。随机matrices.5。多个正交多项式。所有这些问题都将通过黎曼-希尔伯特问题方法进行研究，黎曼-希尔伯特问题方法已经显示出解决各种广义可积问题的巨大潜力。这个项目分析了几个数学问题，从偏微分方程，随机矩阵，到近似理论。其中一些问题具有纯数学意义，另一些问题在物理和工程中有应用。其中一个应用是摄动非线性薛定谔方程的孤子解。这些孤子解用于光纤传输。

DMS-0071398The project contains five research problems:1. Analytic properties for scattering/inverse scatteringtransforms.2. Perturbation on NLS.3. Perturbation for Toda lattice.4. Random matrices.5. Multiple orthogonal polynomials. All these problems will be studied via the Riemann--Hilbertproblem approach, which has shown tremendous potential insolving various integrable problems in broad sense.This project analyzes several mathematical problems rangingfrom partial differential equations, random matrices, to the approximation theory. Some of these problems have pure mathematicalinterest and others have applications in physics and engineering.One of such applications is the soliton solutions of the perturbednonlinear Schrodinger equation. These soliton solutions are usedin the fiber optical transmission.