Inverse Scattering Theory

DMS-0071398项目包含五个研究问题：1.散射/逆散射变换的分析性质. NLS上的扰动。3.户田晶格的微扰.随机矩阵。多个正交多项式。 所有这些问题都将通过Riemann-Hilbert问题方法进行研究，该方法在解决各种广义可积问题方面显示出巨大的潜力。该项目分析了从偏微分方程、随机矩阵到逼近理论等多个数学问题。这些问题有的是纯理论性的，有的是在物理和工程上有应用的，其中之一就是微扰非线性薛定谔方程的孤子解。这些孤子解可用于光纤传输。

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.