Mathematical Sciences: Inverse Scattering Theory

周9401403 本计画系关于非线性偏微分方程式理论中所产生问题之数学研究。 将讨论几个主题。 其中包括对聚焦非线性薛定谔方程零色散极限的分析。 聚焦方程和许多其他可积系统都有非自伴散射算子。 这种类型的零色散问题还没有得到很好的理解。 工作也将做的非线性薛定谔方程的微扰理论。 这将通过努力获得最大范数和加权二次范数之间的关系的潜力。 一旦这样做了，扰动问题可以通过McKean和Shatah的方法来解决。 第三条调查线将检查高阶方程和孤子。 方程在Gelfand-Dikii层次引起的等谱变形，可以减少到研究的黎曼-希尔伯特问题。 本文将分析跳跃矩阵振荡的Gelfand-Dikii流的长时间行为。 这项研究的新功能包括更大的尺寸的矩阵和跳跃数据表现出不同的订单在原点的奇异性，从这个冲击区可以开发。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。

Zhou 9401403 This project is concerned with mathematical research on problems arising in the theory of nonlinear partial differential equations. Several topics will be addressed. Included in them is an analysis of the zero dispersion limit for the focusing nonlinear Schrodinger equation. The focusing equation and many other integrable systems have nonself-adjoint scattering operators. This type of zero dispersion problem is not well understood. Work will also be done on a perturbation theory for nonlinear Schrodinger equations. This will be done through efforts to obtain relationships between the maximal norm and a weighted quadratic norm of the potential. Once this is done, the perturbed problem can be solved by a method of McKean and Shatah. A third line of investigation will examine higher order equations and solitons. Equations in the Gelfand-Dikii hierarchy give rise to isospectral deformations which can be reduced to the study of a Riemann-Hilbert problem. Work will be done analyzing the long time behavior of those Gelfand-Dikii flows for which the jump matrices are oscillatory. The new features of this study includes the larger size of the matrices and the jump data exhibiting singularities of different orders at the origin; from this shock regions can be developed. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.