Mathematical Sciences: Inverse Scattering Theory

数学科学：逆散射理论

• 批准号: 9401403
• 负责人: Xin Zhou
• 金额: $ 5.5万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401403&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Scattering; Theory

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.

本课题对非线性偏微分方程理论中出现的问题进行数学研究。会议将讨论几个主题。其中包括对聚焦非线性薛定谔方程的零色散极限的分析。聚焦方程和许多其他可积系统都具有非自伴随散射算子。这种类型的零色散问题还没有得到很好的理解。我们还将研究非线性薛定谔方程的摄动理论。这将通过努力获得最大范数和势的加权二次范数之间的关系来完成。一旦这样做了，扰动问题就可以用McKean和Shatah的方法来解决。第三条研究路线将检验高阶方程和孤子。Gelfand-Dikii层次中的方程引起等谱变形，这可以归结为黎曼-希尔伯特问题的研究。我们将分析跳跃矩阵为振荡的Gelfand-Dikii流的长时间行为。该研究的新特点包括：矩阵的尺寸更大，跳跃数据在原点处表现出不同阶的奇点；从这个冲击区域可以发展。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创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。