Mathematical Sciences: Inverse Scattering Theory

点击翻译按钮获取中文摘要

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.