Research on the asymptotic form of the solutions to the Helmholtz equation and the application to mathematical scattering theory

点击翻译按钮获取中文摘要

We studied the solutions to the Helmholtz equation with respect to elastic wave in 3-dimensional half space with free boundary.Especially, we studied the asymptotic forms of infinite direction of space. As the results, we obtained the asymptotic forms concerning P mode, R mode, SV mode (the reflected P wave component only), SH mode and SV+SV0 mode (except for the reflected P wave component), respectively. But, for the asymptotic forms concerning SH mode and SV+SV0 mode, the direction of North pole was excepted. We also obtained the results concerning scattering and inverse scattering problem for wave equations with dissipative terms, inverse problem for nonlinear Schrodinger equations and the regularity (at interface) of the solutions to a system of first order partial differential equation as the related topics.