Mathematical Sciences: Dynamics of Nonlinear Dispersive Systems

Weinstein将研究非线性基态在非线性色散演化方程动力学中的作用。要考虑的具体问题包括非线性色散哈密顿系统的无限时间行为，非线性束缚态的谱稳定性和不稳定性，光纤中的非线性波和耦合光纤阵列的动力学，矢量Zakharov方程和矢量非线性Schroding方程。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。

PI: Weinstein DMS-9500997 Weinstein will study the role of nonlinear ground states in the dynamics of nonlinear dispersive evolution equations. Specific problems to be considered include the infinite time behavior of nonlinear dispersive Hamiltonian systems, the spectral stability and instability of nonlinear bound states, nonlinear waves in optical fibers and the dynamics of coupled fiber arrays, and vector Zakharov and vector nonlinear Schroding equations. 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.