Mathematical Methods for Nonlinear Wave Equations

提案ID：0604546PI：Bernard Deconinck机构：华盛顿大学题目：非线性波动方程的数学方法摘要我将开发两种研究非线性波动方程的通用方法。这些方法涉及可积方程(拟)周期解的显式计算，以及可积或近可积方程的这些解和其他解的数值稳定性分析。第一种方法使用我在过去几年开发的关于黎曼曲面的计算微积分。第二种方法是对Hill方法的扩展和严格检验，我们在2006年全面介绍了Hill方法。这些方法的应用将使我们能够计算和分析描述各种现象的方程的解，如海啸、激光和在极端温度下发生的新的物质状态。这项拟议的研究将涉及不同的研究生和本科生，并让他们走上为美国应用数学研究做出自己重要贡献的道路。

Proposal ID: 0604546PI: Bernard DeconinckInstitution: University of WashingtonTitle: Mathematical Methods for Nonlinear Wave EquationsAbstractI will develop two general methods for the study of nonlinear wave equations. These methods pertain to the explicit computation of (quasi-)periodic solutions of integrable equations, and the numerical stability analysis of these and other solutions of integrable or near-integrable equations. The first method uses the computational calculus on Riemann surfaces I have developed over the last years. The second method extends and rigorously examines Hill's method, which we introduced in its full generality in 2006.The applications of the methods will allow us to compute and analyse solutions of equations describing such diverse phenomena as tsunamis, laserlight, and new states of matter occurring at extreme temperatures. The proposed research will involve different graduate and undergraduate students, and put them on their way to making their own important contributions to applied mathematical research in the United States.