Mathematical Methods for Nonlinear Wave Equations

提案ID：0604546 PI：Bernard Deconinck机构：华盛顿大学标题：非线性波动方程的数学方法摘要我将发展两种研究非线性波动方程的一般方法。这些方法属于明确的计算（准）周期解的可积方程，和数值稳定性分析这些和其他解决方案的可积或近可积方程。第一种方法使用我在过去几年中开发的黎曼曲面上的计算微积分。第二种方法扩展和严格检查希尔的方法，我们在2006年介绍了它的全部通用性。该方法的应用将使我们能够计算和分析方程的解，这些方程描述了海啸，激光和极端温度下发生的新的物质状态。拟议的研究将涉及不同的研究生和本科生，并把他们的方式作出自己的重要贡献，应用数学研究在美国。

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.