Mathematical Sciences: Waves in Nonlinear Dispersive and Hyperbolic Systems

这个正在进行的关于非线性波的研究项目将集中在以下三个方面：1。函数路径积分。2. 精确调制。3. 三波共振相互作用。前两个主题是研究具有Painleve性质的方程、用逆散射变换方法可解的方程、波调制理论和用路径积分表示偏微分方程的解之间的关系。最后一个主题是关于严格双曲系统中波的弱非线性相互作用的研究。非线性波的数学理论是物理波传播的数学模型。从这个理论推导出的结论对我们理解许多自然现象和技术现象通常是有用的。目前对非线性波的研究计划旨在更好地理解共振和不稳定性，它们是结构和流体系统失效的原因。研究结果有望为许多此类工程系统的更好设计做出贡献。

This ongoing research program on nonlinear waves will focus on the following three topics: 1. Functional path integrals. 2. Exact modulation. 3. Three wave resonant interactions. The first two topics are concerned with the study of some relationships among equations with the Painleve property, equations solvable by the Inverse Scattering Transform method, the theory of wave modulation and representation of solution of partial differential equations by path integrals. The last topic is concerned with the study of weakly nonlinear interactions of waves in strictly hyperbolic systems. The mathematical theory of nonlinear waves is a mathema- tical model of physical wave propagation. The conclusions deduced from the theory are generally useful to our under- standing of many natural and technological phenomena. The present research program on nonlinear waves is aimed toward a better understanding of resonance and instability which are causes for failures in structural and fluid systems. The research results are expected to contribute to better designs for many such engineereing systems.