Mathematical Sciences: Waves in Nonlinear Dispersive and Hyperbolic Systems

主要研究人员将继续研究非线性波理论的三个主要领域：非线性色散波、弱激波在焦散线处的聚焦和方波爆轰中的灾难性不稳定性。研究方法包括使用解析和渐近技术，辅之以数值分析。之所以需要这样一种综合方法，是因为所研究的现象发生在现有理论的边界上。自然界中的许多现象不能用线性模型来充分描述；自然界天生就是“非线性的”。建议的研究涉及由非线性微分方程组描述的波动的研究。一个很好的例子是，当海浪接近海滩时，海浪“破裂”。为了描述这种非线性波的行为，首席研究人员将结合数值分析使用渐近分析，以获得定性和定量信息。

The principal investigator will continue his important studies of three major areas of nonlinear wave theory: nonlinear dispersive waves, focussing of weak shock waves at caustics and catastrophic instabilities in square-wave detonations. The methods of study involve the use of analytic and asymptotic techniques supplemented with numerical analysis. Such a combined approach is called for because the phenomena under study occur at the boundaries of existing theories. Many phenomena in nature cannot be described adequately with linear models; nature is inherently "nonlinear". The proposed research involves the study of wave motions that are described by nonlinear differential equations. A good example is the "breaking" of an ocean wave as it approaches the beach. In order to describe the behavior of such a nonlinear wave the principal investigator will use asymptotic analysis in concert with numerical analysis to obtain qualitative and quantitative information.