Mathematical Sciences: Research in Nonlinear Wave Motion

8822444 Segur 该项目包括三个独立的研究领域， 演化系统中的非线性波动 三个想法 基本上是独立的。 A部分涉及一个 一种有效的方法来确定特定的 演化方程的解对于下列参数的微小变化是稳定的： 初始数据。 稳定性理论在 Arnol'd（1965）和其他人发现了一个系统的 构造李雅普诺夫泛函的方法。 即便如此， 目前可用的是无法确定是否流作为 简单的稳定分层，平行剪切流是一个稳定的 二维欧拉方程的解 一个目标是 设计一个通用的方法， 以下问题 B部分的重点是Kadomstev-Peviashvili（KP） 具有周期边界条件的方程 该方程 已知是完全可积的，所以有理由 认为KP方程的初值问题， 周期边界条件是可解的。 KP方程也 提供了一类水波的精确模型， 与黎曼曲面理论密切相关， 代数几何 事实上，问题是如此丰富， 几乎任何方向的进展都是有价值的。 C部分的目标是构建一个新的动力学 一个理论的大集合波包，相互作用 主要是通过共振三和弦。 如果成功，理论上 会导致玻尔兹曼型方程， 解将提供波的平衡谱 能量，类似于麦克斯韦-玻尔兹曼分布， 分子。 这样一个理论的早期版本是由 Segur（1984）;项目这一部分的工作将 纠正了模型中的缺陷。

8822444 Segur This project includes three separate lines of research in nonlinear wave motion in evolutionary systems. The three ideas are largely independent. Part A concerns the development of an effective method to determine whether or not a particular solution of an evolution equation is stable to small changes in initial data. Stability theory advanced significantly after the discovery by Arnol'd (1965) and others of a systematic method to construct Lyapunov functionals. Even so, the methods currently available are unable to determine whether a flow as simple as stably statified, paralled shear flow is a stable solution of Euler's equations in two dimensions. One goal is to devise a generalization of current methods that might settle the question. The focus in part B is on the Kadomstev-Peviashvili (KP) equation with periodic boundary conditions. The equation is known to be completely integrable, so there is reason to believe that the initial value problem for the KP equation with periodic boundary conditions is solvable. The KP equation also provides accurate models of a class of water waves, and it is intimately connected with the theory of Riemann surfaces in algebraic geometry. In fact, the problem is so rich that progress in almost any direction would valuable. The objective in part C is to construct a new kinetic theory for a large collection of wavepackets that interact primarily through resonant triads. If successful, the theory would result in a Boltzmann-type equation, whose equilibrium solutions would provide an equilibrium spectrum of wave energies, analogous to the Maxwell-Boltzmann distribution for molecules. An earlier version of such a theory was given by Segur (1984); the work in this part of the project would correct a deficiency in that model.