Mathematical Sciences: Research in Nonlinear Wave Motion

8822444 Segur这个项目包括三个独立的研究方向，研究演化系统中的非线性波动。这三种思想在很大程度上是独立的。A部分涉及一种有效方法的发展，以确定演化方程的特解对初始数据的微小变化是否稳定。稳定性理论在Arnol'd（1965）等人发现了构造Lyapunov泛函的系统方法后取得了重大进展。尽管如此，目前可用的方法还无法确定像稳定静态、平行剪切流这样简单的流动是否是二维欧拉方程的稳定解。一个目标是设计一种可能解决这个问题的现有方法的概括。B部分的重点是具有周期边界条件的Kadomstev-Peviashvili （KP）方程。已知方程是完全可积的，因此有理由相信具有周期边界条件的KP方程的初值问题是可解的。KP方程还提供了一类水波的精确模型，它与代数几何中的黎曼曲面理论密切相关。事实上，这个问题是如此丰富，以至于几乎任何方向的进展都是有价值的。C部分的目标是为主要通过共振三元组相互作用的大量波包构建一个新的动力学理论。如果成功，该理论将产生一个玻尔兹曼型方程，其平衡解将提供波能量的平衡谱，类似于分子的麦克斯韦-玻尔兹曼分布。Segur（1984）给出了这种理论的早期版本；项目这一部分的工作将纠正该模型中的一个缺陷。

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.