Mathematical Sciences: Nonlinear Wave Motion

这项研究包括三个独立的研究路线：a） 超越所有阶的渐近性; B）动力学理论和共振 三元组，以及c）空间中曲线和曲面的运动模型。 这些结果具有重要的应用潜力， 非线性动力学和KAM理论的应用。 锦 定理断言，在适当的情况下，大多数 系统的轨迹在混沌开始时保持规则; 只有少数轨迹实际上变得不规则或混乱。 到目前为止，还没有可用的理论可以确定 轨道在小扰动下变得混乱。 如果 成功的话，这里提出的工作将使科学家能够工作， 同时包含混沌和规则的动力系统 轨迹 这个问题在许多领域都很重要，其中之一就是 等离子聚变装置的设计，比如托卡马克。

This research encompasses three separate lines of study: a) Asymptotics beyond all orders; b) Kinetic theory and resonant triads, and c) Models for motion of curves and surfaces in space. The results have potential for important applications in nonlinear dynamics and applications of the KAM theory. The KAM theorem asserts that under appropriate circumstances most of the trajectories of the system remain regular at the onset of chaos; only a few trajectories actually become irregular, or chaotic. So far there is no available theory that can identify which trajectories become chaotic under small perturbations. If successful, the work proposed here will enable scientists to work with dynamical systems containing both chaotic and regular trajectories. This issue is important in many fields, one of which is the design of plasma fusion devices like tokomaks.