类粒子波包系统的非线性动力学
Nonlinear dynamics of a system of particle-like wavepackets
摘要：
这是我们对波包系统非线性演化研究的继续。本文研究了由常系数双曲型偏微分方程组成的非线性系统控制的波的传播问题。根据定义，一般波包有一个明确的主波矢量，并且，正如我们在以前的工作中证明的那样，非线性动力学保持波包及其主波矢量的系统。本文研究一类特殊的波包，即类粒子波包的非线性演化。类粒子波包具有双重性质：一方面，它是具有明确定义的主波矢量的波，另一方面，它是粒子，因为它可以在空间中被分配一个明确定义的位置。我们证明了在非线性演化下，一般的多粒子波包仍然是高精度的多粒子波包，而构成的每一个单粒子波包不仅保持其主波数，而且具有一个定义明确的空间位置，以一个恒定的速度（即群速度）演化。值得注意的是，尽管所涉及的单粒子波包在空间中经历非线性相互作用和多次碰撞，所描述的性质仍然存在。我们还证明了，如果多粒子波包的主波矢是通用的，则不同波包之间的非线性相互作用的结果是小的，并且近似线性叠加原理关于波包的初始空间位置一致成立。
英文摘要：
This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.
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