非线性水波与随机环境的相互作用

Study on motion and interaction of the widely existed nonlinear waves is of certain importance. As typical nonlinear waves, solitary waves are temporally and spatially localized wave structures that can exist stably. By using the analytical method and numerical simulation, this project will investigate some water-wave models with random perturbations. We will focus on the motion and interaction of solitary waves in random environments, in order to further understand the interplay between nonlinearity and randomness. The originalities of this project include the following three aspects: (1) The theory of diffusion will be employed to analyze the dynamics of solitary waves in random environment, and systematic numerical simulations will be performed to support and improve our analysis; (2) Some water-wave models will be simulated with consideration of the random perturbations. Our attention will be concentrated in the inelastic interaction of solitary waves, as well as the breakup of solitary waves; (3) By systematically simulating interaction of the multiple solitary waves in random environment, we will try to capture the nonlinear waves with high amplitudes, and discuss the relevant generation mechanism.

非线性波动在自然界中广泛存在，对其传播和相互作用的研究具有一定的重要性和实用性。作为典型的非线性波，孤立波是一类在局域时空能相对稳定存在的波动结构。本项目拟以受随机扰动的水波模型为研究对象，综合利用解析分析和数值模拟的手段加以研究，试图阐明孤立波在随机环境中运动和相互作用的规律，以达到进一步了解非线性效应和随机效应作用机理的目的。本研究的创新点主要包括以下三个方面：（1）利用扩散理论来分析和描述孤立波在随机环境下的动力学行为，并通过系统性的数值模拟验证和改进理论分析；（2）模拟水波模型中孤立波在随机环境下的相互作用，重点研究可能出现的非弹性相互作用和波的破裂等现象；（3）通过大量系统性的数值实验，试图捕获多孤立波和随机环境作用时可能产生的大振幅波动，并讨论相应的产生机制。

本项目利用解析分析和数值模拟的手段重点研究了受随机扰动的水波模型（以无序非线性薛定谔类模型为主），在一定条件下阐明了单孤立波在随机环境中反常扩散的动力学规律，并进一步针对不同随机环境模拟分析了近可积湍流中多孤立波相互作用下典型参数的统计分布，同时展示了非线性波动的分岔流动并揭示了极端大振幅波动产生的一些机制。我们的研究成果发表在Phys. Rev. E和Chaos等非线性领域的权威期刊，得到了国内外学者的正面引用。项目的研究有助于水波领域和相关交叉学科领域中非线性波动与随机环境相互作用的理解及潜在应用。