水波问题及其逼近模型的多峰广义孤立波

Applying the theory of the dynamical system and in particular the invariant manifold reduction theorem, the normal form analysis and the bifurcation theory, this project will study the existence of multi-hump generalized solitary waves (multi-hump solitary waves with same amplitude exponentially tending to a periodic solution with small amplitude at infinity) of water-wave problems and their approximate models. Especially, we will focus on . (1) the existence of multi-hump generalized solitary waves of the approximate models of 2D and 3D water-wave problems,. (2) the existence of multi-hump generalized solitary waves of 2D and 3D water-wave problems,. (3) the existence of multi-hump generalized homoclinic solutions of ordinary differential systems with high dimensions, and. (4) the existence of multi-hump generalized solitary waves of a nonlinear lattice equation.. The results of this project can theoretically explain the results obtained from numerical simulations and the phenomena observed in the experiments, and answer some open questions raised in some papers. The method here can be used to investigate the existence of multi-hump generalized solitary waves in other mathematical models.

本项目主要利用动力系统理论，特别是不变流形约化定理、正规形、分岔理论，来研究水波问题及其逼近模型多峰广义孤立波(孤立波有多个振幅相同的波峰且在无穷远处指数趋于一个小振幅的周期解)的存在性，特别是. (1) 二维、三维水波问题的逼近模型的多峰广义孤立波，. (2) 二维、三维水波问题的多峰广义孤立波，. (3) 高维常微分系统的多峰广义同宿轨，. (4) 非线性晶格方程的多峰广义孤立波。. 本项目的结果能从理论上解释数值模拟得到的结果和实验中观察到的现象，并回答一些文章提出的公开问题。所用方法可以用来研究其他数学物理模型多峰广义孤立波的存在性。

一些数学物理模型的多峰广义同宿轨(多峰同宿轨指数趋于一个小振幅的周期解)、水波问题中多峰广义孤立波(多峰孤立波指数趋于一个小振幅的周期解)已在数值模拟和实验中得到。本项目利用泛函分析、偏微分方程和动力系统理论等从理论上首次证明KdV-KdV系统多峰广义同宿轨的存在性，二维和三维水波问题多峰广义孤立波的存在性。同时把这些方法应用到差分模型中，分析其定性性质和分岔，特别是研究了反弹球模型、离散约化Lorenz系统、具有合作狩猎的离散捕食-被捕食模型、Ricker竞争模型、食叶-食草动物模型等系统的1:1共振、1:2共振、1:3共振、折-翻转分岔问题，所得结果解释了一些已观察到的生物现象。