数学物理模型孤立波的存在性及稳定性: 空间动力学方法

Solitary wave solutions have been found in mathematics, fluid mechanics, biology and many other fields. It has remarkable applications in a series of high-tech fields such as nonlinear optics, flux quantum, fiber communications and so on. The study of its existence and stability in the mathematical-physical models has always been one of the hottest topics in the world. Applying the spatial dynamic approach, we will study the mathematical-physical models and in particular the following problems: (1) the existence of the solitary wave solutions of the two-dimensional water wave problems near the critical point, the existence of the multi-solitary (or solitary) wave solutions of the three-dimensional water wave problems (or with a moving bottom); (2) the Swift-Hohenberg equation and its generalized equations: the existence of the homoclinic (heteroclinic) orbits, generalized homoclinic orbits or generalized heteroclinic orbits (homoclinic or heteroclinic orbit exponentially approaching a nonzero periodic solution), one-dimensional or two-dimensional (uniformly translating in a horizontal propagation direction x and periodic in a transverse direction y) solitary wave solutions or generalized solitary wave solutions (solitary wave solution exponentially approaching a nonzero periodic solution as x tends to positive and negative infinity), the multi-solitary wave solutions; (3) the stability of solitary wave solutions of the Swift-Hohenberg equation and its generalized equations.

孤立波在数学、流体力学、生物等许多领域里被发现。它已在非线性光学、磁通量子器 件及光纤通讯等一系列高科技领域有了令人瞩目的应用。对它在数学物理模型中存在性及稳定性的研究一直是国内外同行关注的热门方向之一。 本项目利用空间动力学方法来研究数学物理模型，特别是： （1）二维水波在临界点附近孤立波的存在性，三维水波多孤立波的存在性，底部移动的三维水波孤立波的存在性； （2）Swift-Hohenberg方程及其广义方程：同(异)宿轨、广义同宿轨或广义异宿轨(同宿轨或异宿轨指数趋于一个非零周期解)、一维或二维(沿一个方向x传播而沿另一方向y是周期)孤立波或广义孤立波(当x趋于正负无穷时，孤立波指数趋于一个非零周期解)、多孤立波的存在性； （3）Swift-Hohenberg方程及其广义方程孤立波的稳定性。

本项目主要利用泛函分析、偏微分方程和动力系统理论来研究:. （1） Swift-Hohenberg方程及其广义方程的周期解、同宿轨、波前解、多峰广义同宿轨（带有多个峰的同宿轨指数趋于一个小振幅的周期解）等的存在性；. （2） 其他连续数学物理模型如耦合Shrodinger系统的广义异宿轨（异宿轨指数趋于一个小振幅的周期解）、广义同宿轨、波前解、行波解等；. （3） 具有自引力的二维气态星球广义音叉分岔；. （4） 一些有实际意义的差分方程如离散Guzowska-Luis-Elaydi模型的定性性质；. （5） Gronwall-Bellman不等式及相关不等式的推广。. 本项目所得的一些结果从理论上验证了数值模拟的结果，回答了一些文章提出的未解决的问题，所用方法可以用来研究其他数学物理模型的定性性质。