Hamiltonian偏微分方程的共振动力与高频扩散

Many partial differential equations arising in physics can be seen as infinite dimensional Hamiltonian systems, such as the nonlinear Schrödinger equation and the KdV equation. The energy exchange between resonant Fourier modes in nonlinear Schrödinger equations and complex KdV equations is studied. We prove the existence of unbounded Sobolev orbits and long-time instability for the high order and coupled nonlinear Schrödinger equations on two-dimensional torus by analyzing the heteroclinic orbits of the truncation ODEs and the growth of Sobolev norms. By using the spectral asymptotic properties of differential operators and almost resonant conditions, we prove the existence of unbounded Sobolev orbits and long-time instability for the nonlinear Schrödinger equations with multiplicative potential. By using the Birkhoff norms and Shilnikov method, we give estimates for the time with respect to the growth of the Sobolev norms. The research of this project will enrich the theory of Hamiltonian systems and promote the development of related disciplines.

物理学中许多偏微分方程都可以看作无穷维Hamiltonian系统，例如非线性Schrödinger方程和KdV方程。本项目研究非线性Schrödinger方程和复KdV方程的共振傅里叶模之间的能量交换。根据对截断的常微分系统的异宿轨道和解的Sobolev范数增长的分析，证明二维环面上高次非线性Schrödinger方程和耦合的非线性Schrödinger方程无界Sobolev轨道的存在性和长时间不稳定性。关于具有乘积势的非线性Schrödinger方程，通过使用微分算子的谱渐近性质和几乎共振条件，证明无界Sobolev轨道的存在性和长时间不稳定性。利用Birkhoff标准型和Shilnikov方法，给出Sobolev范数增长的时间估计。本项目的研究将丰富Hamiltonian系统的理论，促进相关学科的发展。

物理学中许多偏微分方程都可以看作无穷维Hamiltonian系统，例如非线性Schrödinger方程和KdV方程。本项目研究五次非线性Schrödinger 方程的非横截异宿链的存在性, 基于一个约化的有限维常微分系统, 构造了该系统的非横截异宿链, 给出了非横截异宿轨道的显式表达式. 研究了一个扰动mKdV方程的周期波和异宿解的存在性，并通过分析阿贝尔积分证明了波速是单调减少的. 证明了改变的Camassa-Holm 方程的椭圆周期peakon的轨道稳定性. 通过使用 Picard–Fuchs 方程，证明了Friedmann–Robertson–Walker 模型有两个等时中心，从而有两族等周期的周期解。本项目的研究丰富了Hamiltonian系统的理论，促进相关学科的发展。