Mathematical Sciences: Attractors, Weak-Turbulence and Nonexistence of Breathers

这个项目将发展三个关于偏微分方程解的长时间行为的主题。首先，尝试用耗散强迫非线性偏微分方程组的低维吸引子的描述理论来研究弱湍流。这一理论适用于一维受阻的Sine-Gordon方程；PI将其推广到更高维的Klein-Gordon方程和Ginzburg-Landau方程。第二个主题是非线性守恒双曲型偏微分方程解的不存在性。第三个主题是描述气泡云的方程和长波正则化的Boussinesq方程的孤立波的稳定性。这些问题将以数值和解析的方式进行研究。他们的解决方案将有助于我们理解基本的数学和物理问题。除其他外，大气科学和水动力学将受到该项目成果的影响。

This project will develop three topics concerning long time behavior of solutions to partial differential equations. First a descriptive theory of low dimensional attractors of dissipative forced nonlinear PDEs will be attempted in order to investigate weak turbulence. This theory exists for the damped and driven sine-Gordon equation in one dimension; the PI will extend it to the Klein-Gordon equations in higher dimensions and the Ginzburg- Landau equation. The second topic is the nonexistence of breather solutions to nonlinear conservative hyperbolic PDEs. The third topic is the stability of solitary waves for equations describing a bubble cloud and long-wave regularized Boussinesq equations. These problems will be studied numerically and analytically. Their solutions will accrue to our understanding of basic mathematical and physical issues. Atmospheric sciences and hydrodynamics, among others, will be impacted by the results of this project.