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

本课题将发展三个关于偏微分方程解的长时间行为的课题。首先，为了研究弱湍流，我们尝试了耗散强迫非线性偏微分方程的低维吸引子的描述理论。该理论适用于一维阻尼驱动正弦-戈登方程；PI将把它扩展到更高维度的Klein-Gordon方程和Ginzburg- Landau方程。第二个主题是非线性保守双曲偏微分方程呼吸解的不存在性。第三个主题是描述气泡云方程和长波正则Boussinesq方程的孤立波的稳定性。这些问题将进行数值和分析研究。他们的解决方案将增进我们对基本数学和物理问题的理解。大气科学和流体力学等学科将受到该项目的影响。

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.