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

这个项目将发展三个主题， 偏微分方程解的性质。 首先 耗散系统低维吸引子描述理论 强迫非线性偏微分方程将试图研究 弱湍流 这个理论存在于阻尼和驱动 sine-Gordon方程; PI将其扩展为 更高维度的Klein-Gordon方程和Ginzburg方程 朗道方程。 第二个主题是不存在 非线性保守双曲型偏微分方程的呼吸子解 第三个题目是方程孤立波的稳定性 描述了一个气泡云和长波正则化Boussinesq 方程 这些问题将被研究数值和分析。 他们的解决方案将有助于我们理解 数学和物理问题。 大气科学和 水动力学，除其他外，将受到影响的结果， 这个项目

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.