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

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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.