Critical Nonlinear Dispersive Equations

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This project will study dispersive partial differential equations. Such equations model many different phenomena, among them the propagation of various kinds of waves, such as water waves and laser light. These equations also model certain phenomena in particle physics. This project attempts to understand long-time behavior of such equations and related systems.There are a number of problems that will be studied in the course of this endeavor. The study will mainly revolve around the three well-known dispersive partial differential equations: wave, Schrodinger, and Korteweg de Vries. A great deal is unknown for the focusing Schrodinger and Korteweg de Vries (KdV) problems, particularly for the mass-critical problem. The PI intends to study the focusing, mass-critical Schrodinger problem for mass above the mass of the ground state, as well as the focusing gKdV problem for large mass below the mass of the ground state. The PI also plans to extend recent work with the I-method to the wave and Klein-Gordon equations. Finally, the PI will study the ultra hyperbolic Schrodinger equation.