Hamiltonian Partial Differential Equations

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Continuum vibrations in physical settings, like the motions of a lake's surface, light waves in a fiber optic channel or spacetime metric oscillations caused by the ringdown of a binary star system are mathematically modeled by equations like the Euler and Maxwell systems, the nonlinear Schrödinger and Einstein equations. Even though they arise in descriptions of wildly diverse physical phenomena, the equations that model continuum vibrations without friction enjoy rich mathematical similarities (e.g. conserved quantities like energy and mass, Hamiltonian formulation, symplectic phase space, ...) which define the conceptual framework for the field of Hamiltonian partial differential equations. The goal of the proposed research is to contribute to the general understanding of the maximal-in-time behavior of solutions of the initial value problem for Hamiltonian PDE. The aim is to produce new answers to the main question about evolution equations: what happens?