Partial differential equations, Hamiltonian dynamical systems, and their applications

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The overall goals of this research proposal are to analyse the behavior and to address the singularities of solutions of a class of partial differential equations (PDEs) that govern a variety of physical phenomena involving wave propagation. Most of the principal equations which describe fundamental physical processes are differential equations; conservation laws in the form of evolution equations which describe the future state of a physical systems whose initial state is prescribed among a phase space of possible states. When a process involves wave propagation, these equations are generally PDEs, and those of physical or of geometric significance can often be formulated as Hamiltonian dynamical systems, whose phase space is naturally an infinite dimensional space of states. Examples include the nonlinear Schroedinger equation, many versions of nonlinear wave equations, the Euler equations for inviscid fluid dynamics, the problem of ocean waves, and the myriad of nonlinear dispersive equations such as the Korteweg deVries equation and the Boussinesq system that arise in fluid dynamics and plasmas.