This project will study dynamical systems with infinitely many degrees of freedom which arise in mathematical physics through techniques such as the Kolmogorov-Arnold-Moser theory, invariant manifold theorems, and Liapunov exponents. First, the project will focus on the relationship between techniques like the Kolmogorov-Arnold-Moser theory, and the accelerated convergence methods developed to study localized states in Schrodinger operators with random or quasi-periodic potentials, and will attempt to exploit these relationships to prove the existence of periodic orbits in partial differential equations. Second, it will investigate how ideas like center manifold theorem can be combined with a Hamiltonian structure to study partial differential equations like the Boussinesq equation and other equations related to hydrodynamics. Finally, it will study the existence of quantities like the Liapunov exponents for infinite dimensional dynamical systems arising in statistical mechanics. A better understanding of such quantities would give one a dynamical analogue of the thermodynamic limit in equilibrium statistical mechanics.
