Infinite Dimensional Dynamical Systems and Partial Differential Equations

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Professor Wayne will study the behavior of infinite dimensional dynamical systems such as the Fermi-Pasta-Ulam model, the Navier-Stokes equations and the Euler equations. He will use methods from dynamical systems theory to make qualitative and quantitative predictions about the solutions of these systems and will focus on four main areas: (i) The stability and interactions of solitary wave solutions in infinite dimensional dispersive Hamiltonian systems, and the geometry of the phase space of such systems; (ii) The derivation and justification of approximate equations for the evolution of wave packets on a fluid surface and the relation of such results to normal form theorems for Hamiltonian systems in which the linear part has continuous spectrum; (iii) Metastable behavior in the nearly inviscid Navier-Stokes equations and other weakly dissipative systems; and (iv) The use of invariant manifold theorems to analyze singularly perturbed partial differential equations. The geometrical properties of objects like invariant manifolds have been a great aid in illuminating the behavior of finite dimensional dynamical systems and this project will develop similar methods and insights into the behavior of infinite dimensional systems, particularly those defined on unbounded spatial regions where the linear problem has continuous spectrum.The differential equations that Professor Wayne will study arise in a variety of different physical circumstances and are characterized by the fact that while the equations themselves are well known they are too complicated to solve except in special and/or unrealistic cases. Nonetheless, applications require at least a qualitative understanding of the behavior of their solutions and this project will develop such an understanding for the equations enumerated above. As an example related to point (i) in the preceding paragraph, the equations that describe waves on the ocean, which are of importance both for understanding climate and weather and for predicting events such as tsunamis, have a family of solutions known as ``solitary waves'' which represent a single wave traveling across theocean. In practice, however, many waves are inevitably present and it becomes necessary to understand how these waves interact with each other. This project will study the types of interactions that can occur in such systems and their consequences. The remaining three sections of the research project will aim, in a similar fashion, to extend the understanding of special or limiting cases of equations of physical importance to more realistic conditions.