Mathematical Sciences: Invariant Manifolds and Attracting Sets of Nonlinear Partial Differential Equations

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The central theme of this mathematical research project is the analysis of nonlinear partial differential equations. Particular emphasis will be placed on extending methods which describe invariant manifolds and attracting sets of perturbations of integrable nonlinear equations to the case of higher dimensional semilinear differential equations. Work will also be done in developing methods for determining global coordinates that best describe the finite dimensional attracting sets. Application of these investigations to problems of physical importance include studies of the damped and driven sine-Gordon equation describing the Josephson junction and the analysis of a system of Euler-Raleigh-Plesset equations describing a cloud of gas bubbles in liquid. Additional work includes studies of perturbations that change the spatial structure of the perturbed equations; in particular the question of existence of breather solutions to these equations will be taken up. The bearing these answers have on the symplectic geometry of the phase space containing the solutions will also be considered. Long term objectives of this research include the application of bifurcation techniques developed in the study of perturbed integrable differential equations to improved understanding of general fluid equations.***