Mathematical Sciences: Mathematical Physics and Continuum Mechanics

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This research project is devoted to various algebraic and geometric aspects of the equations of continuum mechanics and mathematical physics. The principal areas of concentration include the equations of linear and nonlinear elasticity, Lie algebras and quasi-exactly solvable quantum problems, the Cartan equivalence method, bi-Hamiltonian systems, solitary waves, and nonlinear dissipation. Specific applications include: 1. Conservation laws, symmetries, canonical forms and invariants for the variational problems of linear and nonlinear elasticity, 2. The classification of Lie algebras of differential operators and quasi-exactly solvable Schrodinger operators, 3. Canonical forms and integrability of bi-Hamiltonian systems, 4. Nonlinear dissipation and normal forms for both ordinary and partial differential equations, 5. Equivalence problems from the calculus of variations, and 6. Existence and non-existence of solitary wave solutions to higher order model evolution equations. These research topics are relevant to the studies of important problems arising from fluid mechanics, elasticity, and quantum mechanics. Particular problems to be addressed include the investigation of solitary wave solutions to certain model equations for water waves, classification and application of a new and physically important class of quantum mechanical systems, understanding dissipation (friction and viscosity) in mathematical models of nonlinear systems, and the analysis of symmetry and conservation laws for linear and nonlinear elasticity.