Symmetries, conserved integrals, Hamiltonian flows, and integrable systems.

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Symmetries, conserved integrals, Hamiltonian structures, and related aspects of PDEs have attracted much activity in recent years. At the same time, there has been a surge of geometric approaches to the study of integrable PDE systems. My proposal builds significantly on these directions of research in my current grant. Many parts are well suited for MSc projects, PhD theses, and postdoctoral work, as well as contributions by undergrad research students.Bi-Hamiltonian integrable systems and curve/surface flows:Using geometric frame methods, I recently derived universal multi-component integrable systems which are group-invariant generalizations of nonlinear Schrodinger (NLS), modified Korteveg de Vries (mKdV), sine-Gordon (SG) equations with a bi-Hamiltonian structure, arising from curve flows in Riemannian symmetric spaces and semisimple Lie groups.In work underway with MSc students, we are deriving new types of NLS/mKdV eqns. from Hermitian and Lorentzian symmetric spaces. I plan to derive new quaternion and matrix types of eqns.from quaternion and Grassmannian spaces. I also plan to apply similar methods to an important open problem of how to derive group-invariant integrable systems from curve flows in homogeneous spaces. I intend to extend my method to derive Lax pairs. This will lead to a geometric understanding of Drinfeld&Sokolov's universal construction of Lax pairs in affine Lie algebras, which is a central result in the theory of integrable systems. I also intend to lift the Lax pairs and bi-Hamiltonian structures from the frame variables back to the curve flow. This will give an explicit formulation of abstract results known on Lax pairs and Poisson brackets.Generalizing work I have done in R^3, I plan to derive 2+1 dimensional integrable systems from geometric surface flows in symmetric spaces/Lie groups. The results will have a large impact because only a few types of such systems are presently known.Multi-component solitons:In work with students, we are studying interaction of 2-component mKdV solitons. Compared to the 1-component case, these solutions exhibit interesting, new features, including formation of "rogue waves", which is a very active topic in applied math. We will go on to n-component solitons. Conserved integrals:An open problem is to determine all conserved integrals for fundamental equations of fluid flow. In work with a PhD student on inviscid compressible fluid eqns., we derived all kinematic and vorticity integrals on moving domains in R^n. I recently extended the results to moving surfaces, yielding interesting generalizations of helicity and entropy circulation. I plan to obtain all such conserved integrals on moving domains/surfaces for several other important fluid eqns. in R^n and in Riemannian manifolds. The results will be of basic interest in the geometrical/topological study of n-dim. hydrodynamics elaborated by Arnold&Khesin. Symmetries, conservation laws, and exact solutions of nonlinear PDEs:Symmetry-invariant solutions are very useful for physical and mathematical reasons. With collaborators, I developed a novel symmetry method based on geometrical group foliations to find explicit invariant and non-invariant solutions to PDEs with power nonlinearities. For the nonlinear wave/heat equations we considered, these solutions have interesting analytical behavior related to blow-up, attractors, decay/dispersion, including cases of critical powers. I plan to continue developing this method and apply it to other physically and analytically interesting PDEs. This will have a large impact in symmetry methods and analysis of PDEs.I also plan to explore new connections between symmetries and conservation laws, stemming from work with Bluman summarized in our recent book.