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 mathematical activity in recent years. At the same time, there has been a surge of geometric approaches to the study of integrable systems in applied mathematics and mathematical physics. 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. (1) Integrable systems and curve/surface flows Using geometric frame methods, I have previously derived 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. This work will be extended and applied to obtain new integrable equations involving quaternionic and octonionic variables, which will be a large advance in the theory of integrable systems. I also will generalize some other earlier work on surface flows in Euclidean space to symmetric spaces and Lie groups,, which will lead to new integrable systems in 2+1 dimensions. (2) Peakon equations Peakons are travelling waves that have a peaked exponential profile. They arise in nonlinear dispersive wave equations connected to water wave theory. I have begun to work on several aspects of peakon equations, which includes finding a new peakon system related to the NLS equation. I have also shown that a very general family of wave equations possesses multi peakon solutions. I will continue to work on these two topics to understand and look at interesting features of peakon interactions. (3) Conserved integrals in fluid flow An open problem is to determine all conserved integrals for the fundamental equations of fluid flow. In recent work, I settled this problem for two important types of conserved integrals for compressible fluid flow in n>1 dimensions. I also extended the results by finding new conserved integrals on advected surfaces, which gave interesting generalizations of helicity and circulation. I plan to obtain carry out similar work for fluid flow with free boundaries, and for compressible magnetohydrodynamics. (4) Symmetries and conservation laws of nonlinear PDEs Conservation laws are very important in the study of nonlinear PDEs. For PDEs that have a Lagrangian, all conservation laws can be obtained from symmetries through Noether's theorem, but this connection fails for PDEs without a Lagrangian. In previous work published in several papers and a co-authored book, I have developed an algorithmic method to find the conservation laws admitted by any given PDE whether or not it has a Lagrangian. I plan to continue to develop and apply this method to PDEs of importance in applied mathematics and physics.