The Mathematical Physics of Completely Integrable Hamiltonian Partial Differential Equations

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Among the many partial differential equations governing mathematical models in physics, a small number stand out by being completely integrable - a special class of equations that possess a large number of conserved quantities, making them analytically solvable. While these models have always been central figures of the scientific landscape, there has been dramatic progress on some fundamental mathematical questions in recent years. One of the foundations for the current project is joint work of the Principal Investigator (PI) that constructed solutions for greatly enlarged classes of initial states and showed that such solutions are well-behaved. A second foundation is a new kind of explicit representation of solutions introduced by Gerard and collaborators. In very recent work, the PI has already demonstrated the great potential of combining these approaches. The main thrust of this project is to employ these tools in tackling two families of classical questions about physical systems governed by completely integrable equations. First, the project will seek to elucidate long-time behavior, with particular emphasis on models that describe interfacial waves. The long-time behavior of such waves has resisted traditional techniques in the study of integrable systems. Thus, it provides a most impactful opportunity in which to employ the new methods. The second major avenue of research is in the statistical mechanics of integrable systems. A major premise of this project is that the favorable mathematical structures of integrable systems provide an excellent opportunity to make progress on the foundational questions of the subject. To this end, the PI will develop dynamical theories of integrable systems in thermal equilibrium and begin to probe their ergodic properties. The project provides significant research training for graduate students and postdoctoral scholars; indeed, junior researchers are integrated into every major activity of the project. It also provides opportunities for the early-career participants to disseminate their work, to learn from other researchers, and to hasten their scientific development. The long-time behavior of the Benjamin-Ono and continuum Calogero-Moser models will be investigated by synthesizing ideas from the method of commuting flows and the explicit formulae introduced by Gerard and collaborators. This includes understanding both multi-soliton and radiative solutions as well as solutions combining both behaviors. The separatrix between solitonless and soliton-bearing solutions will also be investigated. The project seeks to construct dynamics in the Gibbs state for a variety of completely integrable systems, including the Continuum Calogero-Moser and Landau-Lifshitz equations. Such constructions form the foundation for the investigation of the ergodic properties of such states. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.