Nonlinear Interactions and Dynamics in Problems From Fluids and Optics

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The goal of this project will be to study theoretically and computationally quasilinear dispersive partial differential equations related to Hamiltonian models in mathematical physics from nonlinear optics, water waves, many body quantum systems and plasma physics. The equations of interest come from families of Schrödinger, Dirac, Korteweg-de Vries, and gravity-capillary wave equations. In addition, the PI hopes to continue to explore the strongly nonlinear effects relating to frequency cascades in Hamiltonian models on compact domains. The bulk of the project will focus on understanding theoretically the existence and regularity of solutions to strongly nonlinear models, as well as studying local and global dynamics of special solutions within these complex models. However, a component will also consist of using modern functional analytic techniques to study convergence of numerical methods and to study validity of discrete approximations to these models. Such analysis can serve as motivation for dynamics, as well as a means of testing asymptotic limits where direct analysis may no longer predict precise dynamics. In addition, the PI will work to develop a functional analytic framework for studying stability under stochastic perturbations of quasilinear models and computational approximations. Quasilinear partial differential equations arise in models where curvature has a strong influence on the underlying physics. Hence, surface tension in fluids, surface energy in crystals or relativistic effects in optics can introduce interaction terms that strongly depend upon higher order derivatives of the solution. It is the aim of this project to shed light on in what sense these models have solutions and in particular understand precise asymptotic descriptions of the solutions when possible. Capillary waves might be useful for measuring the surface signature of internal water waves, crystal relaxation plays a role in semiconductor fabrication and ultra-short pulse lasers have appeared in many recent physics applications and plasma generation. From a human resources standpoint, quasilinear problems provide a large pool of problems that can be used for training purposes in work with researchers of all levels, from undergraduate to postdoctoral. In particular, undergraduates can learn some of the innate difficulties in quasilinear equations by numerically analyzing simple 1,2-dimensional toy-model systems, graduate students can learn the functional analysis framework and develop the techniques through applications to reduced models or critical equations, and postdocs can collaborate on projects to fully explore the models, develop techniques and work towards optimal results in stability theory and phenomenology. Such approaches give trainees and the PI a broad class of analytic and computational tools to use for solving many interesting problems, allowing them to work towards a full enough understanding to consider complex nonlinear inverse problems and a large set of open nonlinear scattering problems to work on well into their careers.