Nonlinear Wave Models in Domains with a Boundary

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

Physical phenomena in water waves, optics, and other fields of science naturally occur in spatially confined regions. Capturing in the modeling of such phenomena the behavior at the boundary of the region presents fundamental challenges, which in mathematical terms translates into questions of existence and uniqueness of solutions of dispersive equations and of sensitivity of these solutions to changes in initial and boundary conditions. This project will develop a methodology for deciphering the complex behavior of the equations modeling optics and water waves phenomena in bounded regions and will provide tools to understand and design physical and numerical experiments, especially in the direction of capturing novel unexplored behaviors. The investigator will engage both undergraduate and graduate students in the project, in line with a commitment to the training of future generations of mathematicians and the promotion of diversity in the field of mathematics and beyond. The project involves three main components. The first component considers nonlinear Schrödinger-type equations in higher than one space dimensions. The second focuses on equations in higher dimensional bounded domains with an emphasis on completely integrable models. This direction will exploit the rich mathematical structure of integrable equations and provide rigorous results on a currently largely unexplored area. The third studies multi-component systems of nonlinear shallow water wave equations in domains with a boundary. Such systems arise as approximations to the Euler equations of hydrodynamics; therefore, results achieved in this direction will be of value to the broader area of hydrodynamics. 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.