New Boundary-Value Problem Techniques for Nonlinear Wave Problems

University Of Washington, Seattle WA

Investigators

Abstract

The PI and his students and collaborators will further develop the new methods of Athanassios Fokas for boundary value problems to investigate two different problems. First, the periodic problem for integrable differential equations will be revisited using this new approach, with the goal of obtaining a more efficient (from the point of view of asymptotics and numerics) answer than the current approach. Second, the PI and his students will use the new approach to analytically investigate the linear stability of different problems posed on non-standard domains such as the half-line or finite intervals with prescribed boundary conditions. In addition, the new formulation of the water wave problem due to Ablowitz, Fokas and Muslimmani will be used to initiate the solution of the inverse water wave problem: the problem of recovering the bottom topography using measurements of the water wave surface. Using and expanding on new methods for studying problems with prescribed input at their boundaries, the PI and his collaborators and students will investigate problems describing a wide range of phenomena. Some of these application areas are relevant for waves on the surface of the ocean (tsunamis and rogue waves), while others impact engineering applications with wide sociological impact such as high-speed communication (nonlinear optics) or help with our understanding of current environmental problems (smog formation and dispersal). Lastly, the PI and his research group will initiate a new approach to determine the surface at the bottom of the near-shore ocean from remote sensing measurements of the surface. This work has direct commercial and defense naval consequences. The PI is committed to the training of the next generation of US scientists, on a local scale by working with students, and on a more global scale by preparing a text book describing the techniques used.