Nonlinear dispersive wave equations,in particular the nonlinear Schrodinger, nonlinear Klein-Gordon, and Korteweg-de Vries equations, can exhibit coherent and stable solutions called solitons, as well as solutions that experience blow-up in finite time. The principal investigator will prove rigorous results that give quantitative descriptions of solutions to these equations in a number of scientifically relevant contexts. Among these are results describing the effect of a small random external field on the dynamics of solitons or blow-up solutions, the effect of interaction forces between multiple solitons on their dynamics, the stabilizing influence of a rapidly oscillating nonlinear coefficient on a soliton, and the extent to which a strong confining external field can reduce the effective dimension of the dynamics. The principal investigator has previously developed new techniques based on symplectic projection, and its relation to the Hamiltonian structure of these equations, to study related problems. He will continue to refine and extend his methods to address these more sophisticated problems.
The equations studied in this project arise as important physical models. Solitons appear as well-localized stable structures, while blow-up is associated with a sharp focusing of a wave and ultimate break-down of the physical model. For example, the nonlinear Schrodinger equation is the wave function for a collection of ultracold atoms in a Bose-Einstein condensate. Since its Nobel-prize-winning realization in the laboratory in 1995, further experiments have produced solitons and finite-time blow-up, and there is now a large body of physics literature motivating many of the problems this project will consider. This work brings together pure mathematics, numerical analysis, and physics. The principal investigator collaborates with faculty at several institutions in the U.S. and overseas and supervises several graduate students on work related to the project. Certain aspects of the project can be adapted to be accessible to undergraduates, especially those components involving computer simulation and the production of web demonstrations.