Dynamics of Nonlinear Dispersive Equations

Dispersive equations are ubiquitous in nature, arising in areas such as water waves, optics, lasers, ferromagnetism, particle physics, general relativity, nonlinear elasticity, and many others. Examples are nonlinear Schrödinger equations that govern Bose–Einstein condensates, a fascinating phenomenon predicted by quantum statistical mechanics, for which the 2001 Nobel Prize in Physics was awarded following experimental verification. Solitons, or coherent solitary waves, are an extraordinary and still mysterious feature of solutions to dispersive equations. The project’s overarching goal is to establish (in)stability results for solitons, further study their dynamics even in the presence of (in)elastic collisions and blow-up phenomena and understand the soliton resolution conjecture for general solutions. The project provides research training opportunities for graduate students. Combining techniques from partial differential equations (PDE), harmonic analysis, asymptotic analysis, dynamical systems, and spectral theory, this project explores qualitative descriptions of the dynamics of dispersive waves from three distinct perspectives. The first objective is to understand the dynamics of multi-solitons in dispersive equations, focusing on their stability, uniqueness, and (in)elastic collisions. The second objective is to examine the asymptotic stability of (topological) solitons under the influence of long-range scattering and internal modes. The third objective is to utilize integrable structures to investigate the dynamics of physically important integrable systems and their perturbations. 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.