Research in Integrable Nonlinear Waves and Related Topics

This project concerns properties of solutions of nonlinear evolution equations and related ordinary differential equations that model the propagation of large-amplitude waves in physical media such as surface and internal water waves or electromagnetic waves in optical fibers. The equations studied are completely integrable systems, for which there are many more analytical techniques available than for more general equations. However, at the same time integrable equations are known to arise from more general equations in certain limiting cases. Each time that a new result is established for a completely integrable model that arises in this way, a version of that result immediately applies to all of the more general models. These properties make the study of integrable models both mathematically compelling and also of prime physical relevance. This project uncovers new solutions of integrable equations of recognized importance and studies their exact and asymptotic properties. Knowledge of the solutions obtained impacts several application areas such as marine engineering (prediction/properties of large-amplitude surface water waves or "freak waves" and deformations of the free surface between layers in a density-stratified ocean) and condensed matter physics (Bose-Einstein condensation). As part of the project, computer codes and a textbook are being produced for public consumption. Three PhD students are being partially supported to work on the project. Specifically, this project addresses open problems in the theory of internal waves in fluids and related evolution equations that are fundamentally nonlocal. Long-time asymptotics for the Benjamin-Ono equation are being established, and soliton gases are being constructed for this equation. A nonlocal model of nonlinear Schroedinger type is being studied. Pivoting to more classical nonlinear Schroedinger equations and their relatives, dynamical stability of slowly-decaying rogue waves on the zero background ("rogue waves of infinite order") is being analyzed, uniformity of asymptotic behavior of such solutions in the large is being determined, a mechanism for generating such waves from extreme self-focusing is being explored in several models, a family of solutions continuously interpolating between rogue waves and solitons is being analyzed, and a universality result is being proved for the first time in the setting of the defocusing nonlinear Schroedinger equation. The project also addresses open questions in the theory of Painleve equations, establishing an isomonodromic representation for general special-function solutions and studying asymptotic limits in the parameter space for which in some cases there are outstanding conjectures in the literature. 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.