Analysis of Regular and Random Soliton Gases in Integrable Dispersive Partial Differential Equations.

The University Of Central Florida Board Of Trustees, Orlando FL

Investigators

Abstract

Nonlinear wave interactions describe phenomena observed throughout science and engineering, from the motion of water waves to fiber optic transmissions to the physics of plasmas relevant to star and fusion reactors. Despite the disparate physical contexts, similar wave patterns emerge, and their behavior can be modeled by the same nonlinear partial differential equations. In applications and real-world measurements, the nonlinear interactions lead to waves patterns of exceeding complexity and apparent randomness. This project will further the understanding of these waveforms by developing techniques to model them as accumulations of special isolated traveling wave solutions, i.e., solitons. The resulting ensembles of many solitons, known as soliton gases, will be studied directly and statistically, when their properties are allowed to behave randomly. The aim is to precisely describe, and ultimately control, the nonlinear wave dynamics observed in applications, for example in fiber optic transmission. The project will provide research training opportunities for undergraduate and graduate students. The project has three main mathematical goals: 1) to build a rigorous analytical description of soliton gases, via characterization of their spectral properties, and use this description to study their statistical properties; 2) to analyze the small dispersion semiclassical limit of the focusing nonlinear Schrodinger equation, as a mechanism for generating integrable turbulence and rogue waves from initial data; 3) to study the long-time behavior of spectrally singular solutions of integrable partial differential equations. The study will involve using and further developing tools from complex and asymptotic analysis, with an emphasis in the Inverse Scattering Transform method. The double scaling limit of the focusing nonlinear Schrodinger equation has the potential to introduce a new class of universality at wave breaking and further the understanding of rogue wave generation. Mathematically, the work on spectrally singular solutions and fat-tailed waves would lead to an extension of Inverse Scattering Transform techniques to broader classes of initial data inaccessible by existing techniques. 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.