CAREER: Emerging Challenges in Wave Turbulence Theory

Southern Methodist University, Dallas TX

Investigators

Abstract

Wave turbulence (WT) is a general physical phenomenon describing the nonlinear dynamical interactions of waves far from thermal equilibrium. Examples of wave turbulence occur in classical surface water waves and also in quantum dynamical systems involving a Bose-Einstein condensate (BEC). Even though wave fields describing the processes of random wave interactions in nature are enormously diverse, there is a common mathematical framework that models and describes the dynamics of spectral energy transfer through probability densities associated with weakly non-linear interactions in quantum or classical wave systems. The probability densities are solutions of wave kinetic (WK) equations, whose nonlocal interaction operators are of kinetic type. This project aims to tackle open challenges in the theory of nonlinear waves via the study of the associated wave turbulence theory. An integral part of the project is its educational component and the opportunities to involve students from all levels in the research. To this end, the principal investigator will organize summer schools for underrepresented and disabled K-12 students, design graduate courses on Partial Differential Equations, Wave Turbulence and Statistical Physics, and organize an undergraduate and graduate research internship program of excellence and a mathematics-physics conference for young researchers. In addition, the principal investigator will participate in the NSF RTG SMU summer undergraduate research program, with participants from both SMU and Texas Rio Grande Valley University (a Hispanic-serving institution) as well as the SMU Hamilton Undergraduate Research Scholars and the SMU Undergraduate Research Assistant Programs. This project concentrates on three main topics in WT theory. The first topic is to study the rigorous justification of 3-wave kinetic equations, by the Feynman diagrammatic approach. The second topic is to establish a mathematical foundation for the Garrett-Munk spectrum of the ocean, using the renormalization group method. The third topic is the proof of the finite time formation of singularities of solutions to the finite temperature BEC system, with the addition of a new, previously missing, collision operator derived by Yves Pomeau and the principal investigator. Several ideas from kinetic theory, dispersive equations, oceanography, and quantum mechanics will be combined to study the proposed problems. 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.