Ziad Musslimani, Department of Mathematics, Florida State University

Thermalization of the discrete nonlinear Schrodinger equation: Theory, computation and experiments

I will present a comprehensive study related to the connection between thermalization of cubic nonlinear lattices with nearest-neighbor coupling and the structure of the mixing tensor that arises due to the presence of nonlinearities. The approach is based on rewriting the underlying lattice system as a nonlinear evolution equation governing the dynamics of the modal amplitudes (or projection coefficients). In this formulation, the linear coupling become diagonalizable, whereas all cubic nonlinear terms transform into a combinatorial sum over a product of three modal amplitudes weighted by a fourth-order mixing tensor. The exact structure of several mixing tensors (corresponding to different types of cubic nonlinearities) is found, and their symmetry properties are connected with thermalization or lack thereof. Furthermore, through direct numerical simulations, it is found that the modal occupancies of lattices preserving these tensorial symmetries approach a Rayleigh-Jeans distribution at thermal equilibrium. In addition, few examples that indicate that cubic lattices with broken tensorial symmetries end up not equilibrating to a Rayleigh-Jeans distribution are presented. Finally, an inverse approach to the study of thermalization of cubic nonlinear lattices is developed. The idea is to establish a trade-off between the type of nonlinearities in local base and their respective interactions in supermode base. With this at hand, we were able to identify a large class of nonlinear lattices that are embedded in the modal space and admit a simple form that can be used to shed more light on the role that localization of the supermodes plays in thermalization processes.