Asymptotic nonlinear spreading in chaotic billiards

Establish the asymptotic time behavior of nonlinear spreading toward high-energy linear modes for the nonlinear Schrödinger equation in chaotic billiards at moderate nonlinearity, specifying whether spreading persists indefinitely and characterizing the associated rates and scaling laws with respect to the nonlinearity strength and initial energy distribution.

Background

The paper studies dynamical thermalization of the nonlinear Schrödinger equation (NSE) in a D-shape (chaotic) billiard and contrasts regimes above and below a chaos border. While the KAM regime is expected at weak nonlinearity, rigorous mathematical results for partial differential equations in billiards are challenging.

In this context, the authors note that for chaotic billiards at moderate nonlinearity the long-time behavior of energy transfer (spreading) to high-energy modes is not established, highlighting a need for a precise characterization of asymptotic dynamics.