Growth or decay of a coherent structure interacting with random waves.

Abstract

Solitary waves interacting with random Rayleigh-Jeans distributed waves of a nonintegrable and noncollapsing nonlinear Schrödinger equation are studied. Two opposing types of dynamics are identified: First, the random thermal waves can erode the solitary wave; second, this structure can grow as a result of this interaction. These two types of behavior depend on a dynamical property of the solitary wave (its angular frequency), and on a statistical property of the thermal waves (the chemical potential). These two quantities are equal at a saddle point of the entropy that marks a transition between the two types of dynamics: high-amplitude coherent structures whose frequency exceeds the chemical potential grow and smaller structures with a lower frequency decay. Either process leads to an increase of the wave entropy. We show this using a thermodynamic model of two coupled subsystems, one representing the solitary wave and one for the thermal waves. Numerical simulations verify our results.