Bound States in a Gas of Solitons Supported by a Randomly Fluctuating Force

Abstract

A non-linear Schrödinger equation with small terms accounting for dissipation and a driving force randomly varying in time is considered. Physical applications of this model are, e.g., Langmuir waves in a plasma driven by a random electric field, or a randomly pumped non-linear optical fibre. The analysis is developed for the “high-temperature” case, when the drive is essentially stronger than the dissipation. In this case, it is possible to introduce a mean potential of the soliton-soliton interaction, defined as the known usual potential (containing an oscillatory tail generated by the dissipative term) averaged over an equilibrium distribution of the soliton's amplitude, which is produced by the corresponding Fokker-Planck equation. It is demonstrated that the mean potential contains a set of local minima, which should give rise to bound states in the rarefied gas of solitons supported by the random drive. An equilibrium separation between the solitons in the bound states depends, in the “high-temperature” approximation, only on the dissipative constant, but not on the “temperature” (mean-squared amplitude of the random drive).