Condensation with two constraints and disordered Discrete Non Linear Schr\"odinger breathers

Abstract

Motivated by the study of breathers in the disordered Discrete Non Linear Schr\"odinger equation, we study the uniform probability over the intersection of a simplex and an ellipsoid in $n$ dimensions, with quenched disorder in the definition of either the simplex or the ellipsoid. Unless the disorder is too strong, the phase diagram looks like the one without disorder, with a transition separating a fluid phase, where all variables have the same order of magnitude, and a condensed phase, where one variable is much larger than the others. We then show that the condensed phase exhibits "intermediate symmetry breaking": the site hosting the condensate is chosen neither uniformly at random, nor is it fixed by the disorder realization. In particular, the model mimicking the well-studied Discrete Non Linear Schr\"odinger model with frequency disorder shows a very weak symmetry breaking: all variables have a sizable probability to host the condensate (i.e. a breather in a DNLS setting), but its localization is still biased towards variables with a large linear frequency. Throughout the article, our heuristic arguments are complemented with direct Monte Carlo simulations.