The formation of a soliton gas condensate for the focusing nonlinear Schrödinger equation

Abstract In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schrödinger equation as the number, N , of solitons grows to infinity. We discover configurations of N -soliton solutions which exhibit the formation (as $N \to \infty$ ) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov–Shabat operator for the nonlinear Schrödinger equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from 0, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of ( x , t ). We then consider more complex solutions with an extra soliton component, and we show that, in this deterministic setting, the kinetic theory of solitons applies. This is to be distinguished from previous analyses of soliton gases where the norming constants were tending to zero with N , and the asymptotic description only included elliptic waves in the long-time asymptotics.