Macroscopic dynamics of incoherent soliton ensembles: Soliton gas kinetics and direct numerical modelling

We undertake a detailed comparison of the results of direct numerical simulations of the soliton gas dynamics for the Korteweg-de Vries equation with the analytical predictions inferred from the exact solutions of the relevant kinetic equation for solitons. Two model problems are considered: i) the propagation of a "trial" soliton through a one-component "cold" soliton gas consisting of randomly distributed solitons of approximately the same amplitude; and ii) the collision of two cold soliton gases of different amplitudes (the soliton gas shock tube problem) leading to the formation of an expanding incoherent dispersive shock wave. In both cases excellent agreement is observed between the analytical predictions of the soliton gas kinetics and the direct numerical simulations. Our results confirm the relevance of the kinetic equation for solitons as a quantitatively accurate model for macroscopic non-equilibrium dynamics of incoherent soliton ensembles.