Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

TL;DR

This paper studies the large-scale behavior of dense soliton gases in the KdV equation, showing that their kinetic description simplifies to classical modulation equations and exploring complex wave phenomena through explicit solutions and numerical simulations.

Contribution

It demonstrates that the kinetic equation for soliton condensates reduces to the KdV-Whitham modulation equations in the condensate limit, linking kinetic theory with classical integrable wave modulation.

Findings

Reduction of kinetic equation to KdV-Whitham equations in the condensate limit

Explicit solutions for Riemann problems showing rarefaction and shock waves

Numerical simulations revealing rich incoherent behaviors in diluted condensates

Abstract

We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special "condensate" limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the $N$-phase KdV-Whitham modulation equations derived by Flaschka, Forest and McLaughlin (1980) and Lax and Levermore (1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for "diluted" soliton condensates exhibiting rich incoherent behaviours associated with integrable turbulence.