Riemann problem for polychromatic soliton gases: a testbed for the spectral kinetic theory

TL;DR

This paper validates the spectral kinetic theory for soliton gases in KdV and fNLS equations by comparing solutions of the kinetic equation with direct numerical simulations of dense, polychromatic soliton gases, using a new synthesis method.

Contribution

It introduces a new numerical method to synthesize dense polychromatic soliton gases and validates the spectral kinetic theory against direct simulations.

Findings

Spectral kinetic theory accurately predicts macroscopic observables.

New synthesis method effectively creates dense soliton gases.

Validation confirms the theory's applicability to complex soliton interactions.

Abstract

We use Riemann problem for soliton gas as a benchmark for a detailed numerical validation of the spectral kinetic theory for the Korteweg-de Vries (KdV) and the focusing nonlinear Schr\"odinger (fNLS) equations. We construct weak solutions to the kinetic equation for soliton gas describing collision of two dense "polychromatic" soliton gases composed of a finite number of "monochromatic" components, each consisting of solitons with nearly identical spectral parameters of the scattering operator in the Lax pair. The interaction between the gas components plays the key role in the emergent, large-scale hydrodynamic evolution. We then use the solutions of the spectral kinetic equation to evaluate macroscopic physical observables in KdV and fNLS soliton gases and compare them with the respective ensemble averages extracted from the "exact" soliton gas numerical solutions of the KdV and fNLS equations. To numerically synthesise dense polychromatic soliton gases we develop a new method which combines recent advances in the spectral theory of the so-called soliton condensates and the effective algorithms for the numerical realisation of $n$-soliton solutions with large $n$.