Equilibrium states of Burgers and KdV equations

TL;DR

This paper investigates the equilibrium states of the KdV and Burgers equations through simulations with random initial conditions, showing they maintain zero energy flux and exhibit Gaussian and Boltzmann distributions, confirming their equilibrium nature.

Contribution

It demonstrates that both equations reach equilibrium states characterized by specific statistical distributions and introduces energy flux as a criterion for equilibrium.

Findings

Energy flux remains zero in both equations.

Equilibrium states follow Gaussian and Boltzmann distributions.

Single soliton in KdV also exhibits zero energy flux.

Abstract

We simulate KdV and dissipation-less Burgers equations using delta-correlated random noise as initial condition. We observe that the energy fluxes of the two equations remain zero throughout, thus indicating their equilibrium nature. We characterize the equilibrium states using Gaussian probability distribution for the real space field, and using Boltzmann distribution for the modal energy. We show that the single soliton of the KdV equation too exhibits zero energy flux, hence it is in equilibrium. We argue that the energy flux is a good measure for ascertaining whether a system is in equilibrium or not.