Equilibrium states of Burgers and Korteweg-de Vries equations

We simulate Korteweg-de Vries (KdV) and dissipationless Burgers equations using δ-correlated random noise as an 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.