Soliton gas in bidirectional dispersive hydrodynamics

The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the properties are determined by overtaking elastic pairwise interactions between solitons. In this paper, we extend to gases bidirectional Eulerian systems where both head-on and collisions solitons take place. We distinguish two qualitatively different types gases: isotropic gases, position shifts accompanying have same sign, anisotropic opposite signs. construct kinetic equations solve respective shock-tube problems collision ``monochromatic'' beams consisting approximately amplitude velocity. corresponding weak solutions differing uniform states separated contact discontinuities mean flow constructed. Concrete examples defocusing nonlinear Schr\"odinger (NLS) equation resonant NLS considered. is shown be equivalent that shallow-water described Kaup-Boussinesq equations. analytical results Riemann excellent agreement with direct numerical simulations.