Kinetic equation for a soliton gas, its hydrodynamic reductions and symmetries

We study a new class of kinetic equations describing nonequilibrium macroscopic dynamics of soliton gases with elastic collisions. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of N -component ‘cold-gas’ hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong indication to integrability of the full kinetic equation. We derive explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the soliton gas component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated.