Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. 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 evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the ‘cold-gas’ component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and Communicated by T. Fokas. G.A. El ( ) Department of Mathematical Sciences, Loughborough University, Loughborough, UK e-mail: [email protected] A.M. Kamchatnov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, Russia M.V. Pavlov Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia S.A. Zykov SISSA, Trieste, Italy S.A. Zykov Institute of Metal Physics, Urals Division of Russian Academy of Sciences, Ekaterinburg, Russia 152 J Nonlinear Sci (2011) 21: 151–191 investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.