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Kinetic equation for a soliton gas and Its hydrodynamic reductions. (English) Zbl 1252.37056
Summary: 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 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.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35L60 First-order nonlinear hyperbolic equations
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