EL, G.A. ... et al., 2011. Kinetic equation for a soliton gas and its hydrodynamic reductions. Journal of Nonlinear Science, 21 (2), pp. 151-191.
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.
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 (quasi-periodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed the 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
The final publication is available at Springer via http://dx.doi.org/10.1007/s00332-010-9080-z.
The work has been partially supported by EPSRC (UK)(grant EP/E040160/1) and London Mathematical Society
(Scheme 4 Collaborative Visits Grant). Work of M.V.P. has been also supported by the
Programme "Fundamental problems of nonlinear dynamics" of Presidium of RAS. M.V.P.
and S.A.Z. also acknowledge partial financial support from the Russian-Taiwanese grant
95WFE0300007 (RFBR grant 06-01-89507-HHC).