Journal of Nonlinear Science

, Volume 21, Issue 2, pp 151–191 | Cite as

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

  • G. A. ElEmail author
  • A. M. Kamchatnov
  • M. V. Pavlov
  • S. A. Zykov


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.


Soliton gas Thermodynamic limit Kinetic equation Hydrodynamic reduction Integrability 

Mathematics Subject Classification (2000)

37K10 37K40 35L60 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • G. A. El
    • 1
    Email author
  • A. M. Kamchatnov
    • 2
  • M. V. Pavlov
    • 3
  • S. A. Zykov
    • 4
    • 5
  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK
  2. 2.Institute of SpectroscopyRussian Academy of SciencesTroitskRussia
  3. 3.Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia
  4. 4.SISSATriesteItaly
  5. 5.Institute of Metal PhysicsUrals Division of Russian Academy of SciencesEkaterinburgRussia

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