

International Workshop «Geometric Structures in Integrable Systems»
October 30, 2012 10:10–10:50, Moscow, M.V. Lomonosov Moscow State University






Kinetic equation for a soliton gas – a new integrable system?
G. A. El^{} ^{} Department of Mathematical Sciences, Loughborough University

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Abstract:
In 1971 V.E. Zakharov introduced a kinetic equation describing dynamics of a
spectral distribution function in a rarefied soliton gas – an infinite
random ensemble of KdV solitons distributed on the line with nonzero small
density [1]. The finitedensity generalisation [2] of Zakharov's equation
represents a nonlinear integrodifferential equation, which was shown in [3]
to be related to the infinitegenus, thermodynamic limit of the Whitham
modulation systems associated with finitegap solutions of the KdV equation.
Recent studies [4], [5] have revealed a number of remarkable properties of
the new kinetic equation, which, in particular, has been shown to possess an
infinite number of integrable hydrodynamic reductions. This is a strong
evidence in favour of integrability of the full kinetic equation in the
sense yet to be understood. Construction of kinetic theory of soliton gases
is part of the general programme of the development of turbulence theory in
integrable systems [6].
References:
[1] V.E. Zakharov, Kinetic equation for solitons. Sov. Phys. JETP 33
(1971) 538–541. [2] G.A. El, A.M. Kamchatnov, Kinetic equation for a dense
soliton gas, Phys. Rev. Lett. 95 (2005) 204101. [3] G.A. El, The
thermodynamic limit of the Whitham equations, Phys. Lett. A, 311
(2003) 374383. [4] G.A. El, A.M. Kamchatnov, M.V. Pavlov, S.A. Zykov,
Kinetic equation for a soliton gas and its hydrodynamic reductions, J.
Nonlin. Sci. 21 (2011) 151191. [5] G.A. El, M.V. Pavlov, V.B.
Taranov, Generalised hydrodynamic reductions of the kinetic equation for
soliton gas, Theor. Math. Phys. 171 (2012) 675–682. [6] V.E.
Zakharov, Turbulence in integrable systems. Stud. Appl. Math. 122
(2009) 219–234.
Materials:
el_maks50_2.pdf (236.4 Kb)
Language: English

