Kinetic equation for a soliton gas – a new integrable system?

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 finite-density generalisation [2] of Zakharov's equation represents a nonlinear integro-differential equation, which was shown in [3] to be related to the infinite-genus, thermodynamic limit of the Whitham modulation systems associated with finite-gap 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) 374-383. [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) 151-191. [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.