Macroscopic dynamics of incoherent soliton ensembles: soliton-gas kinetics and direct numerical modelling

We undertake a detailed comparison of the results of direct numerical simulations of the soliton gas dynamics for the Korteweg – de Vries equation with the analytical predictions inferred from the exact solutions of the relevant kinetic equation for solitons. Two model problems are considered: (i) the propagation of a ‘trial’ soliton through a one-component ‘cold’ soliton gas consisting of randomly distributed solitons of approximately the same amplitude; and (ii) the collision of two cold soliton gases of different amplitudes (the soliton gas shock tube problem) leading to the formation of an expanding incoherent dispersive shock wave. In both cases excellent agreement is observed between the analytical predictions of the soliton gas kinetics and the direct numerical simulations. Our results confirm relevance of the kinetic equation for solitons as a quantitatively accurate model for macroscopic non-equilibrium dynamics of incoherent soliton ensembles. Introduction. – Dynamics of incoherent nonlinear dispersive waves have been the subject of very active research in nonlinear physics for several decades, most notably in the contexts of ocean wave dynamics and nonlinear optics (see e.g. [1–3]). Two major areas where statistical properties of random ensembles of nonlinear waves play essential role are wave turbulence and rogue wave studies (see [4, 5] and references therein). A very recent direction in the statistical theory of nonlinear dispersive waves introduced by V.E. Zakharov is turbulence in integrable systems [6]. It was suggested in [6] that many questions pertinent to a classical turbulent motion can be meaningfully formulated in the framework of completely integrable systems. The physical relevance of integrable turbulence theory has been supported by recent fibre optics experiments [7]. Solitons play the key role in the characterisation of nonlinear wave fields in dispersive media, therefore the theory of soliton gases in integrable systems comprises an important part of the general theory of integrable turbulence [6]. The very recent observations of dense statistical ensembles of solitons in shallow water wind waves in the ocean [8] well modelled by the KdV equation provide further physical motivation for the development of the theory of soliton turbulence in integrable systems. In this paper we shall be using the KdV soliton gas as a simplest analytically accessible model yielding a major insight into general properties of soliton gases in integrable systems. Macroscopic dynamics of a KdV soliton gas are determined by the fundamental ‘microscopic’ properties of two-soliton interactions [9]: (i) soliton collisions are elastic, i.e. the interaction does not change the soliton amplitudes (or, more precisely, the discrete spectrum levels in the associated linear spectral problem for the quantummechanical Schrödinger operator); (ii) after the interaction, each soliton acquires an additional phase shift; (iii) the total phase shift of a ‘trial’ soliton acquired during a certain time interval can be calculated as a sum of the ‘elementary’ phase shifts in pairwise collisions of this soliton with other solitons during this time interval. These fundamental properties of two-soliton interactions enabled Zakharov in 1971 to introduce the kinetic equation for a rarefied gas of KdV solitons [19]. The generalisation of Zakharov’s equation to finite densities derived in [10] has required the consideration of the thermodynamic-type limit for finite-gap potentials and the associated Whitham modulation equations [11]. A straightforward, physical derivation of the kinetic equation was made in [12]. The effect