Abstract
We review the spectral theory of soliton gases in integrable dispersive hydrodynamic systems. We first present a phenomenological approach based on the consideration of phase shifts in pairwise soliton collisions and leading to the kinetic equation for a non-equilibrium soliton gas. Then, a more detailed theory is presented in which soliton gas dynamics are modelled by a thermodynamic type limit of modulated finite-gap spectral solutions of the Korteweg–de Vries and the focusing nonlinear Schrödinger (NLS) equations. For the focusing NLS equation the notions of soliton condensate and breather gas are introduced that are related to the phenomena of spontaneous modulational instability and the rogue wave formation. The integrability properties of the kinetic equation for soliton gas are discussed and some physically relevant solutions are presented and compared with direct numerical simulations of dispersive hydrodynamic systems.
Highlights
1.1 Integrable turbulence and soliton gas Random nonlinear dispersive waves have been the subject of an active research in nonlinear physics for more than five decades, most notably in the contexts of water wave dynamics
In this article we have reviewed the results on the spectral theory of soliton gases obtained by the author and his collaborators over the last two decades, many of the developments are relatively recent
It has been realised that soliton gas represents a ubiquitous physical phenomenon, a kind of strongly nonlinear turbulent wave motion, that can observed in the environmental conditions [21], [39] and realised in laboratory experiments [23], [24]
Summary
1.1 Integrable turbulence and soliton gas Random nonlinear dispersive waves have been the subject of an active research in nonlinear physics for more than five decades, most notably in the contexts of water wave dynamics. One of the early and most significant results of the wave turbulence theory was the analytical determination by Zakharov [2] of the analogs of the Kolmogorov spectra describing energy flux through scales in dissipative hydrodynamic turbulence These spectra, called Kolmogorov-Zakharov spectra, were obtained as solutions of the kinetic equations for the evolution of the Fourier spectra of random weakly nonlinear dispersive waves in multidimensional non-integrable systems. Analytical description of soliton gases in nonlinear dispersive wave systems was initiated in the Zakharov’s 1971 paper [28], where a spectral kinetic equation for KdV solitons was introduced using an IST-based phenomenological ‘flea gas’ reasoning enabling the evaluation of an effective adjustment to the soliton’s velocity in a rarefied gas due to the interactions (collisions) between individual solitons, accompanied by the well-defined phase-shifts. There has been a recent surge of related activity in generalised hydrodynamics (see [55], [56], [57] and references therein), where the equations analogous to those arising in the soliton gas theory became pivotal for the understanding of large-scale, hydrodynamic properties of quantum many-body systems
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