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We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg -de Vries (KdV) and nonlinear Schrödinger (NLS) equations. As a simple physical example we construct an explicit solution for the case of interaction of two cold NLS soliton gases.

In the framework of the Gross-Pitaevskii mean field approach it is shown that the supersonic flow of a Bose-Einstein condensate can support a new type of pattern-an oblique dark soliton. The corresponding exact solution of the Gross-Pitaevskii equation is obtained. It is demonstrated by numerical simulations that oblique solitons can be generated by an obstacle inserted into the flow.PACS numbers: 03.75.KkIntroduction. It is known that the nonlinear and dispersive properties of a Bose-Einstein condensate (BEC) can lead to the formation of various nonlinear structures (see, e.g., [1]). Until recently, most research has been focused on experimentally observed vortices and bright and dark solitons. Furthermore, the formation of dispersive shock waves in BECs with repulsive interactions between atoms was considered theoretically in [2,3] and studied experimentally in rotating [4] and non-rotating [5] condensate, where it was shown that dispersive shocks are generated as a result of the evolution of large disturbances in the BEC. However, another important type of nonlinear structure, namely a spatial dark soliton, can also be realized in a BEC. The first experimental evidence of their generation has recently appeared [6]. In fact, the existence of oblique spatial solitons in a BEC has a natural physical basis if the Cherenkov generation of dispersive sound waves by a small obstacle in the supersonic flow of a BEC is considered and the effect of increasing the size of the obstacle (i.e. the amplitude of the waves) is determined. Evidently, along with dispersion, nonlinear effects become equally important at finite distances from the obstacle, so that the Cherenkov cone breaks-up into a spatial structure consisting of one or several dark solitons. Such a structure represents the dispersive analog of the well-known steady spatial shock generated in the supersonic flow of a viscous compressible fluid past an obstacle. In this sense, it is the spatial counterpart of the one-dimensional expanding dispersive shock [2]-[5] generated in the evolution of large disturbances in a BEC. In the simplest case, the nonlinear wave structure would consist of a single spatial dark soliton given by a steady solution of the equations governing the BEC flow. Motivated by this physical consideration and the results of experiments [6], in this Letter we shall develop the theory of spatial dark solitons in the framework of the Gross-Pitaevskii (GP) mean field approach.

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg--de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.Comment: 25 pages, 24 figures, slightly revised and corrected versio

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 Ncomponent '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 (quasi-periodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed the 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.