Spatial dispersive shock waves generated in supersonic flow of Bose–Einstein condensate past slender body

Supersonic flow of Bose–Einstein condensate past macroscopic obstacles is studied theoretically. It is shown that in the case of large obstacles the Cherenkov cone transforms into a stationary spatial shock wave which consists of a number of spatial dark solitons. Analytical theory is developed for the case of obstacles having a form of a slender body. This theory explains qualitatively the properties of such shocks observed in recent experiments on nonlinear dynamics of condensates of dilute alkali gases

Generation of oblique dark solitons in supersonic flow of Bose-Einstein condensate past an obstacle

Nonlinear and dispersive properties of Bose-Einstein condensate (BEC) provide a possibility of formation of various nonlinear structures such as vortices and bright and dark solitons (see, e.g., [1]). Yet another type of nonlinear wave patterns has been observed in a series of experiments on the BEC flow past macroscopic obstacles [2]. In [3] these structures have been associated with spatial dispersive shock waves. Spatial dispersive shock waves represent dispersive analogs of the the well-known viscous spatial shocks (oblique jumps of compression) occurring in supersonic flows of compressible fluids past obstacles. In a viscous fluid, the shock can be represented as a narrow region within which strong dissipation processes take place and the thermodynamic parameters of the flow undergo sharp change. On the contrary, if viscosity is negligibly small compared with dispersion effects, the shock discontinuity resolves into an expanding in space oscillatory structure which transforms gradually, as the distance from the obstacle increases, into a \fan" of stationary solitons. If the obstacle is small enough, then such a \fan" reduces to a single spatial dark soliton [4]. Here we shall present the theory of these new structures in BEC

On Whitham theory for perturbed integrable equations

Whitham theory of modulations is developed for periodic waves described by nonlinear wave equations integrable by the inverse scattering transform method associated with $2\times2$ matrix or second order scalar spectral problems. The theory is illustrated by derivation of the Whitham equations for perturbed Korteweg-de Vries equation and nonlinear Schr\"odinger equation with linear damping.Comment: 17 pages, no figure

Formation of soliton trains in Bose-Einstein condensates as a nonlinear Fresnel diffraction of matter waves

The problem of generation of atomic soliton trains in elongated Bose-Einstein condensates is considered in framework of Whitham theory of modulations of nonlinear waves. Complete analytical solution is presented for the case when the initial density distribution has sharp enough boundaries. In this case the process of soliton train formation can be viewed as a nonlinear Fresnel diffraction of matter waves. Theoretical predictions are compared with results of numerical simulations of one- and three-dimensional Gross-Pitaevskii equation and with experimental data on formation of Bose-Einstein bright solitons in cigar-shaped traps.Comment: 8 pages, 3 figure

On generating functions in the AKNS hierarchy

It is shown that the self-induced transparency equations can be interpreted as a generating function for as positive so negative flows in the AKNS hierarchy. Mutual commutativity of these flows leads to other hierarchies of integrable equations. In particular, it is shown that stimulated Raman scattering equations generate the hierarchy of flows which include the Heisenberg model equations. This observation reveals some new relationships between known integrable equations and permits one to construct their new physically important combinations. Reductions of the AKNS hierarchy to ones with complex conjugate and real dependent variables are also discussed and the corresponding generating functions of positive and negative flows are found. Generating function of Whitham modulation equations in the AKNS hierarchy is obtained.Comment: 11 pages, no figure

Dynamics of ring dark solitons in Bose-Einstein condensates and nonlinear optics

Quasiparticle approach to dynamics of dark solitons is applied to the case of ring solitons. It is shown that the energy conservation law provides the effective equations of motion of ring dark solitons for general form of the nonlinear term in the generalized nonlinear Schroedinger or Gross-Pitaevskii equation. Analytical theory is illustrated by examples of dynamics of ring solitons in light beams propagating through a photorefractive medium and in non-uniform condensates confined in axially symmetric traps. Analytical results agree very well with the results of our numerical simulations.Comment: 10 pages, 4 figure

Condition for convective instability of dark solitons

Simple derivation of the condition for the transition point from absolute instability of plane dark solitons to their convective instability is suggested. It is shown that unstable wave packet expands with velocity equal to the minimal group velocity of the disturbance waves propagating along a dark soliton. The growth rate of the length of dark solitons generated by the flow of Bose-Einstein condensate past an obstacle is estimated. Analytical theory is confirmed by the results of numerical simulations

Kinetic equation for a dense soliton gas

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 Schr¨odinger (NLS) equations. As a simple physical example we construct an explicit solution for the case of interaction of two cold NLS soliton gases

Spatial dispersive shock waves generated in supersonic flow of Bose-Einstein condensate past slender body

Supersonic flow of Bose-Einstein condensate past macroscopic obstacles is studied theoretically. It is shown that in the case of large obstacles the Cherenkov cone transforms into a stationary spatial shock wave which consists of a number of spatial dark solitons. Analytical theory is developed for the case of obstacles having a form of a slender body. This theory explains qualitatively the properties of such shocks observed in recent experiments on nonlinear dynamics of condensates of dilute alkali gases

Wave breaking and the generation of undular bores in an integrable shallow-water system

The generation of an undular bore in the vicinity of a wave-breaking point is con- sidered for the integrable Kaup-Boussinesq shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the Gurevich-Pitaevskii type of problem for a generic “cubic” breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear Euler-Poisson equation. The motion of the undular bore edges is investigated in detail