A. M. Kamchatnov

Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length.

Evolution of periodic matter waves in one-dimensional Bose-Einstein condensates with time-dependent scattering length is described. It is shown that variation of the effective nonlinearity is a powerful tool for controlled generation of bright and dark solitons starting with periodic waves.

Oblique Dark Solitons in Supersonic Flow of a Bose-Einstein Condensate

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.

Dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms

We consider formation of dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms. It is shown that big enough initial inhomogeneity of density leads to wave breaking phenomenon followed by generation of a train of dark solitons. Analytical theory is confirmed by numerical simulations.

DYNAMICS OF BRIGHT MATTER WAVE SOLITONS IN A BOSE EINSTEIN CONDENSATE

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

Undular bore theory for the Gardner equation

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), 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.

Stabilization of Solitons Generated by a Supersonic Flow of Bose-Einstein Condensate Past an Obstacle

The stability of dark solitons generated by supersonic flow of a Bose-Einstein condensate past an obstacle is investigated. It is shown that in the reference frame attached to the obstacle a transition occurs at some critical value of the flow velocity from absolute instability of dark solitons to their convective instability. This leads to the decay of disturbances of solitons at a fixed distance from the obstacle and the formation of effectively stable dark solitons. This phenomenon explains the surprising stability of the flow picture that has been observed in numerical simulations.

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

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 N-component ‘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 (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed 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.

Analytic model for a weakly dissipative shallow-water undular bore

We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by viscosity. This is derived in Riemann variables using a modified finite-gap integration technique for the Ablowitz-Kaup-Newell-Segur (AKNS) scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup-Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.

Asymptotic soliton train solutions of Kaup-Boussinesq equations

Asymptotic soliton trains arising from a ‘large and smooth’ enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup–Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr–Sommerfeld quantization rule which generalizes the usual rule to the case of ‘two potentials’ h0(x) and u0(x) representing initial distributions of height and velocity, respectively. The influence of the initial velocity u0(x) on the asymptotic stage of the evolution is determined. Excellent agreement of numerical solutions of the Kaup–Boussinesq equations with predictions of the asymptotic theory is found.

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.