Polarization hydrodynamics in a one-dimensional polariton condensate

We study the hydrodynamics of a nonresonantly-pumped polariton condensate in a quasi-one-dimensional quantum wire taking into account the spin degree of freedom. We clarify the relevance of the Landau criterion for superfluidity in this dissipative two-component system. Two Cherenkov-like critical velocities are identified corresponding to the opening of different channels of radiation: one of (damped) density fluctuations and another of (weakly damped) polarization fluctuations. We determine the drag force exerted onto an external obstacle and propose experimentally measurable consequences of the specific features of the fluctuations of polarization

Wave pattern induced by a localized obstacle in the flow of a one-dimensional polariton condensate

Motivated by recent experiments on generation of wave patterns by a polariton condensate incident on a localized obstacle, we study the characteristics of such flows under the condition that irreversible processes play a crucial role in the system. The dynamics of a non-resonantly pumped polariton condensate in a quasi-one-dimensional quantum wire is modeled by a Gross-Pitaevskii equation with additional phenomenological terms accounting for the dissipation and pumping processes. The response of the condensate flow to an external potential describing a localized obstacle is considered in the weak-perturbation limit and also in the nonlinear regime. The transition from a viscous drag to a regime of wave resistance is identified and studied in detail

Nonlinear waves of polarization in two-component Bose-Einstein condensates

Waves with different symmetries exist in two-component Bose-Einstein condensates (BECs) whose dynamics is described by a system of coupled Gross-Pitaevskii (GP) equations. A first type of waves corresponds to excitations for which the motion of both components is locally in phase. In the second type of waves the two components have a counter-phase local motion. In the case of different values of inter- and intra-component interaction constants, the long wave-length behavior of these two modes corresponds to two types of sound with different velocities. In the limit of weak nonlinearity and small dispersion the first mode is described by the well-known Korteweg-de Vries (KdV) equation. We show that in the same limit the second mode can be described by the Gardner (modified KdV) equation, if the intra-component interaction constants have close enough values. This leads to a rich phenomenology of nonlinear excitations (solitons, kinks, algebraic solitons, breathers) which does not exist in the KdV description.Comment: 10 pages, 5 figure

Mixed-isotope Bose-Einstein condensates in Rubidium

We consider the ground state properties of mixed Bose-Einstein condensates of 87Rb and 85Rb atoms in the isotropic pancake trap, for both signs of the interspecies scattering length. In the case of repulsive interspecies interaction, there are the axially-symmetric and symmetry-breaking ground states. The threshold for the symmetry breaking transition, which is related to appearance of a zero dipole-mode, is found numerically. For attractive interspecies interactions, the two condensates assume symmetric ground states for the numbers of atoms up to the collapse instability of the mixture.Comment: Revised; 21 pages, 5 figures, submitted to Physical Review

On dissipationless shock waves in a discrete nonlinear Schr\"odinger equation

It is shown that the generalized discrete nonlinear Schr\"odinger equation can be reduced in a small amplitude approximation to the KdV, mKdV, KdV(2) or the fifth-order KdV equations, depending on values of the parameters. In dispersionless limit these equations lead to wave breaking phenomenon for general enough initial conditions, and, after taking into account small dispersion effects, result in formation of dissipationless shock waves. The Whitham theory of modulations of nonlinear waves is used for analytical description of such waves.Comment: 15 pages, 9 figure

Linear "ship waves" generated in stationary flow of a Bose-Einstein condensate past an obstacle

Using stationary solutions of the linearized two-dimensional Gross-Pitaevskii equation, we describe the ``ship wave'' pattern occurring in the supersonic flow of a Bose-Einstein condensate past an obstacle. It is shown that these ``ship waves'' are generated outside the Mach cone. The developed analytical theory is confirmed by numerical simulations of the flow past body problem in the frame of the full non-stationary Gross-Pitaevskii equation.Comment: 5 pages, 4 figure

Matter sound waves in two-component Bose-Einstein condensates

The creation and propagation of sound waves in two-component Bose-Einstein condensates (BEC) are investigated and a new method of wave generation in binary BEC mixtures is proposed. The method is based on a fast change of the inter-species interaction constant and is illustrated for two experimental settings: a drop-like condensate immersed into a second large repulsive condensate, and a binary mixture of two homogeneous repulsive BEC's. A mathematical model based on the linearized coupled Gross-Pitaevskii equations is developed and explicit formulae for the space and time dependence of sound waves are provided. Comparison of the analytical and numerical results shows excellent agreement, confirming the validity of the proposed approach.Comment: 16 pages, 9 figure

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.Comment: 4 pages, 1 figure, final version published in Phys. Rev. Let

Stationary wave patterns generated by an impurity moving with supersonic velocity through a Bose-Einstein condensate

Formation of stationary 3D wave patterns generated by a small point-like impurity moving through a Bose-Einstein condensate with supersonic velocity is studied. Asymptotic formulae for a stationary far-field density distribution are obtained. Comparison with three-dimensional numerical simulations demonstrates that these formulae are accurate enough already at distances from the obstacle equal to a few wavelengths.Comment: 7 pages, 3 figure

Lagrangian and Hamiltonian two-scale reduction

Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.Comment: 40 page