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arXiv:1501.01229 · Physical Review A

Nonlinear waves in two-component Bose-Einstein condensates: Manakov system and Kowalevski equations

A. M. Kamchatnov, Vladimir Vyacheslavovich Sokolov

Published Apr 15, 2015• 12 citations
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Abstract

Traveling waves in two-component Bose-Einstein condensates whose dynamics is described by the Manakov limit of the Gross-Pitaevskii equations are considered in a general situation with relative motion of the components when their chemical potentials are not equal to each other. It is shown that in this case the solution is reduced to the form known in the ``Kowalevski top'' theory of motion. Typical situations are illustrated by the particular cases when the general solution can be represented in terms of elliptic functions and their limits. Depending on the parameters of the wave, both density waves (with in-phase motions of the components) and polarization waves (with counterphase motions) are considered.

DOI
10.1103/physreva.91.043621

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Traveling waves in two-component Bose-Einstein condensates whose dynamics is described by the Manakov limit of the Gross-Pitaevskii equations are considered in a general situation with relative motion of the components when their chemical potentials are not equal to each other. It is shown that in this case the solution is reduced to the form known in the ``Kowalevski top'' theory of motion. Typical situations are illustrated by the particular cases when the general solution can be represented in terms of elliptic functions and their limits. Depending on the parameters of the wave, both density waves (with in-phase motions of the components) and polarization waves (with counterphase motions) are considered.
Periodic waves in a two-component Bose-Einstein condensate are considered with allowance for mixing of components admitted by the interaction between atoms. The possibility of relative motion of the components leads to significant variability of the system dynamics as compared to the case of a single-component condensate. In particular, it was experimentally established that, if one component flows relative to another, waves with counterphase oscillations of the components can be excited in the two-component condensate. A theory of one-phase periodic waves is developed for the case of weak mixing of the components, in which the condensate dynamics is described with sufficiently good precision by a system of Gross-Pitaevskii equations with equal constants of nonlinear interaction between atoms both of the same and different kinds. Both condensates with repulsive and attractive interactions between atoms are considered. The limiting case of waves with constant amplitude is studied in detail. The structure of soliton analogs in these systems is considered. The possibility of applying the proposed theory to description of experimentally observed wavy periodic structures in two-component Bose-Einstein condensate is discussed.
We study the dynamics of so-called simple waves in a two-component Bose-Einstein condensate. The evolution of the condensate is described by Gross-Pitaevskii equations which can be reduced for these simple wave solutions to a system of ordinary differential equations which coincide with those derived by Ovsyannikov for the two-layer fluid dynamics. We solve the Ovsyannikov system for two typical situations of large and small difference between interspecies and intraspecies nonlinear interaction constants. Our analytic results are confirmed by numerical simulations.
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.
We investigate the expansion dynamics of a Bose–Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the nonuniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.
We investigate the localized nonlinear matter waves in the two-component Bose-Einstein condensates with time- and space-modulated nonlinearities analytically and numerically. The similarity transformations are developed to solve the coupled Gross-Pitaevskii equations and two families of explicitly exact solutions are derived. Our results show that not only the attractive spatiotemporal inhomogeneous interactions but the repulsive ones support novel localized nonlinear matter waves in two-component Bose-Einstein condensates. The dynamics of these matter waves, including the breathing solitons, quasibreathing solitons, resonant solitons, and moving solitons, is discussed. We confirm the stability of the exact solutions by adding various initial stochastic noise and study the general cases of the interaction parameters numerically. We also provide the experimental parameters to produce these phenomena in future experiments.