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Physical Review E

Exact solitonic solutions of the Gross-Pitaevskii equation with a linear potential

Usama Al Khawaja

Published Jun 19, 2007• 23 citations
Publisher

Abstract

We derive classes of exact solitonic solutions of the time-dependent Gross-Pitaevskii equation with repulsive and attractive interatomic interactions. The solutions correspond to a string of bright solitons with phase difference between adjacent solitons equal to pi. While the relative phase, width, and distance between adjacent solitons turn out to be a constant of the motion, the center of mass of the string moves with a constant acceleration arising from the inhomogeneity of the background.

DOI
10.1103/physreve.75.066607

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We derive classes of exact solitonic solutions of the time-dependent Gross-Pitaevskii equation with repulsive and attractive interatomic interactions. The solutions correspond to a string of bright solitons with phase difference between adjacent solitons equal to pi. While the relative phase, width, and distance between adjacent solitons turn out to be a constant of the motion, the center of mass of the string moves with a constant acceleration arising from the inhomogeneity of the background.
We derive classes of exact solitonic solutions of the time-dependent Gross-Pitaevskii equation with repulsive and attractive interatomic interactions. The solutions correspond to a string of bright solitons with phase difference between adjacent solitons equal to πππ. While the relative phase, width, and distance between adjacent solitons turn out to be a constant of the motion, the center of mass of the string moves with a constant acceleration arising from the inhomogeneity of the background.
We derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation with time-dependent strengths of the harmonic external potential and the interatomic interaction. The time-dependence of the external potential and interatomic interaction are given in terms of a general function of time. For an oscillating strength of the external potential, the solutions correspond to breathing single and multiple solitons. The amplitude and frequency of the oscillating potential can be used to control the dynamics of the center of mass of the solitons. For certain values of these parameters, the solitons can be {\it trapped} at the center of the harmonic potential.
We derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation with time-dependent strengths of the harmonic external potential and the interatomic interaction. The time-dependence of the external potential and interatomic interaction are given in terms of a general function of time. For an oscillating strength of the external potential, the solutions correspond to breathing single and multiple solutions. The amplitude and frequency of the oscillating potential can be used to control the dynamics of the centre of mass of the solitons. For certain values of these parameters, the solitons can be trapped at the centre of the harmonic potential.
We derive exact solitonic solutions of a class of Gross–Pitaevskii equations with a time-dependent harmonic trapping potential and interatomic interaction. We find families of exact single-solitonic, multi-solitonic and solitary wave solutions. We show that, with the special case of an oscillating trapping potential and interatomic interaction, a soliton can be localized indefinitely at an arbitrary position. The localization is shown to be experimentally possible for sufficiently long time even with only an oscillating trapping potential and a constant interatomic interaction.
We present a family of exact solutions of the one-dimensional nonlinear Schro dinger equation which describes the dynamics of a bright soliton in Bose-Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential. Our results show that, under a safe range of parameters, the bright soliton can be compressed into very high local matter densities by increasing the absolute value of the atomic scattering length, which can provide an experimental tool for investigating the range of validity of the one-dimensional Gross-Pitaevskii equation. We also find that the number of atoms in the bright soliton keeps dynamic stability: a time-periodic atomic exchange is formed between the bright soliton and the background.
A family of analytical solutions to a quasi one-dimensional time-dependent Gross–Pitaevskii equation with inverted parabolic potentials and time-varying nonlinear two-body interactions are presented, which simulate the dynamical evolutions of nonlinear matter-wave bright solitons. Synchronous behaviour between the variations of nonlinear interaction and compression or expansion of the solitons is demonstrated. For the case of higher order solitons, collision features are investigated. For travelling two solitons, different kinds of collisions are discussed for π and 0 phase difference between the two solitons at the collision point. Furthermore, a moving two-soliton 'bound' state is demonstrated. During the propagation, the solitons oscillate and collide many times.