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

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.

Soliton localization in Bose–Einstein condensates with time-dependent harmonic potential and scattering length

“…The solid and dashed curves show the corresponding trajectories in the presence of the oscillating interatomic interaction as described by Eq. (33). The difference between the solid and filled circles curves shows the important role played by the oscillations in the interatomic interactions in stabilizing the soliton.…”

Wave chaos in the nonequilibrium dynamics of the Gross-Pitaevskii equation

“…For a few special cases, integrability has been demonstrated. These cases include the harmonic potential, U (x) ∝ x 2 , with time-dependent nonlinearities [49][50][51], and linear potentials, U (x) ∝ x, with time-independent nonlinearity [52,53].…”

Regular and chaotic Bose-Einstein condensate in an accelerated Wannier-Stark lattice

“…(17) for the modulus R. Applying θ ξ = J 0 /R 2 of Eq. (20) to Eq. (17), we arrive at the decoupled equation…”

“…On the other hand, it is well-known that the BEC governed by a Gross-Pitaevskii equation (GPE) without external potential is an integrable system and the integrability could be easily broken by external potentials of different forms [11]. So previously, only few analytical works concern exact solutions of the system, where one-dimensional (1D) stationary systems with some simple potentials are treated, such as the infinite or finite square-wells [12,13,14,15], the step-potentials [16], δ or δ comb potentials [17,18,19], linear ramp potential [20,21] and optical lattice potentials [22,23]. Under some rigorous conditions on the interaction intensities or external potentials, several exact nonstationary-state solutions were constructed [24,25], including the exact soliton solutions [26,27,28,29,30].…”