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Physical Review A

Fermi-Ulam problem for solitons

Nikolay N. Rosanov, N. V. Vysotina

Published Jan 21, 2015• 17 citations
Publisher

Abstract

Dynamics of solitons in a one-dimensional trap with motionless and moving walls is studied by the example of bright solitons of an atomic Bose-Einstein condensate. Newton-type equations of the soliton's center motion are proposed and analyzed, and regimes of periodic, almost-periodic, and deterministically chaotic dynamics are presented. The predictions based on these approximate equations are in excellent agreement with that based on the Gross-Pitaevskii equation in the case of sufficiently narrow and slow solitons.

DOI
10.1103/physreva.91.013622

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Dynamics of solitons in a one-dimensional trap with motionless and moving walls is studied by the example of bright solitons of an atomic Bose-Einstein condensate. Newton-type equations of the soliton's center motion are proposed and analyzed, and regimes of periodic, almost-periodic, and deterministically chaotic dynamics are presented. The predictions based on these approximate equations are in excellent agreement with that based on the Gross-Pitaevskii equation in the case of sufficiently narrow and slow solitons.
Systems of solitary waves in the one-dimensional Gross-Pitaevskii equation, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. To analyze the soliton nature of these solitary waves, a particle analogy for the solitary waves is formulated. Exact soliton solutions exist in the absence of an external trapping potential, which behave in a particlelike manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present up to a gradual shift in the trajectories when the harmonic trap period is short compared with the collision time of the solitons. We find that the collision time of the solitons is dependent on the relative phase of the solitons as they collide. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. This phase preservation can be used to find regimes where there is agreement between the wave and particle models. This also implies that soliton regimes may be found in three-dimensional geometries where solitary waves can be made to repeatedly collide out of phase, stabilizing the condensate against collapse. The extension to three particles supports both regular and chaotic regimes. The trajectory shift observed for two solitons carries over to the case of three solitons. This shift aside, the agreement between the particle model and the wave dynamics remains good, even in chaotic regimes.
The motion of the center of a soliton in a trap with oscillating walls is studied analytically and numerically for the case in which the intrinsic frequency of small soliton oscillations in the equilibrium state considerably exceeds the frequency of wall oscillations. this problem can be solved either by applying the gross–pitaevskii equation, which most exactly describes the behavior of the soliton in the trap, or by using the approximate, “mechanical,” equation of motion of the newtonian type for the center of the soliton. an approximate analytical solution of the mechanical equation is obtained and is compared with the numerical solution of the newton equation, while the latter solution is compared with the numerical solution of the gross–pitaevskii equation. good agreement between the first two solutions is revealed. it is also shown that there is a range of parameters in which the numerical solutions of the newton and gross–pitaevskii equations are closest to each other. the frequency-sweeping effect of soliton center oscillations is revealed. an approximate analytical formula for the limiting frequency of these oscillations is obtained and the numerical analysis of this phenomenon is performed.
Collisions between bright solitary waves in the 1D Gross-Pitaevskii equation with a harmonic potential, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. A particle analogy for the solitary waves is formulated and shown to be integrable for a two-particle system. The extension to three particles is shown to support chaotic regimes. Good agreement is found between the particle model and simulations of the full wave dynamics, suggesting that the dynamics can be described in terms of solitons both in regular and chaotic regimes, presenting a paradigm for chaos in wave mechanics.
Motion of a soliton of Bose–Einstein condensate of atoms captured by a trap with rapidly oscillating walls has been studied. This motion can be described using both the Gross–Pitaevskii equation for a condensate wave function and an approximate equation in the form of the Newton equation for the soliton center coordinate. An analytical approach for solving the Newton equation has been developed. This approach is based on the multiscale method where the solution is sought for in the form of small-parameter expansion. This parameter is a ratio of the frequency of intrinsic slow soliton oscillations around the equilibrium position to the frequency of fast oscillations of the trap walls. In the first part of the study, an approach based on two time scales is described and the case of extremely fast wall oscillations is investigated. The calculation performed within the zero approximation shows a very good coincidence with the numerical solution of the Newton equation with respect to all parameters. A good agreement with the numerical solutions of the Gross–Pitaevskii equation is also demonstrated for calculations of the parameters such as oscillation frequency and shift of the soliton equilibrium position under the action of the wall motion. In the second part, the role of corrections to the obtained solution is analyzed for a decreasing wall-oscillation frequency and the range of applicability of the used analytical approach is discussed.
We study the motion of bright matter-wave solitons in nonlinear potentials, produced by periodic or random spatial variations of the atomic scattering length. We obtain analytical results for the soliton motion, the radiation of matter wave, and the radiative soliton decay in such configurations of the Bose-Einstein condensate. The stable regimes of propagation are analyzed. The results are in remarkable agreement with the numerical simulations of the Gross-Pitaevskii equation with periodic or random spatial variations of the mean field interactions.