Abstract In this paper, we theoretically investigate the dynamics and stability of matter‐wave solitons (MWS) in a cigar‐shaped Bose–Einstein condensate (BEC) consisting of atoms with attractive interatomic interactions trapped by Pöschl–Teller (PT) potential. In this scenario, we propose the idea of the BEC being dragged by PT potential at a constant speed. The BEC was modeled by the quasi‐one‐dimensional Gross–Piteaevskii Equation (1D‐GPE). In addition, an analytical (time‐dependent variational method) and numerical (split‐step Crank–Nicolson method) approach has been proposed to investigate the dynamic properties of solitons during the temporal evolution of this system. Both analytical and numerical results demonstrated that MWSs dragged by PT potential can be stable. From a general point of view, the results obtained in this paper can motivate both theoretical and experimental investigations on the dynamics and stability of solitons in dilute and ultracold unitary Bose gases trapped by PT potential, nonlinear optical properties in PT quantum wells, and propagation of solitons in nonlinear media.
We propose a stochastic description of the dynamics of a Bose-Einstein condensate within the context of Nelson stochastic mechanics. We start from the
N interacting conservative diffusions, associated with the
N Bose particles, and take an infinite particle limit. We address several aspects of this formulation. First, we consider the problem of extending to a system with self-interaction the variational formulation of Nelson stochastic mechanics due to Guerra and Morato. In this regard we discuss two possible extensions, one based on a doubling procedure and another based on a constraint Eulerian type variational principle. Then we consider the infinite particle limit from the point of view of the
N-particles Madelung equations. Since conservative diffusions can be identified with proper infinitesimal characteristics pairs
(ρN(t),vN(t)), a time marginal probability density and a current velocity field, respectively, we consider a finite Madelung hierarchy for the marginals pairs
(ρN,n(t),vN,n(t)), obtained by properly conditioning the processes. The infinite Madelung hierarchy arises from the finite one by performing, for each fixed
n, a mean-field scaling limit in
N. Finally, we introduce a
n-particle conditioned diffusions which naturally parallels the quantum mechanical approach and is a new approach within the context of Nelson stochastic mechanics. We then prove the convergence, in the infinite particle limit, of the law of such a conditioned process to the law of a self-interacting diffusion which describes the condensate.