Stationary solutions for the1+1nonlinear Schrödinger equation modeling attractive Bose-Einstein condensates in small potentials

Abstract

Stationary solutions for the 1+1 cubic nonlinear Schrödinger equation (NLS) modeling attractive Bose-Einstein condensates (BECs) in a small potential are obtained via a form of nonlinear perturbation. The focus here is on perturbations to the bright soliton solutions due to small potentials which either confine or repel the BECs: under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of the bright solitons. Importantly, we do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This is useful, in that it allows us to consider perturbations of bright solitons of arbitrary size. In some cases, exact solutions can be recovered, and these agree with known results from the literature. Several special cases are considered which involve confining potentials of specific relevance to BECs. We make several observations on the influence of the small potentials on the behavior of the perturbed bright solitons. The results demonstrate the difference between perturbed bright solitons in the attractive NLS and those results found in the repulsive NLS for dark solitons, as discussed by Mallory and Van Gorder, [Phys. Rev. E 88 013205 (2013)]. Extension of these results to more spatial dimensions is mentioned.

Stationary solutions for the nonlinear Schrödinger equation modeling three-dimensional spherical Bose-Einstein condensates in general potentials

“…In three-dimensions, this radial symmetry means that the potentials will exhibit spherical symmetry. Our analysis here extends our results for 1D [23,24] and 2D [25] BECs under general potentials. However, owing to the fact that we now consider three spatial dimensions, there will be some differences between the present analysis and that which we have considered in previous work.…”

“…When g > 0, we have the repulsive case, while when g < 0 we have the attractive case. General perturbation results (of the type we seek here) were recently given for the 1 + 1 model in [23] for the repulsive case and [24] for the attractive case.…”

“…In one spatial dimension, the repulsive case holds dark solitons as a special solution. Similarly, the attractive 1 + 1 model will have bright solitons as a special class of solutions [24]. In [25], solutions in the particular case of two spatial dimensions were discussed via a form of perturbation.…”

Effective nonlinear model for electron transport in deformable helical molecules

“…The perturbation approach, introduced by Mallory and Van Gorder in Ref. [30], confirms that the discrete nature of the array of dipoles has a negligible impact on the bright solitons of the extended Davydov model. In addition, some other relevant perturbations, such as the out-of-axis motion of electrons and an effective spin-dependent electron-lattice interaction, do not alter much the bright soliton solutions of the extended Davydov model.…”

“…This inhomogeneous equation can be straightforwardly solved by means of Green's function techniques [30]. To proceed, we consider two independent solutions of the homogeneous…”