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

Abstract

Stationary solutions for the 1+1 cubic nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates (BEC) in a small potential are obtained through a form of nonlinear perturbation. In particular, for sufficiently small potentials, we determine the perturbation theory of stationary solutions, by use of an expansion in Jacobi elliptic functions. This idea was explored before in order to obtain exact solutions [Bronski, Carr, Deconinck, and Kutz, Phys. Rev. Lett. 86, 1402 (2001)], where the potential itself was fixed to be a Jacobi elliptic function, thereby reducing the nonlinear ODE into an algebraic equation, (which could be easily solved). However, in the present paper, we outline the perturbation method for completely general potentials, assuming only that such potentials are locally small. 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 allows us to consider even poorly behaved potentials, so long as they are bounded locally. We demonstrate the effectiveness of this approach by considering a number of specific potentials: for the simplest potentials, and we recover results from the literature, while for more complicated potentials, our results are new. Dark soliton solutions are constructed explicitly for some cases, and we obtain the known one-soliton tanh-type solution in the simplest setting for the repulsive BEC. Note that we limit our results to the repulsive case; similar results can be obtained for the attractive BEC case.

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.…”

“…The stationary solutions studied here correspond to radially symmetric BECs in three spatial dimensions. Such results generalize the 1D and 2D results considered by the authors previously [23][24][25]. Numerical solutions have also been discussed in a number of cases where the potential was permitted to be larger, and in the attractive case in the degenerate limit ǫ → 0.…”

“…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.…”

“…Since I 1 (r) is independent of the potential, it will appear in the order-one term of each of the following examples. Thus, we shall use I 1 (r) in place of the explicit expression (23) wherever it simplifies notation. Having calculated I 1 (r), we have that…”

Multicomplex Wave Functions for Linear And Nonlinear Schrödinger Equations

“…However, we will give one solution, to demonstrate that we recover the famous bright and dark soliton solutions for the one-dimensional spatial domain. For a more thorough summary of solutions to one-dimensional cubic nonlinear Schrödinger equations, see references [27,28]. We consider equation (5.5) with the real-valued potential…”