Modulation of breathers in cigar-shaped Bose–Einstein condensates

Abstract

We present new solutions to the nonautonomous nonlinear Schrödinger equation that may be realized through convenient manipulation of Bose-Einstein condensates. The procedure is based on the modulation of breathers through an analytical study of the one-dimensional Gross-Pitaevskii equation, which is known to offer a good theoretical model to describe quasi-one-dimensional cigar-shaped condensates. Using a specific Ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Numerical simulations confirm stability of the breathers against random perturbation on the input profile of the solutions.

Bright solitons from the nonpolynomial Schrödinger equation with inhomogeneous defocusing nonlinearities

“…This subject has been greatly upheld by the use of the Feshbachresonance (FR) technique, i.e., the control of the strength of the inter-atomic interactions by externally applied fields [23][24][25], which opens the possibility to implement sophisticated nonlinear patterns. In particular, the management of localized solutions of the Gross-Pitaevskii equation (GPE) [26] by means of the spatially inhomogeneous nonlinearity, which may be created by external nonuniform fields that induce the corresponding FR landscape, has attracted a great deal of interest in theoretical studies [27][28][29][30][31][32][33][34][35][36].In this vein, the existence of bright solitons in systems with purely repulsive, alias self-defocusing (SDF) nonlinearity, in the absence of external linear potentials, was recently predicted [37]. This result is intriguing because the existence of such solutions, supported by SDF-only nonlinearities, without the help of a linear potential, was commonly considered impossible.…”

Modulation of breathers in the three-dimensional nonlinear Gross-Pitaevskii equation

“…In this way, we extend our recent work [11] to the more realistic 3D case. To this end, we use an Ansatz that changes the 3D GP equation into specific 1D equation with constant coefficients, which is easier to solve.…”

“…The presence of nonuniform and time-dependent parameters opens interesting perspectives not only from the theoretical point of view, for investigation of nonautonomous nonlinear equations, but also from the experimental point of view, for the study of the physical properties of the systems. In this context, in a recent work we have considered modulation of genuine breather solutions in cigar-shaped Bose-Einstein condensates (BECs) with potential and nonlinearity depending on both space and time, in the one-dimensional (1D) case [11].The study of BECs of dilute gases of weakly interacting bosons, realized for the first time in 1995 on vapors of rubidium [12] and sodium [13], constitutes a very interesting scenario to modulate breathers, since they are well described by a threedimensional (3D) Gross-Pitaevskii (GP) equation arising from a mean-field dynamics [14]. In the BEC context, one finds high experimental flexibility to control nonlinearity via Feshbach resonance, and confinement profile via optical lattices and harmonic and dipole traps [15], and there we can also investigate the effects of dimensionality reduction on the soliton solution.…”

Airy–Bessel modulated self-similar rogue waves in a nonlinear Schrödinger equation model

“…The parameters a 0 , b 0 , c 0 and d 0 facilitate control over nonlinearity. Introducing a self-similar transformation, the solution assumes the form [23][24][25][26][27][28] …”