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

Abstract

In this paper we present analytical breather solutions of the three-dimensional nonlinear generalized GrossPitaevskii equation. We use an Ansatz to reduce the three-dimensional equation with space-and time-dependent coefficients into an one-dimensional equation with constant coefficients. The key point is to show that both the space-and time-dependent coefficients of the nonlinear equation can contribute to modulate the breather excitations. We briefly discuss the experimental feasibility of the results in Bose-Einstein condensates. [8,9], and semiconductor quantum wells [10], the breather excitations play an important role, directly affecting the electronic, magnetic, optical, vibrational and transport properties of the systems.In the above mentioned studies, one usually considers genuine breathers, i.e., solutions which oscillate in time when the nonlinear equation presents constant coefficients (i.e., without modulation). However, in a more realistic scenario the several parameters that characterize the physical systems may depend on both space and time, leading to breather solutions that can be modulated in space and time. 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.In the case of a strong trapping in two spatial directions, the 3D GP equation reduces to the simpler one-dimensional (1D) form, giving rise to the so-called cigar-shaped configuration. The 1D GP equation is a nonlinear Schrödinger equation, which can also be used to investigate pulse propagation in bulk crystals or optical fibers [16]. In a former work, however, the search for analytical solutions of the 1D GP equation with stationary inhomogeneous coefficients has been implemented via similarity transformation [17]. More recently, however, the case of space-and time-dependent coefficients were considered for the cubic [18], the cubic-quintic [19], the quintic [20], and also the GP equation in higher dimensions [21].The similarity transformation wa...

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.…”

“…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].…”

Modulation of localized solutions in quadratic-cubic nonlinear Schrödinger equation with inhomogeneous coefficients

“…Also, the modulation of breathers and rogue waves were investigated in Refs. [40,42] and Refs. [47,50,58], respectively.…”

“…Exact solutions to three-dimensional generalized NLS equations with varying potential and nonlinearities were studied in Refs. [38][39][40]. Also, the modulation of breathers and rogue waves were investigated in Refs.…”

Controlled self-similar matter waves in PT-symmetric waveguide

“…Authors have also studied the dynamics of BEC in the presence of competing cubic-quintic nonlinearity [9,10,11,12,13,14,15] and quadratic-cubic nonlinearity [16,17,18,19,20]. Over the past several years, there is a considerable interest on the existence of matter wave solutions for GP equation with time-dependent coefficients or generalized nonlinear Schrödinger equation (GNLSE) [21,22,23,24,25]. Earlier, Paul and his collaborators [26,27,28] numerically studied the resonant transport of interacting BEC through a symmetric double barrier potential in a waveguide for the modified GP equation.…”