Class of solitary wave solutions of the one-dimensional Gross-Pitaevskii equation

Abstract

We present a large family of exact solitary wave solutions of the one dimensional GrossPitaevskii equation, with time-varying scattering length and gain/loss, in both expulsive and regular parabolic confinement regimes. The consistency condition governing the soliton profiles is shown to map on to a linear Schrödinger eigenvalue problem, thereby enabling one to find analytically the effect of a wide variety of temporal variations in the control parameters, which are experimentally realizable. Corresponding to each solvable quantum mechanical system, one can identify a soliton configuration. These include soliton trains in close analogy to experimental observations of Strecker et al., [Nature 417, 150 (2002)], spatio-temporal dynamics, solitons undergoing rapid amplification, collapse and revival of condensates and analytical expression of two-soliton bound states, to name a few.PACS numbers: 03.75. Lm, 05.45.Yv, Coherent atom optics is the subject of much current interest due to its relevance to both fundamental aspects of physics, as well as to technology [1,2,3,4,5,6]. For that purpose, lower dimensional condensates e.g., cigarshaped Bose-Einstein condensates (BECs) have been the subject of active study in the last few years [7,8,9,10,11]. Observations of dark and bright solitons [12,13,14], particularly the latter one, since the same is a condensate itself, have generated considerable interest in this area. This has spurred intense investigations about the behavior of condensates in the presence of time-varying control parameters. These include nonlinearity, achievable through Feshbach resonance [15,16,17], gain/loss and the oscillator frequency [18]. The fact that for a condensate in oscillator potential, exact solutions of the Gross-Pitaevskii (GP) equation are not available, makes it extremely difficult to examine the effects of time variation in the aforementioned parameters. In the context of pulse propagation in non-linear optical fibers, a number of authors have recently investigated the effects of variable non-linearity, dispersion and gain or loss. Moores analyzed this problem for constant dispersion and nonlinearity and a distributed gain [19] and obtain a number of exact solutions. Recently, exact solutions of a driven NLSE with distributed dispersion, nonlinearity and gain, which exhibits pulse compression in a twin-core optical fiber, has also been obtained [22].In this Letter, we present a large family of exact solutions of the quasi one-dimensional GP equation, which is the familiar NLSE, with time varying scattering length, gain/loss, in the presence of an oscillator potential, which can be both expulsive or regular. It is shown that, the consistency condition governing the soliton profiles identically maps on to the linear Schrödinger eigenvalue problem, thereby allowing one to solve analytically the GP equation for a wide variety of temporal variations in the control parameters. Corresponding to each solvable quantum mechanical eigenvalue problem, one can identify a soliton-like profile. The...

Manipulating matter rogue waves and breathers in Bose-Einstein condensates

“…Focusing our attention on BECs alone, it is well known that the Gross-Pitaevskii (GP) equation governs the evolution of macroscopic wave function at ultra low temperatures [6,9]. In particular, for cigar shaped BECs, it has been shown that the GP equation can be reduced to the 1D variable coefficient NLS equation [10][11][12] iψ t + 1 2 ψ xx + R(t)|ψ| 2 ψ + 1 2 β(t) 2 x 2 ψ = 0,…”

Excitation states of Bose Einstein condensate coupled by time-dependent potential

“…In addition, if one wants to compare the final results with the experimental results, one has to choose the appropriate time-dependent potential, for further discussions one can see Ref. [15].…”

“…Some authors has written the evolution of the condensate wave function, trapped by three-dimensional anisotropic function, and containing the time-dependent potential describing gain or loss effect as [15] …”

“…In Ref. [15], authors have extended GPE by adding time-dependent potential to explore the gain-loss aspects in pulse propagation in collapse-revival of the condensate. Although the above equation has no analytical solution, they can interpret the final results that it creates the soliton solutions by determining the ansatz solution of wave function and choosing the appropriate the potential.…”

Feshbach resonance and growth of a Bose-Einstein condensate

“…Furthermore, GP gain equation is of special importance in the field of an atom laser, that is a device which produces an intense coherent beam of atoms by a stimulated process [12,13]. In recent years, GP gain equation (1) has also been studied mathematically by some authors [14,15,16]. GP gain equation can also be used in the theory of optical fibers if the harmonic trap potential is assumed to be zero [15,16].…”