Abstract
We consider the one-dimensional (1D) cubic-quintic Gross–Pitaevskii (GP) equation, which governs the dynamics of Bose–Einstein condensate matter waves with time-varying scattering length and loss/gain of atoms in a harmonic trapping potential. We derive the integrability conditions and the compensation condition for the 1D GP equation and obtain, with the help of a cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift, exact analytical solitonlike solutions with the corresponding frequency chirp which describe the dynamics of femtosecond solitons and double-kink solitons propagating on a vanishing background. Our investigation shows that under the compensation condition, the matter wave solitons maintain a constant amplitude, the amplitude of the frequency chirp depends on the scattering length, while the motion of both the matter wave solitons and the corresponding chirp depend on the external trapping potential. More interesting, the frequency chirps are localized and their feature depends on the sign of the self-steepening parameter. Our study also shows that our exact solutions can be used to describe the compression of matter wave solitons when the absolute value of the s-wave scattering length increases with time.
Highlights
First realized experimentally in 1995 for rubidium [1], lithium [2, 3] and sodium [4], Bose–Einstein condensates is a significant, rapidly growing research area at the forefront of contemporary physics
We have considered the cubic-quintic GP equation in a complex potential consisting of parabolic and complex terms which describes the dynamics of the Bose–Einstein condensate (BEC) matter waves with the time-dependent s-wave scattering length, with loss/gain of atoms, and with both the two- and three-body interatomic interactions in a time-dependent harmonic trapping potential
We have demonstrated that the competing cubic-quintic nonlinearity induces propagating solitonlike dark solitons and double-kink solitons in the nonlinear Schrodinger (NLS) equation with self-steepening
Summary
First realized experimentally in 1995 for rubidium [1], lithium [2, 3] and sodium [4], Bose–Einstein condensates is a significant, rapidly growing research area at the forefront of contemporary physics. With the help of the above exact solitonlike solutions of Eq (2), we turn to the analytical investigation of the dynamics of chirped femtosecond solitons and double-kink solitons in BECs described by the GP Eq (2) It follows from different expressions for the wave function ψ(x, t) and the corresponding frequency chirp δω(x, t) found in the previous section that the centre of different solitons corresponding to ψ(x, t) and δω(x, t) is ζ υ g0. This means that the centre of mass of the macroscopic wave packet behaves like a classical particle, and allows us to manipulate the motion of chirped femtosecond solitons and double-kink solitons in BEC systems by controlling the external harmonic trapping potential. As we will see in the below examples, frequency chirp corresponding to femtosecond solitons and double-kink solitons will be localized; chirp associated to double-kink soliton will have a double-kink feature dark or bright dependent on the sign of the self-steepening coefficient α0, while that corresponding to bright femtosecond soliton will have, dependent on the sign of the self-steepening coefficient α0, either a bright or dark soliton feature
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