^{1,2,3}, A. Lakhssassi

^{2}, R. Vaillancourt

^{3}and Wu-Ming Liu

^{1}

### Abstract

We consider a cubic-quintic Gross–Pitaevskii equation which governs the dynamics of Bose–Einstein condensate matter waves with time-dependent scattering length and spatiotemporal complex potential. By introducing phase-imprint parameters in the system, we present the integrable condition for the equation and obtain the exact analytical solutions, which describe the propagation of a solitary wave. By applying specific time-modulated feeding/loss functional parameter, various types of magnetic trap strengths, and phase-imprint parameters, the dynamics of the solutions can be controlled. Solitary wave solutions with breathing and snaking behaviors are reported.

This work was supported by the Chinese Academy of Sciences Visiting Professorship for Senior International Scientists, the NKBRSFC under Grant Nos. 2011CB921502, 2012CB821305, 2009CB930701, and 2010CB922904, the NSFC under Grant Nos. 10934010 and 60978019, and the NSFC-RGC under Grant Nos. 11061160490 and 1386-N-HKU74810.

The first author, E.K., dedicates this work to his father-in-law, Papa SADO Jean Le Prince.

I. INTRODUCTION

II. DERIVATIVE CUBIC-QUINTIC GP EQUATION AND INTEGRABLE CONDITION

III. SYMMETRY REDUCTION METHOD FOR SOLITARY WAVE-LIKE AND JACOBIAN ELLIPTIC FUNCTION SOLUTIONS UNDER THE INTEGRABLE CONDITION (14)

A. Reduction of Eq. (15) to an ODE for Jacobian elliptic functions by a symmetry reduction method

B. Solitary wave-like solutions

C. Periodic Jacobian elliptic function solutions

IV. MODULATIONAL INSTABILITY AND DYNAMICS OF EXACT BRIGHT SOLITARY WAVESOLUTION UNDER THE INTEGRABLE CONDITION

V. CONCLUSION

### Key Topics

- Bose Einstein condensates
- 19.0
- Ginzburg Landau theory
- 17.0
- Atom trapping
- 12.0
- Hydrodynamic waves
- 11.0
- Jacobians
- 11.0

##### H05H1/02

## Figures

The dynamics of a solitary wave in a modulated complex spatiotemporal complicated potential and time-dependent s-wave scattering length. These plots are generated with solution (31) and parameters , g 0 = −1, μ = 1, α0 = 0.2, υ = 0.5, ℓ(0) = 2, and two lens transformation parameters η(t) shown in (a). Plots (b) and (c) are, respectively, realized with γ(t) = 0.1 + cos (1.1t) (red curve in (a)) and γ(t) = −0.1 + cos (1.1t) (blue curve in (a)).

The dynamics of a solitary wave in a modulated complex spatiotemporal complicated potential and time-dependent s-wave scattering length. These plots are generated with solution (31) and parameters , g 0 = −1, μ = 1, α0 = 0.2, υ = 0.5, ℓ(0) = 2, and two lens transformation parameters η(t) shown in (a). Plots (b) and (c) are, respectively, realized with γ(t) = 0.1 + cos (1.1t) (red curve in (a)) and γ(t) = −0.1 + cos (1.1t) (blue curve in (a)).

Effect of the functional phase-imprint parameter h(t) (via the phase-imprint parameter α0) on the dynamics of solitary waves in 1D BEC described by Eqs. (1) and (2) . We use solution (31) and the same parameters as in Fig. 1 to realize plots (a) and (b). Here, we use (a) γ(t) = 0.1 + cos (1.1t) and (b) γ(t) = −0.1 + cos (1.1t).

Effect of the functional phase-imprint parameter h(t) (via the phase-imprint parameter α0) on the dynamics of solitary waves in 1D BEC described by Eqs. (1) and (2) . We use solution (31) and the same parameters as in Fig. 1 to realize plots (a) and (b). Here, we use (a) γ(t) = 0.1 + cos (1.1t) and (b) γ(t) = −0.1 + cos (1.1t).

The dynamics of a solitary wave in a modulated complex spatiotemporal complicated potential and time-dependent s-wave scattering length associated with Eq. (31) in the case of a time-independent strength of the magnetic trap with the same parameters as in Fig. 1 and k = 0.005. (a) γ(t) = 0.1 + cos (1.1t); (b) γ(t) = −0.1 + cos (1.1t).

The dynamics of a solitary wave in a modulated complex spatiotemporal complicated potential and time-dependent s-wave scattering length associated with Eq. (31) in the case of a time-independent strength of the magnetic trap with the same parameters as in Fig. 1 and k = 0.005. (a) γ(t) = 0.1 + cos (1.1t); (b) γ(t) = −0.1 + cos (1.1t).

The dynamics of a bright solitary wave in a spatiotemporal complex potential with a temporal periodic modulation of the strength of the magnetic trap k(t) given by Eq. (18) with parameters A 0 = −0.02, B 0 = 0.1, m = 0.98, . (a) γ(t) = 0.1 + 0.098 cos 1.1t, (b) γ(t) = −0.1 + 0.098 cos 1.1t, and (c) γ(t) = −0.31 + 0.098 cos 1.1t. These plots are associated with the density |ψ(x, t)|2 obtained from solution (31) and with parameters g 0 = μ = 1, , ℓ(0) = 1, υ = 0.75, and α0 = 0.495.

The dynamics of a bright solitary wave in a spatiotemporal complex potential with a temporal periodic modulation of the strength of the magnetic trap k(t) given by Eq. (18) with parameters A 0 = −0.02, B 0 = 0.1, m = 0.98, . (a) γ(t) = 0.1 + 0.098 cos 1.1t, (b) γ(t) = −0.1 + 0.098 cos 1.1t, and (c) γ(t) = −0.31 + 0.098 cos 1.1t. These plots are associated with the density |ψ(x, t)|2 obtained from solution (31) and with parameters g 0 = μ = 1, , ℓ(0) = 1, υ = 0.75, and α0 = 0.495.

Effect of the functional phase-imprint parameter h(t) on the dynamics of solitary waves in the case of a temporal periodic modulation of the strength of the magnetic trap k(t) given by Eq. (18) with the same parameters values as in Fig. 4 . (a) γ(t) = 0.1 + 0.098 cos 1.1t, (b) γ(t) = −0.31 + 0.098cos 1.1t. The two plots are realized with density |ψ(x, t)|2 obtained from solution (31) at time t = 5 and with parameters g 0 = μ = 1, , ℓ(0) = 1, υ = 0.75.

Effect of the functional phase-imprint parameter h(t) on the dynamics of solitary waves in the case of a temporal periodic modulation of the strength of the magnetic trap k(t) given by Eq. (18) with the same parameters values as in Fig. 4 . (a) γ(t) = 0.1 + 0.098 cos 1.1t, (b) γ(t) = −0.31 + 0.098cos 1.1t. The two plots are realized with density |ψ(x, t)|2 obtained from solution (31) at time t = 5 and with parameters g 0 = μ = 1, , ℓ(0) = 1, υ = 0.75.

The density of bright solitary waves given by Eq. (39) for |ψ+(x, t)|2 with the temporal periodic modulation of the strength of the magnetic trap k(t) given by Eq. (18) with the parameters A 0 = 0.1, B 0 = 0.2, m = 0.98, and . The parameters chosen to generate these plots are ℓ(0) = 1, γ(0) = 0, g 0 = 1, ρ cw = 1.1, q 0 = 0.1, ω = −4, υ = 0.75, μ = 1, C 0 = 1, γ(t) = γ0 + 0.098 cos (1.1t): (a) γ0 = 0.1, α0 = 0.1, ; (b) γ0 = 0.1, α0 = 0.1, ; (c) γ0 = −0.1, α0 = 0.1, ; (d) γ0 = −0.1, α0 = 0.1, .

The density of bright solitary waves given by Eq. (39) for |ψ+(x, t)|2 with the temporal periodic modulation of the strength of the magnetic trap k(t) given by Eq. (18) with the parameters A 0 = 0.1, B 0 = 0.2, m = 0.98, and . The parameters chosen to generate these plots are ℓ(0) = 1, γ(0) = 0, g 0 = 1, ρ cw = 1.1, q 0 = 0.1, ω = −4, υ = 0.75, μ = 1, C 0 = 1, γ(t) = γ0 + 0.098 cos (1.1t): (a) γ0 = 0.1, α0 = 0.1, ; (b) γ0 = 0.1, α0 = 0.1, ; (c) γ0 = −0.1, α0 = 0.1, ; (d) γ0 = −0.1, α0 = 0.1, .

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