^{1,a)}and Dingbian Qian

^{2}

### Abstract

In this paper we give a frame for application of the averaging method to Bose-Einstein condensates(BECs) and obtain an abstract result upon the dynamics of BECs. Using the averaging method, we determine the location where the modulated amplitude waves (periodic or quasi-periodic) exist and obtain that all these modulated amplitude waves (periodic or quasi-periodic) form a foliation by varying the integration constant continuously. Compared with the previous work, modulated amplitude waves studied in this paper have nontrivial phases and this makes the problem become more difficult, since it involves some singularities.

The work of Qihuai Liu was supported, in a part, by Grant No. 11071181 from National Science Foundation of China and by Grant No. 10KJB110009 from Universities Foundation in Jiangsu Province. The work of Dingbian Qian was supported by Grant No. 10871142 from National Science Foundation of China.

I. INTRODUCTION

II. COHERENT STRUCTURE AND MODULATED AMPLITUDE WAVE

III. TRANSFORMATION TO STANDARD FORM OF AVERAGING

IV. AN ABSTRACT RESULT OF AVERAGING TO BECs

V. EQUILIBRIUMS AND THE AVERAGED EQUATION

VI. DISCUSSION AND CONCLUSION

### Key Topics

- Bose Einstein condensates
- 12.0
- Eigenvalues
- 6.0
- Spatial analysis
- 5.0
- Wave functions
- 4.0
- Chemical potential
- 3.0

## Figures

A plot of a MAW ψ_{0}(*x*, *t*) corresponding to the equilibrium for unperturbed system of (1.1) on the right-plane. The original system (1.1) has a MAW or QMAW ψ_{ε}(*x*, *t*) such that for all time *t* and the spatial domains of order 1/ε. The parameters we take as follows: *c* = 1, δ = 1, α = 0.30, *b* _{0} = −12.40, *b* _{1} = −20.20. (a) Spatial amplitude *R*(*x*) plot; (b) Nontrivial spatial phase Θ(*x*) plot; (c) A plot of space-time R*e*[ψ(*t*, *x*)]; (d) The density plot of Re[ψ(*t*, *x*)].

A plot of a MAW ψ_{0}(*x*, *t*) corresponding to the equilibrium for unperturbed system of (1.1) on the right-plane. The original system (1.1) has a MAW or QMAW ψ_{ε}(*x*, *t*) such that for all time *t* and the spatial domains of order 1/ε. The parameters we take as follows: *c* = 1, δ = 1, α = 0.30, *b* _{0} = −12.40, *b* _{1} = −20.20. (a) Spatial amplitude *R*(*x*) plot; (b) Nontrivial spatial phase Θ(*x*) plot; (c) A plot of space-time R*e*[ψ(*t*, *x*)]; (d) The density plot of Re[ψ(*t*, *x*)].

A plot of a MAW ψ_{0}(*x*, *t*) corresponding to the equilibrium (π, 10) for unperturbed system of (1.1) on the right-plane. The original system (1.1) has a MAW or QMAW ψ_{ε}(*x*, *t*) such that for all time *t* and the spatial domains of order 1/ε. The parameters we take as follows: *c* = 1, δ = 1, α = 0.30, *b* _{0} = −12.40, *b* _{1} = −20.20. (a) Spatial amplitude *R*(*x*) plot; (b) Nontrivial spatial phase Θ(*x*) plot; (c) A plot of space-time R*e*[ψ(*t*, *x*)]; (d) The density plot of Re[ψ(*t*, *x*)].

A plot of a MAW ψ_{0}(*x*, *t*) corresponding to the equilibrium (π, 10) for unperturbed system of (1.1) on the right-plane. The original system (1.1) has a MAW or QMAW ψ_{ε}(*x*, *t*) such that for all time *t* and the spatial domains of order 1/ε. The parameters we take as follows: *c* = 1, δ = 1, α = 0.30, *b* _{0} = −12.40, *b* _{1} = −20.20. (a) Spatial amplitude *R*(*x*) plot; (b) Nontrivial spatial phase Θ(*x*) plot; (c) A plot of space-time R*e*[ψ(*t*, *x*)]; (d) The density plot of Re[ψ(*t*, *x*)].

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