Shi-min Liu and Da-jun Zhang

Department of Mathematics,Shanghai University,Shanghai 200444,People’s Republic of China

Abstract An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density |u|2 is conserved.We also present an integrable vector Gross–Pitaevskii system with a parabolic potential,where the total particle density is conserved.These equations are related to nonisospectral scalar and vector nonlinear Schr?dinger equations.Infinitely many conservation laws are obtained.Gauge transformations between the standard isospectral nonlinear Schr?dinger equations and the conserved Gross–Pitaevskii equations,both scalar and vector cases are derived.Solutions and dynamics are analyzed and illustrated.Some solutions exhibit features of localized-like waves.

Keywords:Gross–Pitaevskii equation,gauge transformation,nonisospectral,conserved particle density

1.Introduction

2.The conserved GP equation(4)

In this section,we investigate the conserved GP equation(4)and its integrability.

Let us first explain how we are motivated by solutions of the nonconserved GP equation(2)and arrive at the conserved equation(4).Recalling the carrier wave(particle density)of the explicit 1SS of(2)[10],

Figure 1.Shape and motion of 1SS for equation(4)with δ ＞0.(a)Stationary soliton |u|2 with δ=0.36,a1=0.5,b1=h1=0.(b)The 2D plot of(a)at t=0(red dashed curve),t=1(blue dashed curve),t=2(black dashed curve).

Figure 2.Shape and motion of 1SS for equation(4)with δ ＜0.(a)A moving soliton|u|2 with δ=?0.36,a1=0.5,b1=0.5,h1=0.(b)The 2D plot of(a)at t=0(red dashed curve),t=10(blue dashed curve),t=20(black dashed curve).

3.The conserved vector GP equation(5)

3.1.Lax pair

3.2.Conservation laws

3.3.Gauge equivalence and NSS

3.4.Dynamics for the conserved vector GP equation(5)

In this subsection,we investigate dynamics for the conserved vector GP equation(5).

3.4.1.Solutions for the caseδ ＞0.Taking N=1 in the formula(34)and combined with the transformation(31),it is straightforward to obtain 1SS for the conserved vector GP equation(5).Let us consider the two-component case,i.e.

3.4.2.Solutions for the caseδ ＜0.In the case of δ ＜0,1SS for equation(5)with δ ＜0 can be written as

Figure 3.Shape and motion of 1SS given by(38)for equation(5)with δ=0.36.(a)A moving wave runs like with (b)A moving wave runs likemoving wave runs like(d)A stationary wave with.

4.Conclusions and remarks

In this paper,we have introduced a way to obtain a conserved GP equation with a parabolic potential.This idea was explained and illustrated in section 2,where we started from the non-conserved GP equation(2)with a parabolic potential,by calculating its total particle number N(see equation(7))associated with 1SS,we were led to the transformation(9),and the resulting GP equation(4)turns out to be conserved in terms of total particle number N.The conserved GP equation(4)contains a time-dependent coefficient g(t)to measure inter-particle interactions.This idea was then extended to the vector GP equation in section 3 and the conserved version is given in equation(5).The dynamics of some solutions are illustrated.

We remark that the conserved GP equation(4)has been included in[7]as one of the GP equations that are gauge equivalent to the standard NLS equation(see table 1 in[7]).In our paper,we corrected an inaccurate statement given in[7]about the periodic singularities appearing in the case δ ＜0.

Figure 4.(a)1SS of |u1(x,t)| given by(45)for equation(5)with δ=?0.36,a1=1,b1=1,β1,1=1,β1,2=1.(b)Density plot of(a).

The idea of this paper might be applied to the differential-difference GP models,which will be investigated elsewhere.

Acknowledgments

The authors are grateful to the referees for their invaluable comments.This project is supported by the NSF of China(Nos.11 875 040,12 126 352,12 126 343).