Covariant Prolongation Structure , Conservation Laws and Soliton Solutions of the Gross-Pitaevskii Equation in the Bose-Einstein Condensate

In this paper, we investigate the Gross-Pitaevskii (GP) equation which describes the propagation of an electron plasma wave packet with a large wavelength and small amplitude in a medium with a parabolic density and constant interactional damping by the Covariant Prolongation Structure Theory. As a result, we obtain general forms of Lax-Pair representations. In addition, some hidden structural symmetries that govern the dynamics of the GP equation such as ( ) SL 2,R , ( ) SL 2,C , Virasoro algebra, ( ) SU 1,1 and ( ) SU 2 are unearthed. Using the Riccati form of the linear eigenvalue problem, infinite number of conservation laws of the GP equation is explicitly constructed and the exact analytical soliton solutions are obtained by employing the simple and straightforward Hirota’s bilinear method.


Introduction
Nonlinear evolution equations (NLEEs) have been studied in diverse areas in physics and applied mathematics such as plasma physics, nonlinear optical fibers, condensed matter etc [1] [2] [3].The inhomogeneous nonlinear Schrödinger equation among those (NLEEs) describes the propagation of an electron plasma wave packet with a large wavelength and small amplitude in a medium with a parabolic density and constant interactional damping [4] [5].
As it is known, the investigation of integrability of certain kinds of (NLEEs) by many researchers has generated a great deals of attention over the past years and now many methods to analyze the complete integrability of nonlinear evolution equations are developed.Among them, Wahlquist and Estabrook's prolongation technique [6]- [23] is a powerful and systematic method to test the integrability of the physically important nonlinear evolution equations.By virtue of these techniques, Morris [23] has analyzed the multi-component nonlinear Schrödinger equations.In 1980's, based upon the nonlinear connection theory proposed by Lu et al. [24] Guo et al. [25] [26] [27] have proposed a covariant geometry theory for the prolongation structure of the nonlinear evolution equation.
Since the experimental realization of the Bose-Einstein condensate (BEC) for rubidium and sodium [28] [29], study on the properties of the BEC systems has attracted peoples attention [30] [31].The BECs have also been observed in certain atomic gases such as the lithium, hydrogen, helium and potassium atomic gases [32] [33].For a cigar-shaped BEC at a relatively low density, when the energy of the two-body interaction is much less than the kinetic energy in the transverse direction, the BEC system can be treated as quasi-one-dimensional [34] [35].In the following paper, we plan to investigate the following Gross-Pitaevskii equation in the Bose-Einstein condensate [36] [37] by the covariant prolongation structure theory: ( ) where ( ) , q x t represents the mean-field wave function of the Bose-Einstein condensate; x and t are the normalized distance and retarded time, respectively; α and β are all the real numbers; µ is the nonlinearity para- meter; iβ is the gain ( ) x β accounts for the harmonic external potential.
In soliton theory and other fields of science and engineering, the language of technical computing played a very important role in analytically dealing with large amounts of complicated and tedious algebraic calculations [38] [39].In this paper, we will employ symbolic computation to study the integrability aspects and relevant soliton structures of Gross-Pitaevskii equation in the Bose-Einstein condensate [36] [37].
However, to our knowledge, for Equation (1), Lax-Pair, Conservation laws, multiple soliton solutions via Hirota's method and symbolic computation have not been discussed yet.Motivated by the above, a Lax-Pair based on the generators of some hidden structural symmetries governed the dynamics of the (GP) equation will be got in section 3.In section 4, an infinite sequence of conservation laws of Equation ( 1) are obtained.In addition, in section 5 we present the exact one and two soliton solutions of the Gross-Pitaevskii Equation

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in the Bose-Einstein condensate.Finally, the conclusion will be addressed in section 6.

Covariant Theory for Prolongation Structure of Nonlinear Evolution Equations
For a given ( ) where i is the dimension of the representation space of the prolongation algebra, Then we introduce the following connection 1-forms: a cb C are the structure constants of the prolongation algebra g .Using the induced connection i kr L , we can define the following covariant exterior derivative:  is an extended closed ideal, we may derive the following equation from Equation (4) ( ) where i f δ and i l η are the zero and one forms on the base manifold X , respectively.Equation ( 6) can decomposed into the following equations: ( ) ( ) ( ) Equation ( 7) is called the fundamental equation of the prolongation structure [26] [27].In general, we may completely determine the prolongation structure of a given nonlinear system when the solution of the one fundamental equation can be found.

Covariant Prolongation Structure of Gross-Pitaevskii Equation in the Bose-Einstein Condensate
In order to express Equation ( 1) in differential forms, we add the conjugate equation of Equation ( 1) to Equation ( 1) and obtain the following system : ( ) We define the independent variables as , , , , , , , , , , x x X x t q q q q x x x x x x * * = = . The Gross-Pitaevskii equation can then be expressed in the following set of two-forms given by [26] [27] where the letter d denotes the exterior derivative and the symbol ∧ represents the exterior product.In order to ensure complete equivalence between the forms (9) and the Gross-Pitaevskii Equation ( 8), the ideal I must be closed, i.e., dI I ⊂ .In this closed ideal any local surface element which annuls the j σ also annuls their exterior derivatives d j σ .In order to establish the prolongation structure, we extend the above ideal by adding to it a connection 1-forms, defined by [26] [27] ( ) ( ) where , , , , , X x x x x x x = , and l y are the prolongation variable.For some suitably chosen prolongation variables and imposing the closed condition of the extended ideal Then, substituting the first equation of Equation ( 5) into Equation ( 11), we S. Abbagari et al.
have the following over-determined difference equations Solving the over-determined difference equations Equation ( 12), we obtain the following solutions ( ) with λ as the hidden spectral parameter and the other components are zero.
Let us use the two dimensional linear representation of ( ) given by, Setting the transformation we therefore derive the following Lax-Pairs, given by where On the other hand, by selecting the matrix representation of a generators of a ( ) SL 2, C symmetry, the Lax-representation associated to such an algebra is then given by where ( e ± , h ) are the generators of ( ) SL 2, C Lie algebra [8].
Another Lax-representation can be derived in the form ( ) ( ) and ( )

( )
SL 2, R .Such symmetries have some physical implications.For example, the ( ) SU 1,1 -symmetries show that the system (1) possesses some conserved quantities that are rotationally and hyperbolically invariant, respectively.

Conservation Laws of the Gross-Pitaevskii Equation in the Bose-Einstein Condensate
In the following, we will prove the existence of infinitely-many conservation laws, which further verifies the integrability of Equation (1).
By means of the one dimensional linear representation of ( ) SL 2, R , we derive the Riccati equations Then setting [40], ( ) and substituting it into Equation ( 21), then comparing the coefficient of k ρ from which we obtain ( ) ( ) From the compatibility condition, the infinitely-many conservation laws for Equation (1) can be expressed as [41] , 1, , where the conserved density j D and the conserved flow j F are the following ( ) ( ) , Using the vanishing boundary condition, we can give the three constants of motions from the obtained conservation laws,

Exact multisoliton solutions of the Gross-Pitaevskii Equation in the Bose-Einstein Condensate
In order to derive the analytical soliton solutions to Equation (1), we will employ the Hirota bilinear method [42] [43] [44], which is an efficient and direct approach to construct soliton solutions to nonlinear evolution equations via the bilinear forms from the dependent variables transformation.
To get the bilinear forms for Equation (1) we introduce the dependent variable transformation ) where ( ) , g x x is the complex differentiable function, and ( ) , f x x is a real one.Substituting relation (28) into Equation ( 1), the bilinear equations of Equation ( 1) turns out to be in the following forms ( ) where D denotes the Hirota's derivative [42] [43] [44].
To construct the soliton solutions of Equation (1), we expand ( ) , f x x with respect to a formal expansion parameter  as 1 , .
i i g g g g where ( ) is the complex differentiable function, and ( ) To derive the one-soliton solutions to Equation (1), we truncate expressions Equation ( 30) as and substituting then into Bilinear forms Equation (29).We obtain the one-soliton solutions to Equation (1) as where Figure 1.The one-soliton solution via expression (31) with parameters 5 α = ; it is depicted in Figure 1.
Similarly, in order to derive the two-soliton solutions, we can choose and the corresponding depiction is shown in Figure 2.
If one and two-soliton solutions are calculated, then it is possible to generate the multi-soliton solution in the same way.

Conclusion
Throughout the present paper, we investigated the prolongation structure of the -symmetries show that the system (1) possesses some conserved quantities that are rotationally and hyperbolically invariant, respectively.Thus, we have shown that Equation ( 1) is Lax integrable by giving its corresponding Lax-Pair Equations ( 16)- (20).In addition, infinite number of conservation Laws, one and two soliton solutions using Hirota bilinear method have been constructed.The prolongation structure analysis performed in the present study to the system (1) has revealed an infinite number of conserved quantities which stand as strong proof of integrability of this equation.
of the generators of the prolongation algebra g .
coefficients on P and the torsion coefficients in the fiber space Y , respectively, and given by and centreless Virasoro Lie algebra [9] [10].From the previous discussion, it appears that the dynamics of the the Gross-Pitaevskii (GP) equation modeled by Equation (1), are basically governed by internal structural symmetries, including the Virasoro algebra,

Gross-
Pitaevskii equation which describes the propagation of an electron plasma wave packet with a large wavelength and small amplitude in a medium with a parabolic density and constant interactional damping from the viewpoint of covariant prolongation structure.As a result, we have unearthed some hidden structural symmetries governing the dynamics of the Gross-