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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.

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 [

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 [

In 1980’s, based upon the nonlinear connection theory proposed by Lu et al. [

Since the experimental realization of the Bose-Einstein condensate (BEC) for rubidium and sodium [

i q t + q x x + 2 μ 2 | q | 2 q + ( i β − α x + β 2 x 2 ) q = 0, (1)

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, res- pectively; α and β are all the real numbers; μ is the nonlinearity para- meter; i β is the gain ( β < 0 ) or loss ( β > 0 ) term; − α x represents the linear external potential, while β 2 x 2 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 [

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

For a given ( 1 + 1 ) -dimensional nonlinear evolution equation with two independent variables x and t , we can introduce a set of new variables

X = { x 1 , x 2 , x 3 , ⋯ , x n } = { x , t , x 3 , ⋯ , x n } and define a set of 2-forms I = { σ j }

such that it constitutes a differential closed ideal, which lead to the ( 1 + 1 ) -dimensional nonlinear evolution equation if the ideal is restricted on the solution space S = { x , t , x 3 ( x , t ) , ⋯ , x n ( x , t ) } .

Now we take X as the base space, Y = { y } = { y 1 , ⋯ , y i } named prolongation variables as the fiber space and G as the structure group generated by the prolongation algebra g . We can consider a principle bundle P ( X , G ) and the bundle E ( X , Y , G , P ) associated with P . Define the local cross-section on E , τ : X → E , and its covariant derivatives:

ω i = d y i + Γ r i ( X , y ) d x r = d y i + Γ r a ( X ) λ a i ( y ) d x r , (2)

where i is the dimension of the representation space of the prolongation algebra, Γ r a ( X ) are the coefficients of the connection on the principal bundle P and λ a i ( y ) are the coefficients of the generators of the prolongation algebra g .

Then we introduce the following connection 1-forms:

L k i = L k r i d x r = [ λ a i ( y ) ∂ λ k a ( y ) ∂ x r + C c b a Γ r b ( x ) λ k c ( y ) λ a i ( y ) ] d x r , (3)

C c b a are the structure constants of the prolongation algebra g . Using the induced connection L k r i , we can define the following covariant exterior derivative:

D ∗ ω i = d ω i + L j i ∧ ω j = − 1 2 F r s a λ a i d x r ∧ d x s + 1 2 M j k i λ a i ω j ∧ ω k , (4)

where F r s a and M j k i are the curvature coefficients on P and the torsion coefficients in the fiber space Y , respectively, and given by

F r s a ( X ) = ∂ Γ r a ( X ) ∂ x s − ∂ Γ s a ( X ) ∂ x r + Γ r b ( X ) Γ s c ( X ) C c b a ,

M j k i ( Y ) = λ j a ( y ) ∂ λ a i ( y ) ∂ y k − λ k a ( y ) ∂ λ a i ( y ) ∂ y j . (5)

Requiring I ′ = { σ j , ω 1 , ⋯ , ω i } is an extended closed ideal, we may derive the following equation from Equation (4)

1 2 ( F r s a λ a i d x r ∧ d x s + M l k i ω k ∧ ω l ) = f δ i σ δ + η l i ∧ ω l , (6)

where f δ i and η l i are the zero and one forms on the base manifold X , respectively. Equation (6) can decomposed into the following equations:

1 2 F r s a ( X ) λ a i ( y ) d x r ∧ d x s = f δ i σ δ , 1 2 M l k i ( Y ) ω k ∧ ω l = η l i ∧ ω l , (7)

Equation (7) is called the fundamental equation of the prolongation structure [

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 :

− i q t ∗ + q x x ∗ + 2 μ 2 | q | 2 q ∗ + ( − i β − α x + β 2 x 2 ) q ∗ = 0,

i q t + q x x + 2 μ 2 | q | 2 q + ( i β − α x + β 2 x 2 ) q = 0. (8)

We define the independent variables as X = { x , t , q , q x , q ∗ , q x ∗ } = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } . The Gross-Pitaevskii equation can then be expressed in the following set of two-forms given by [

σ 1 = d x 3 ∧ d x 2 − x 4 d x 1 ∧ d x 2 ,

σ 2 = d x 1 ∧ d x 3 − i d x 4 ∧ d x 2 − i [ 2 μ 2 x 3 x 5 + ( i β − α x 1 + β 2 x 1 2 ) ] x 3 d x 1 ∧ d x 2 ,

σ 3 = d x 5 ∧ d x 2 − x 6 d x 1 ∧ d x 2 ,

σ 4 = − d x 1 ∧ d x 5 − i d x 6 ∧ d x 2 − i [ 2 μ 2 x 3 x 5 + ( − i β − α x 1 + β 2 x 1 2 ) ] x 5 d x 1 ∧ d x 2 , (9)

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., d I ⊂ I . In this closed ideal any local surface element which annuls the σ j also annuls their exterior derivatives d σ j . In order to establish the prolongation structure, we extend the above ideal by adding to it a connection 1-forms, defined by [

ω l = d y l + Γ r a ( X ) λ a l ( y ) d x r , (10)

where X = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } , and y l are the prolongation variable. For some suitably chosen prolongation variables and imposing the closed condition of the extended ideal I ′ = { σ j , ω 1 , ⋯ , ω l } under covariant exterior derivative, it leads to the covariant fundamental equations.

Substituting the above two forms σ j | j = 1 , ⋯ , 4 into the fundamental equation Equation (7), we have [

F 1,2 l − x 4 F 2,3 l − x 6 F 2,5 l + i [ 2 μ 2 x 3 x 5 + ( i β − α x 1 + β 2 x 1 2 ) ] x 3 F 1,3 l − i [ 2 μ 2 x 3 x 5 + ( − i β − α x 1 + β 2 x 1 2 ) ] x 5 F 1,5 l = 0,

F 2 , 4 l − i F 1 , 3 l = 0 , F 2 , 6 l + i F 1 , 5 l = 0 , F 1 , 4 l = F 1 , 6 l = 0 , F r , s l = 0 , ( r , s = 4 , 5 , 6 ) . (11)

Then, substituting the first equation of Equation (5) into Equation (11), we have the following over-determined difference equations

∂ Γ 1 1 ∂ x 4 = 0 , ∂ Γ 1 2 ∂ x 4 = 0 , ∂ Γ 1 3 ∂ x 4 = 0 , ∂ Γ 1 1 ∂ x 6 = 0 , ∂ Γ 1 2 ∂ x 6 = 0 , ∂ Γ 1 3 ∂ x 6 = 0 ,

∂ Γ 2 1 ∂ x 4 + i ∂ Γ 1 1 ∂ x 3 = 0 , ∂ Γ 2 2 ∂ x 4 + i ∂ Γ 1 2 ∂ x 3 = 0 , ∂ Γ 2 3 ∂ x 4 + i ∂ Γ 1 3 ∂ x 3 = 0 ,

∂ Γ 2 1 ∂ x 6 − i ∂ Γ 1 1 ∂ x 5 = 0 , ∂ Γ 2 2 ∂ x 6 − i ∂ Γ 1 2 ∂ x 5 = 0 , ∂ Γ 2 3 ∂ x 6 − i ∂ Γ 1 3 ∂ x 5 = 0 ,

∂ Γ 1 1 ∂ x 2 − ∂ Γ 2 1 ∂ x 1 − x 4 ∂ Γ 2 1 ∂ x 3 − x 6 ∂ Γ 2 1 ∂ x 5 + 2 Γ 1 2 Γ 2 3 − 2 Γ 3 1 Γ 2 2 + i [ 2 μ 2 x 3 x 5 + ( i β − α x 1 + β 2 x 1 2 ) ] x 3 ∂ Γ 1 1 ∂ x 3 − i [ 2 μ 2 x 3 x 5 + ( − i β − α x 1 + β 2 x 1 2 ) ] x 5 ∂ Γ 1 1 ∂ x 5 = 0 ,

∂ Γ 1 2 ∂ x 2 − ∂ Γ 2 2 ∂ x 1 − x 4 ∂ Γ 2 2 ∂ x 3 − x 6 ∂ Γ 2 2 ∂ x 5 + Γ 1 1 Γ 2 2 − Γ 1 2 Γ 2 1 + i [ 2 μ 2 x 3 x 5 + ( i β − α x 1 + β 2 x 1 2 ) ] x 3 ∂ Γ 1 2 ∂ x 3 − i [ 2 μ 2 x 3 x 5 + ( − i β − α x 1 + β 2 x 1 2 ) ] x 5 ∂ Γ 1 2 ∂ x 5 = 0 ,

∂ Γ 1 3 ∂ x 2 − ∂ Γ 2 3 ∂ x 1 − x 4 ∂ Γ 2 3 ∂ x 3 − x 6 ∂ Γ 2 3 ∂ x 5 − Γ 1 1 Γ 2 3 + Γ 1 3 Γ 2 1 + i [ 2 μ 2 x 3 x 5 + ( i β − α x 1 + β 2 x 1 2 ) ] x 3 ∂ Γ 1 3 ∂ x 3 − i [ 2 μ 2 x 3 x 5 + ( − i β − α x 1 + β 2 x 1 2 ) ] x 5 ∂ Γ 1 3 ∂ x 5 = 0. (12)

Solving the over-determined difference equations Equation (12), we obtain the following solutions

Γ 1 1 = − i α 2 β − 2 i λ exp ( − 2 β x 2 ) ,

Γ 1 2 = μ x 3 exp ( − i β x 1 2 / 2 ) ,

Γ 1 3 = − μ x 5 exp ( i β x 1 2 / 2 ) ,

Γ 2 1 = − 4 i ( α 4 β + λ exp ( − 2 β x 2 ) ) 2 + 4 i β λ x 1 exp ( − 2 β x 2 ) + 2 i μ 2 x 3 x 5 ,

Γ 2 2 = [ α μ 2 β x 3 + 2 μ λ x 3 e x p ( − 2 β x 2 ) − μ β x 1 x 3 + i μ x 4 ] e x p ( − i β x 1 2 / 2 ) ,

Γ 2 3 = [ − α μ 2 β x 5 − 2 μ λ x 5 e x p ( − 2 β x 2 ) + μ β x 1 x 5 + i μ x 6 ] e x p ( i β x 1 2 / 2 ) , (13)

with λ as the hidden spectral parameter and the other components are zero.

Let us use the two dimensional linear representation of SL ( 2, R ) [

X 1 = − 1 2 Y 1 ∂ ∂ Y 1 + 1 2 Y 2 ∂ ∂ Y 2 , X 2 = − Y 2 ∂ ∂ Y 1 , X 3 = − Y 1 ∂ ∂ Y 2 . (14)

Setting the transformation X m = λ m i ( Y ) ∂ ∂ Y i , which leads to

λ 1 1 = − 1 2 Y 1 , λ 1 2 = 1 2 Y 2 , λ 2 1 = − Y 2 , λ 2 2 = 0 , λ 3 1 = 0 , λ 3 2 = − Y 1 , (15)

we therefore derive the following Lax-Pairs, given by

Y x = − [ Γ 1 2 T − 1 + Γ 1 1 T 0 + Γ 1 3 T 1 ] Y ,

Y t = − [ Γ 2 2 T − 1 + Γ 1 2 T 0 + Γ 2 3 T 1 ] Y , (16)

where T i | i = − 1 , ⋯ , 1 represent the generators of the SL ( 2, R ) -symmetry [

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

Y x = [ Γ 1 2 e − + Γ 1 1 2 h + Γ 1 3 e + ] Y ,

Y t = [ Γ 2 2 e − + Γ 2 1 2 h + Γ 2 3 e + ] Y , (17)

where ( e ± , h ) are the generators of SL ( 2, C ) Lie algebra [

Besides the previous symmetries, we select the generators of the SU ( 1,1 ) - symmetry [

Y x = [ Γ 1 1 T 1 + ( Γ 1 2 + Γ 1 3 ) T 2 + ( Γ 1 2 − Γ 1 3 ) T 3 ] Y ,

Y t = [ Γ 2 1 T 1 + ( Γ 2 2 + Γ 2 3 ) T 2 + ( Γ 2 2 − Γ 2 3 ) T 3 ] Y , (18)

where T i | i = 1 , ⋯ , 3 are the generators of SU ( 1,1 ) Lie algebra [

Another Lax-representation can be derived in the form

Y x = [ i ( Γ 1 2 + Γ 1 3 ) T 1 − ( Γ 1 2 − Γ 1 3 ) T 2 + Γ 1 1 T 3 ] Y ,

Y t = [ i ( Γ 2 2 + Γ 2 3 ) T 1 − ( Γ 2 2 − Γ 2 3 ) T 2 + Γ 2 1 T 3 ] Y , (19)

and

Y x = ( − Γ 1 2 T − 1 + Γ 1 1 T 0 + Γ 1 3 T 1 ) Y ,

Y t = ( − Γ 2 2 T − 1 + Γ 1 1 T 0 + Γ 2 3 T 1 ) Y , (20)

where T i | i = 1 , ⋯ , 3 and T i | i = − 1 , ⋯ , 1 are the generators of a SU ( 2 ) -symmetry [

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, SL ( 2, C ) , SU ( 2 ) , SU ( 1,1 ) 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. Thus, we have shown that Eqution (1) is Lax integrable by giving its corresponding Lax-Pair Equations (16)-(20).

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

Y x = Γ 1 3 − Γ 1 1 Y − Γ 1 2 Y 2

Y t = Γ 2 3 − Γ 2 1 Y − Γ 2 2 Y 2 . (21)

Then setting [

Y = ρ Y − 1 + Y 0 + ρ − 1 Y 1 + ρ − 2 Y 2 + ρ − 3 Y 3 + ρ − 4 Y 4 + ρ − 5 Y 5 + ⋯ , ( ζ = ρ )

Q = x 3 exp ( − i β x 1 2 / 2 ) , ζ = α 4 β + λ exp ( − 2 β x 2 ) , (22)

and substituting it into Equation (21), then comparing the coefficient of ρ k , we have

k = 2 : Y − 1 ( 2 i − μ Q Y − 1 ) = 0 , k = 1 : Y − 1 , x = Y 0 ( 2 i − 2 μ Q Y − 1 ) , k = 0 : Y 0 , x = − μ Q ⋆ + Y 1 ( 2 i − 2 μ Q Y − 1 ) − μ Q Y 0 2 , k = − 1 : Y 1 , x = 2 i Y 2 − 2 μ Q ( Y − 1 Y 2 + Y 0 Y 1 ) , k = − 2 : Y 2 , x = 2 i Y 3 − μ Q ( Y 1 2 + 2 Y − 1 Y 3 + 2 Y 0 Y 2 ) , ⋮ k = j : Y j , x = ( 2 i − 2 μ Q Y − 1 ) Y j + 1 − μ Q ( ∑ m = 0 j Y m Y j − m ) , (23)

from which we obtain

Y − 1 = 2 i μ Q , Y 0 = 1 μ Q x Q − 2 , Y 1 = 1 2 i μ [ ( Q x Q − 2 ) x + μ 2 ( Q ⋆ + Q x 2 Q − 3 ) ] ,

Y j + 1 = Y j , x + μ Q ( ∑ m = 0 j Y m Y j − m ) ( 2 i − 2 μ Q Y − 1 ) . (24)

From the compatibility condition, the infinitely-many conservation laws for Equation (1) can be expressed as [

∂ D j ∂ x = ∂ F j ∂ t , j = − 1 , ⋯ , ∞ (25)

where the conserved density D j and the conserved flow F j are the following

D − 1 = 2 μ Q Y 0 + ( i μ Q x − 2 μ β Q x ) Y − 1 , D 0 = 2 μ Q Y 1 + ( i μ Q x − 2 μ β Q x ) Y 0 + i μ 2 Q Q ⋆ ,

D 1 = 2 μ Q Y 2+ (i μ Q x − 2 μ β Q x) Y 1 , D j = 2 μ Q Y j + 1 + ( i μ Q x − 2 μ β Q x ) Y j , j = 2 , ⋯ , + ∞

F j = μ Q Y j , j = − 1 , ⋯ , + ∞ . (26)

Using the vanishing boundary condition, we can give the three constants of motions from the obtained conservation laws,

∫ − ∞ + ∞ D − 1 d t = ∫ − ∞ + ∞ D 0 d t = ∫ − ∞ + ∞ D 1 d t = constant . (27)

In order to derive the analytical soliton solutions to Equation (1), we will employ the Hirota bilinear method [

To get the bilinear forms for Equation (1) we introduce the dependent variable transformation

q ( x 1 , x 2 ) = g ( x 1 , x 2 ) f ( x 1 , x 2 ) exp ( i β x 1 2 / 2 ) (28)

where g ( x 1 , x 2 ) is the complex differentiable function, and f ( x 1 , x 2 ) is a real one. Substituting relation (28) into Equation (1), the bilinear equations of Equation (1) turns out to be in the following forms

[ i D x 2 + 2 i β x 1 D x 1 + ( 2 i β − α x 1 ) + D x 1 2 ] g ⋅ f = 0,

D x 1 2 f ⋅ f = 2 μ 2 g ⋅ g ⋆ , (29)

where D denotes the Hirota’s derivative [

To construct the soliton solutions of Equation (1), we expand g ( x 1 , x 2 ) and f ( x 1 , x 2 ) with respect to a formal expansion parameter ϵ as

f = 1 + ϵ 2 f 2 + ϵ 4 f 4 + ⋯ + ϵ 2 i f 2 i + ⋯ ,

g = ϵ g 1 + ϵ 3 g 3 + ⋯ + ϵ 2 i + 1 g 2 i + 1 + ⋯ . (30)

where g 2 i + 1 ( x 1 , x 2 ) is the complex differentiable function, and f 2 i ( x 1 , x 2 ) is a real one

To derive the one-soliton solutions to Equation (1), we truncate expressions Equation (30) as g = ϵ g 1 and f = 1 + ϵ 2 f 2 , setting ϵ = 1 and substituting then into Bilinear forms Equation (29). We obtain the one-soliton solutions to Equation (1) as

q ( x 1 , x 2 ) = A 1 ( ξ 1 + ξ 1 ⋆ ) 2 μ | A 1 | exp [ 1 2 ( θ 1 − θ 1 ⋆ ) + i 2 β x 1 2 − 2 β x 2 ] × sech [ 1 2 ( θ 1 + θ 1 ⋆ ) + 2 β x 2 + 2 ln ( μ | A 1 | ξ 1 + ξ 1 ⋆ ) ] (31)

where

θ j ( x 1 , x 2 ) = [ α 2 i β + ξ j exp ( − 2 β x 2 ) ] x 1 + [ α 2 4 i β 2 − 2 β ] x 2 + ξ j 2 4 i β exp ( − 4 β x 2 ) − α ξ j 2 β exp ( − 2 β x 2 ) (32)

it is depicted in

Similarly, in order to derive the two-soliton solutions, we can choose

g 1 = A 1 e x p ( θ 1 ) + A 2 e x p ( θ 2 ) ,

f 2 = B 1 1 ∗ exp ( θ 1 + θ 1 ∗ ) + B 1 2 ∗ exp ( θ 1 + θ 2 ∗ ) + B 2 1 ∗ exp ( θ 2 + θ 1 ∗ ) + B 2 2 ∗ exp ( θ 2 + θ 2 ∗ ) ,

g 3 = C 11 1 ∗ exp ( 2 θ 1 + θ 1 ∗ ) + C 11 2 ∗ exp ( 2 θ 1 + η 2 ∗ ) + C 12 1 ∗ exp ( θ 1 + θ 2 + θ 1 ∗ ) + C 12 2 ∗ exp ( θ 1 + θ 2 + θ 2 ∗ ) + C 22 1 ∗ exp ( 2 θ 2 + θ 1 ∗ ) + C 22 2 ∗ exp ( 2 θ 2 + θ 2 ∗ ) ,

f 4 = D 11 1 ∗ 1 ∗ exp ( 2 θ 1 + 2 θ 1 ∗ ) + D 22 1 ∗ 1 ∗ exp ( 2 θ 2 + 2 θ 1 ∗ ) + D 11 2 ∗ 2 ∗ exp ( 2 θ 1 + 2 θ 2 ∗ ) + D 22 2 ∗ 2 ∗ exp ( 2 θ 2 + 2 θ 2 ∗ ) , (33)

where

B m n ∗ = μ 2 A m A n ∗ 4 ( θ m , x 1 + θ n , x 1 ∗ ) 2 , C m m p ∗ = − M m m p − ( θ ) A m B m p ∗ M m m p + , ( p = 1 , ⋯ , 2 m = n = 1 , ⋯ , 2 ) ,

C m n p ∗ = − M m n p − ( θ ) A m B n p ∗ + M n p m − ( θ ) A n B p m ∗ M m n p + , ( p = 1 , ⋯ , 2 m = 1 , ⋯ n = 2 ) ,

M m n p ∓ ( θ ) = [ i ( θ m , x 2 ∓ θ n , x 2 ∓ θ p , x 2 ∗ ) + ( θ m , x 1 ∓ θ n , x 1 ∓ θ p , x 1 ∗ ) 2 + 2 i β ( θ m , x 1 ∓ θ n , x 1 ∓ θ p , x 1 ∗ ) + ( 2 i β − α x 1 ) ] ,

D m n p l ∗ = − C m n p ∗ C n p l ∗ B m p ∗ ( θ m , x + θ p , x ∗ ) 2 , ( p = l = 1 , ⋯ , 2 m = n = 1 , ⋯ 2 ) . (34)

The two-soliton solutions to Equation (1) is written as

q ( x 1 , x 2 ) = ( g 1 + g 3 ) e x p ( i β x 1 2 2 ) 1 + f 2 + f 4 . (35)

and the corresponding depiction is shown in

If one and two-soliton solutions are calculated, then it is possible to generate the multi-soliton solution in the same way.

Throughout the present paper, we investigated the prolongation structure of the 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-Pitaevskii equation such as SL ( 2, R ) , SL ( 2, C ) , Virasoro algebra, SU ( 1,1 ) and SU ( 2 ) . 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. 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.

The authors would like to express their sincere thanks to the editors, anonymous reviewers and all the members of our discussion group for their critical comments and appropriate suggestions which have made this paper more precise and readable.

Abbagari, S., Halidou, H., Bouetou, T.B. and Kofane, T.C. (2017) Covariant Prolongation Structure, Conservation Laws and Soliton Solutions of the Gross-Pitaevskii Equation in the Bose- Einstein Condensate. Journal of Applied Mathematics and Physics, 5, 1411-1423. https://doi.org/10.4236/jamp.2017.57116