Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross-Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.


Introduction
Experimental advances in the realization of Bose-Einstein condensation (BEC) in weakly interacting alkali-metal atomic gases [1] have generated great interest in the theoretical study of the BEC. Its states and evolution are described using the Gross-Pitaevskii equation (GPE) [2,3] for the wave function Ψ( x, t) of the condensate confined by external field with potential V ext ( x, t) at zero temperature: Here x ∈ R n x ,ˆ p = −i ∂/∂ x, t ∈ R 1 , ∂ t = ∂/∂t, |Ψ| 2 = Ψ * Ψ, Ψ * is complex conjugate to Ψ, κ is a real nonlinearity parameter, |Ψ( x, t)| 2 is the condensate density, and N [Ψ] = R n |Ψ( x, t)| 2 d x is the number of condensate particles. Following quantum mechanics, we refer to the solutions of the GPE as states.
Equation (1) is of nonlinear Schrödinger equation type with the local cubic nonlinearity κ|Ψ( x, t)| 2 representing the boson interaction in the mean field approximation. Besides the BEC, equation (1) describes a wide spectrum of nonlinear phenomena such as instability of water waves, nonlinear modulation of collisionless plasma waves, optical pulse propagation in nonlinear media and others. In all these cases, space localized soliton-like solutions are of principal interest. However, the wave packets described by equation (1) with V ext = 0 in multidimensional space (n > 1) with focusing nonlinearity (κ < 0) are known to collapse, i.e. the situation where the wave amplitude increases extremely and becomes singular within a finite time or propagation distance (see, for example, [4] for reviews). To eliminate the collapse, which is considered as an artifact of the theory, one has to consider some effects that would render collapse impossible. The nonlocal form of nonlinearity is significant since it can, basically, eliminate collapse in all physical dimensions (n = 2, 3) [5]. Meanwhile, in the derivation of the GPE (1), a nonlocal nonlinearity term R n V ( x, x ′ ) |Ψ( x ′ , t)| 2 d x ′ Ψ( x, t) arises, and it is reduced to κ|Ψ( x, t)| 2 Ψ( x, t), In view of this consideration, of fundamental interest is the study of both the properties and the localized solutions of the nonlocal Gross-Pitaevskii equation. The integrability of the GPE is a nontrivial problem. In the one-dimensional case (n = 1) with V ext = 0, equation (1) (called the nonlinear Schrödinger equation (NLSE)) is known to be exactly integrable by the Inverse Scattering Transform (IST) method [6,7]. This is the only IST integrable case since with V ext = 0 the IST fails even in the one-dimensional case. The same is true for both the nonlocal GPE in all dimensions and the local GPE (1) in a multidimensional case (n > 1). Direct application of symmetry analysis [8,9,10,11,12] to the nonlocal GPE is also hampered by presence of the nonlocal term and external potential in the equation.
In [13,14,15,16] a semiclassical integrability approach was developed for a generalized nonlocal GPE named there a Hartree type equation: Here the linear operatorsĤ(t) = H(ẑ, t) and V (ẑ,ŵ, t) are Weyl-ordered functions [17] of time t and of noncommuting operatorŝ , I = I n×n is an identity (n × n)-matrix. This approach is based on the WKB-Maslov complex germ theory [18,19] and gives a formal solution of the Cauchy problem, asymptotic in formal small parameter ( → 0) accurate to O N/2 , where N is any natural number. The Cauchy problem was considered in the P t class of trajectory concentrated functions (TCFs) introduced in [13,14]. Being approximate one in essence, the semiclassical approach results in some cases in exact solutions.
In the present work we construct an exact solution of the Cauchy problem in the P t class for equation (2) with the linear operators H(ẑ, t) and V (ẑ,ŵ, t) being quadratic inẑ,ŵ: Here, H zz (t), W zz (t), W zw (t), W ww (t) are 2n × 2n matrices, H z (t) is a 2n-vector; ·, · is an Euclidean scalar product of vectors: p, x = n j=1 p j x j ; p, x ∈ R n , z, w = 2n j=1 z j w j , z, w ∈ R 2n .
By solving the Cauchy problem, we obtain a nonlinear evolution operator in explicit form.
With the evolution operator obtained, we formulate a nonlinear superposition principle for the solutions of the nonlocal GPE in the class of TCFs. Also, we give symmetry operators in general form that map each solution of the GPE into another solution. The general constructions are illustrated by examples.

The class of trajectory concentrated functions
To set the Cauchy problem for equations (2), (3), (5), and (6), following [13,14,15], we define a class of functions P t via its generic element Φ( x, t, ): Here, the function ϕ( ξ, t, ) belongs to the Schwartz space S(R n ) with respect to ξ ∈ R n , smoothly depends on t, and is regular in ). The real function S(t, ) and the 2n-dimensional vector function Z(t, ) = ( P (t, ), X(t, )) define the P t class, regularly depend on √ in the neighborhood of = 0, and are to be determined. The functions of the P t class are normalizable with respect to the norm Φ(t) 2 = Φ(t)|Φ(t) , where is a scalar product in the space L 2 (R n x ). At any time t ∈ R 1 the function Φ( x, t, ) ∈ P t is localized in the limit → 0 in the neighborhood of a point of the phase curve z = Z(t, 0). For this reason we call P t the class of trajectory concentrated functions.
For t = 0 the P t class transforms to the P 0 class of functions where Z 0 ( ) = ( P 0 ( ), X 0 ( )) is a point of the phase space R 2n px , and the constant S 0 ( ) can be omitted without loss of generality. The Cauchy problem is formulated for equations (2), (3), (5), and (6) in the P t class of trajectory concentrated functions as
The operators H(ẑ, t), equation (5), and V (ẑ,ŵ, t), equation (6), are self-adjoint, respectively, to the scalar product (8) and to scalar product in the space L 2 (R 2n xy ). Define the mean value Â for a linear operatorÂ(t) = A(ẑ, t) and a state Ψ( x, t, ) as From (2), where [Â,B] − =ÂB−BÂ is the commutator of the operatorsÂ andB. We refer to equation (12) as the Ehrenfest equation for the operatorÂ and function Ψ( x, t, ) as in quantum mechanics [20]. ForÂ = 1, equation (12) gives Ψ(t) 2 = Ψ(0) 2 = Ψ 2 . This implies that the norm of a solution of equations (2) and (3) does not depend on time, and we can use the parameter are the moments of order |α| of the function Ψ( x, t) centered with respect to z Ψ (t, ) = ( p Ψ (t, ), x Ψ (t, )). Here {∆ẑ} α is an operator with a Weyl symbol (∆z Ψ ) α , Along with (13) we use the following notation for the variances of coordinates, momenta, and correlations: The Ehrenfest equations (12) in mean values can be obtained for the operatorsẑ j , {∆ẑ} α and trajectory concentrated functions (7) (see [13,14] for details). However, for solving the equations under consideration, equations (2), (3), (5), and (6), we need only equations for the first-order and second-order momentṡ We call the system (15) the Hamilton-Ehrenfest system (HES) of the second order for equations (3), (5), and (6). Following [13,14], we equate the functional vector-parameter Z(t, ) of the P t class of TCFs (7) with z Ψ (t, ), i.e. p Ψ (t, ) = P (t, ), x Ψ (t, ) = X(t, ). This relates the ansatz (7) to an exact solution of equation (2). Consider a phase space M N , dim M N = N = 3n + 2n 2 , of points g ∈ M N with coordinates Here A ⊺ is a matrix transposed to A. The coordinates of g ∈ M N are written as matrix columns. The HES (15) can be considered a dynamic system in M N : With the substitution equation (17) is rewritten in equivalent form: We call equation (18) a system in variations. Denote by g(t, C) the general solution of equations (16), (17): and byĝ -the operator column Here are arbitrary constants. Given constants C, equation (19) describes a trajectory of a point in the phase space M N .
Proof . By construction, the vector is a partial solution of the system (16), (17), which coincides with g(t, C(ψ)) at the initial time t = 0. In view of uniqueness of the solution of the Cauchy problem for the system (16) and (17), the equality g(t, C(ψ)) = g(t, C(Ψ(t))), is valid. The proof is complete.
We call equation (31) the linear associated Schrödinger equation (LASE) for the nonlinear Gross-Pitaevskii equation (28). More precisely, equation (31) is to be considered as a family of equations parametrized by the constants C of the form (21). Each element of the family (31) is a linear Schrödinger equation with a quadratic Hamiltonian with respect to operators of coordinates and momenta. Such an equation is well known to be solvable in explicit form (see, for example, [21,22]). In particular, partial solutions can be found as Gaussian wave packets and a Fock basis of solutions, and Green function can be constructed.

LASE solutions and GPE solutions
Consider a relationship between the solutions of the LASE and GPE. Let Φ( x, t, , C) be the solution of the Cauchy problem for the LASE (31) The function Φ( x, t, , C) depends on arbitrary parameters C which appear in the LASE (31). Let C are subject to the condition (23) and are functionals C(ψ).
The relationship between the steps described above can be shown diagrammatically: Theorem 1 makes it possible to obtain a nonlinear evolution operator for the GPE (27) in the P t class of TCFs (7). The evolution operator can be written as a nonlinear integral operator using the Green function of the LASE (31) with constants C changed by C(ψ) according to relation (34).
Here the operatorĤ(t, g(t, C)), given by (32), is quadratic in coordinates and momenta. We shall seek for the required Green function under the simplifying assumption det H pp (s) = 0.
For the evolution operator (42), the following properties can be verified by direct computation.
Theorem 4. The operatorsÛ κ t, · =Û κ t, 0, · possess the group propertŷ Proof . The functions Ψ( x, t) =Û κ t + s, ψ ( x) and Ψ( x, t) of (44) are partial solutions of equation (31) with the same trajectory g(t, C) in the extended phase space, and (48) is valid for the evolution operator of the linear equation (31). Then, it is also valid for g(t, C(ψ)) corresponding to the nonlinear evolution operator (42). Substituting t + s → t in (48), we find for a solution Ψ( x, t) of the Cauchy problem for the GPE (27) with the initial condition (50). It is rare for this problem to be solved (see, for example, [23]). The symmetry analysis of differential equations deals mainly with generators of one-parametric families of symmetry operators (symmetries of an equation) determined by linear equations [8,9,11,12,10]. Using the evolution operatorÛ κ (t, s, ·) given by (42), we can formulate a general form for symmetry operators of the Gross-Pitaevskii equation (27).

Symmetry and nonlinear superposition Symmetry operators
Letâ be an operator acting in P 0 , (â : P 0 → P 0 ) and Ψ( x, t) is an arbitrary function of the P t class (Ψ( x, t) ∈ P t ). Consider an operatorÂ(·), such that If Ψ( x, t) is a solution of the GPE (27), then Φ( x, t) is also a solution of equation (27). This follows immediately from Theorems 2 and 3, and Corollary 1. Thus, the operatorÂ(·) determined by (51) is a symmetry operator for the GPE (27). Assume now that operatorb and its operator exponent exp(αb) act in the P 0 class, i.e., b : P 0 → P 0 and exp(αb) : P 0 → P 0 , where α is a parameter.
Define a one-parametric family of operatorsB(α, ·) via their action on an arbitrary function Ψ( x, t) ∈ P t aŝ By analogy with the aforesaid, the operatorsB(α, ·) constitute a one-parametric family of the symmetry operators of equation (27).
It is easy to verify the group propertŷ Differentiating (52) with respect to the parameter α, we obtain for α = 0 The operatorĈ(·) determined by (54) is a generator of the one-parametric family of symmetry operators (52). Note that the operatorĈ(·) is not a symmetry operator for equation (27) since the parameters C in the evolution operatorÛ κ (t, ·) (42) depend on α. Indeed, the parameters (C) found from equation (43) include the parameter α explicitly. Therefore, equation (54) includes the derivatives of the evolution operatorÛ κ (t, ·) with respect to the parameters C, and (54) is different in form from the symmetry operator (51).

Nonlinear superposition
The nonlinear superposition principle for the GPE (27) can be formulated in terms of the evolution operator Let be two partial solutions to the GPE (27) corresponding to the initial functions ψ 1 ( x), ψ 2 ( x) ∈ P 0 , respectively. Then, the function is a solution of equation (27) which corresponds to the initial function c 1 ψ 1 ( x) + c 2 ψ 2 ( x), c 1 , c 2 ∈ R 1 . Therefore, is a solution of equation (27) which corresponds to the solutions Ψ 1 ( x, t), Ψ 2 ( x, t) of the form (55), and equation (56) is the superposition principle for the GPE (27).

1D case
To demonstrate symmetry operators in explicit form consider the one-dimensional case of the GPE (27) following [16]. The operators (57) and (58) for n = 1 take the form Here, x = x ∈ R 1 , y = y ∈ R 1 ; k > 0, m, e, E, a, b, and c are parameters of the potential.

Concluding remarks
The evolution operator (42) obtained in Section 3 in explicit form enables one to look in a new fashion at the problem of construction of semiclassically concentrated solutions to the nonlocal Gross-Pitaevskii equation (2). In particular, the semiclassical asymptotics constructed in [13,14] can be presented in more compact and visual form. In addition, the evolution operator leads to the nonlinear superposition principle (56) and to the general form of symmetry operators (51). The latter can be used for study of symmetries of the Gross-Pitaevskii equation under consideration as long as direct finding of the symmetries by means of solution of the determining equations is a nontrivial problem.