Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.


Introduction
Symmetry operators, which, by definition, leave the set of solutions of an equation invariant, are of essential importance in the symmetry analysis of nonlinear partial differential equations (PDEs). The obvious use of symmetry operators of an equation is to generate new solutions from a known one. A modern symmetry analysis of differential equations (DEs) is based on Lie group theory. For example, if for an ordinary differential equation (ODE) there exists a Lie group of point transformations (point symmetries) which act on the space of independent and dependent variables, then they map any solution to another solution of the equation. In a more general case, an ODE can admit contact transformations (contact symmetries) acting on the independent and dependent variables, and also on the first derivatives of the dependent variables. In other words, point symmetries and contact symmetries provide examples of Lie groups of symmetry operators. The Lie group methods, as well as their applications to ODEs and PDEs, are described in many books and review articles (see, e.g., [7]). The prolongation of the action of a Lie group on the space of independent variables, dependent variables, and partial derivatives of the dependent variables up to any finite order allows to apply the Lie group theory to studying symmetries of PDEs [30]. The fundamental property of a Lie group is that it is completely characterized by its infinitesimal operator (generator). Given a system of DEs, finding the Lie symmetry group is reduced to solving a system of equations that determine the Lie group generators. The principal point is that the determining equations for the generator are linear and homogeneous. For a nonlinear PDE the determining equations take the form of an overdetermined system of linear homogeneous PDEs, which can be solved step-by-step to obtain infinitesimal operators in explicit form (see, e.g., [8,29,30]).
Solving the determining equations for given system of PDEs we can find the generators of point or contact symmetries for the system. Following the Lie theory, we can recover the Lie group of finite (i.e., not infinitesimal) symmetry transformations for given system of PDEs. Solutions of the determining equations, however, may contain not only independent variables, dependent variables, and first order derivatives (as with point and contact symmetries), but also higher-order derivatives. Generators of this type are called higher symmetries and they do not yield finite Lie groups. Higher symmetries are related to the so-called Lie-Bäcklund transformations that are widely used in symmetry analysis (see [2] and also, e.g., [29] and [8]). Note that higher-order symmetries do not generate symmetry operators. However, no general approaches to direct calculation of symmetry operators for nonlinear equations are known other than the use of the Lie group formalism. This is due to that the determining equations for symmetry operators are nonlinear operator equations. Solving them is a complicated mathematical problem which requires special techniques not developed yet. In addition, in order to solve determining equations for symmetry operators, we have to specify the structure of symmetry operators consistent with the determining equations, but there are no recipes for choosing such a structure. Therefore, finding the symmetry operators for nonlinear equations is in general an unrealistic task.
Note that for linear PDEs, symmetry operators which are widely used in quantum mechanics applications can be effectively found from linear determining equations, (see, e.g., [14,21,31] and references therein). This inspired us to seek a special class of nonlinear equations for which symmetry operators could be calculated using the methods applicable to linear equations. As an example of such a class of nonlinear equations we consider nonlinear integro-differential equations (IDEs) with partial derivatives. We call the equations of this class nearly linear equations. Symmetry operators for them can be found by solving linear operator equations (similarly to those for linear PDEs) and additional algebraic equations. We consider a generalized multidimensional integro-differential Gross-Pitaevskii equation (GPE) with partial derivatives and a nonlocal cubic nonlinear interaction term of general form. The WKB-Maslov method of semiclassical asymptotics [4,24] is used to obtain a reduced GPE from the original GPE. The reduced GPE is quadratic in spatial coordinates and derivatives, and it contains a nonlocal cubic nonlinear interaction term of special form. This equation belongs to the class of nearly linear equations and determines the principal term of semiclassical asymptotic solution.
The main result of our work is an approach developed for finding symmetry operators for a reduced GPE by solving linear operator equations. This approach is illustrated by an example of a one-dimensional reduced GPE for which symmetry operators can be found explicitly. Using symmetry operators obtained two families of exact solutions can be generated for the reduced GPE. In Section 2 the integro-differential Gross-Pitaevskii equation is considered and its semiclassical reduction is presented. A method for integrating the reduced GPE is described and the essential idea of the method is realized; namely, the consistent system and the linear equation associated with the reduced GPE are found. In Section 3 we propose an approach to finding the class of symmetry operators of the reduced GPE by constructing intertwining operators. The general ideas are illustrated in Section 4 by the example of a one-dimensional GPE of special type. The symmetry operators for this equation are found explicitly, and two families of exact solutions are generated making use of the operators obtained.

The nonlocal Gross-Pitaevskii equation and the Cauchy problem
We consider here the Gross-Pitaevskii equation with a nonlocal interaction term of general form.
Using the concepts of the semiclassical WKB-Maslov method, we arrive at a reduced nonlocal GPE and briefly explain an algorithm for solving the Cauchy problem. The Gross-Pitaevskii equation and its modifications are widely used in study of coherent matter waves in Bose-Einstein condensates (BECs) [10]. Recent extensions to BEC studies involve long-range effects in the condensates described by a generalized GPE containing integral terms responsible for nonlocal interactions. We refer to equations of this class as nonlocal GPE (which are also known as Hartree-type equations). The nonlocal BEC models may keep the condensate wave function from collapse and stabilize the solutions in higher dimensions (see, e.g., [20], the review [13] and references therein). Nonlocal GPEs also serve as basic equations of models describing many-particle quantum systems, nonlinear optics phenomena [1], collective soliton excitations in atomic chains [28], etc.
Let us write the nonlocal Gross-Pitaevskii equation aŝ is a smooth complex scalar function that belongs to a complex Schwartz space S in the space variable x ∈ R n at each time t.
The linear operatorsĤ(t) = H(ẑ, t) and V (ẑ,ŵ, t) in (2.1) are Hermitian Weyl-ordered functions [16] of time t and of noncommuting operatorŝ x, y ∈ R n , with the commutators where [Â,B] − =ÂB −BÂ, J = J kj 2n×2n is the unit symplectic matrix: J = 0 −I I 0 2n×2n , and I = I n×n is the n×n identity matrix. We use the space S to provide existence of the moments of Ψ( x, t) and convergence of the integral in (2.2). In what follows, we use the norm Ψ , Hermitian inner product of the functions Φ, Ψ ∈ S, and Φ * denotes the complex conjugate to Φ. From equation (2.1) it follows immediately that the squared norm of a solution Ψ( x, t) is conserved, Ψ(t) 2 = Ψ(0) 2 = const.
A specific and attractive feature of the nonlocal GPE (2.1) is that in the semiclassical approximation the input GPE is reduced to an equation containing nonlocal terms which can be expressed as a finite number of moments of the unknown function Ψ( x, t). The reduced equation can be considered as nearly linear. The concept of the nearly linear equations implies that among the solutions of a nonlinear equation there exists a subset of solutions that regularly depend on the nonlinearity parameter [16]. In the multidimensional case, the GPE (2.1) with variable coefficients of general form cannot be integrated by well-known methods, such as the inverse scattering transform [27]. Therefore, analytical solutions to this equation can be constructed only approximately. An effective approach to constructing asymptotic solutions in this case is to find semiclassical asymptotics as → 0.
Note that semiclassical asymptotic expansions can be assigned to the following basic classes. The semiclassical asymptotic solutions of the equation under consideration are constructed in a chosen class of functions K . The functions of the class K are determined by specific features of the problem and singularly depend on the small parameter . In the general case, such a class of functions is constructed as follows: In the phase space of a dynamic system of equations corresponding to the equation with partial derivatives under consideration (the classical equations of motion in the case of a linear quantum mechanics Schrödinger equation), a Lagrangian manifold Λ k , k ≤ n, is defined. Here k is the dimension of Λ k and n is the dimension of the configuration space of the phase space. The manifold Λ k evolves in time for the Cauchy problem and is invariant for the spectral problem, i.e. Λ k is not deformed and does not move in space. On the manifold Λ k a set of functions is defined. The Maslov's canonical operator projects a function defined in the phase space onto a function given in the configuration space. If k = n, then the canonical operator should be a real phase operator [25], whereas if k < n, then the canonical operator should be a complex phase one [4,24]. In constructing projections of Λ k onto the configuration space, caustics can appear.
The solutions of the first class (k = n) are given by the WKB ansatz with a real phase [25], where the leading term of the asymptotics outside the neighborhoods of the focal points can be written as Semiclassical asymptotic solutions of the form (2.3) for the Gross-Pitaevskii equation were constructed in [15,22,23] (see also [17]).
The solutions of the second class (k = 0) are constructed using a complex WKB-Maslov ansatz [4,24]. For the Gross-Pitaevskii equation, asymptotic solutions of this type are considered in [6,9,34].
Note that constructing of semiclassical asymptotic solutions for nonlinear equations engender a number of problems: In general, the evolution law for a manifold Λ k is unknown. In other words, the "classical dynamics" related to the nonlinear equation under consideration depends on the initial conditions for the equation. Moreover, the relevant "classical dynamics equations" are unknown a priori for the nonlinear equation and to deduce them is a real problem. For the Gross-Pitaevskii equation (2.1), this problem was solved for the class of functions concentrated on a zero-dimensional manifold Λ 0 [6] and for the class of functions concentrated on an ndimensional manifold Λ n [15,22,23].
Following [6], we denote the second class of functions by P t (Z(t, ), S(t, )) and define it as where the function ϕ( ξ, t, ) belongs to the Schwarz space S in the variable ξ ∈ R n , smoothly depends on t, and regularly depends on √ as → 0. Here ∆ x = x − X(t, ), and the real function S(t, ) and the 2n-dimensional vector function Z(t, ) = ( P (t, ), X(t, )), which characterize the class P t (Z(t, ), S(t, )), regularly depend on √ in the neighborhood of = 0 and are to be determined. Note that P t (Z(t, ), S(t, )) ⊂ S. If this does not lead to misunderstanding, we use the contracted notation P t for P t (Z(t, ), S(t, )).
Here to construct symmetry operators we use the complex WKB-Maslov asymptotic solutions of the Cauchy problem for the GPE (2.1) The definition of the class of trajectory-concentrated functions contains the phase trajectory Z(t, ) and the scalar function S(t, ) as "free parameters". The functions belonging to the class P t , at any fixed time t ∈ R 1 are concentrated, as → 0 in the neighborhood of a point lying on the phase curve z = Z(t, 0), t ∈ R 1 [3]. Therefore, it is natural to call the functions of the class P t trajectory-concentrated functions.
The WKB solutions of the form (2.3) are concentrated on a family of phase trajectories whose projections on the configuration space may intersect, giving rise to a caustic problem [25]. On the other hand, all semiclassical asymptotics of the class P t are concentrated on the same trajectory. So we do not face problems with caustics and collapse problem in constructing trajectory-concentrated solutions of the GPE.
Let O( ν ) be an operatorF such that for any function Φ belonging to the space P t the following asymptotic estimate is valid: It may be shown (see [3,6]) that for the functions belonging to P t , the following asymptotic estimate is valid: Let us expand the operatorsĤ(t) = H(ẑ, t) andV (t) = V (ẑ,ŵ, t) in (2.1) as Taylor series in the operators ∆ẑ =ẑ − Z(t, ) and ∆ŵ =ŵ − Z(t, ), respectively, and restrict ourselves to quadratic terms. Then, in view of (2.5), the solution of the Cauchy problem (2.1) and (2.4) asymptotic in a formal small parameter ( → 0) can be constructed 1 accurate to O( 3/2 ) (see [6]). The leading-order term of the asymptotics can be found by reducing the GPE (2.1) to a GPE with a quadratic nonlocal operator.
The higher-order corrections to the leading-order term can be found using perturbation theory [6]. Thus the study of GPEs with a quadratic nonlocal operator is crucial for the construction of semiclassical asymptotics for this type of GPE in the class of trajectory concentrated functions. Without loss of generality, we consider a GPE of the form where the linear operators H qu (ẑ, t) and V qu (ẑ,ŵ, t) are Hermitian and quadratic inẑ,ŵ, respectively: Here H zz (t), W zz (t), W zw (t), and W ww (t) are 2n × 2n matrices; H z (t) is a 2n vector; the angle brackets ·, · denote the Euclidean inner product of vectors: We call equation (2.6) with the linear operators H qu and V qu given by (2.7) and (2.8), respectively, a reduced Gross-Pitaevskii equation (RGPE). An RGPE can be integrated explicitly [19,33] and it possesses very rich symmetries. Analysis of these symmetries can provide a wealth of information about the equation and its solutions.
As an RGPE contains a nonlocal nonlinear term, its symmetry properties are of special interest in the symmetry analysis of partial differential equations. The matter is that the application of the standard methods of symmetry analysis [2,14,29,30], developed basically for PDEs, leads to a number of difficulties when applied to equations different from PDEs: For instance, there are no regular rules for choosing an appropriate structure of symmetries for non-differential equations. This problem can be avoided by using an RGPE as its symmetry properties are closely related to the symmetry of the linear equation associated with the input nonlinear equation.
The key factor in symmetry analysis of the nonlinear equationF (Ψ)( x, t) = 0 is the symmetry operatorÂ that makes the set of solutions of the equation invariant (see, e.g., [21]): Generally, it is impossible to find effectively a symmetry operatorÂ for a given nonlinear operatorF by solving the nonlinear operator equation (2.9). This situation is resolved in the group analysis of differential equations [2,29,30] where a symmetryσ (generator of a Lie group of symmetry operators) is the main object of analysis. The symmetries are determined by the linear operator equation HereF (Ψ) is the Freshet derivative ofF calculated for Ψ. For a linear operatorF , we havê F =F and the symmetry operators being the same as the symmetries. We assign the RGPE (2.6) to the class of nearly linear equations, following the definition given in [18]: A nearly linear equation determining a function Ψ has the form of a linear partial differential equation with coefficients depending on the moments of the function Ψ. This type of equation can be associated with a consistent system which includes a system of ordinary differential equations (ODEs) describing the evolution of the moments and RGPE.
Using the RGPE as an example, we can see that the class of symmetry operators for nearly linear equations can be found by solving the corresponding determining linear operator equations. In this sense, the symmetry properties of nearly linear equations are similar in many respects to those of linear equations.
Let us consider briefly a method for solving the Cauchy problem (2.4) for the RGPE (2.6), following the scheme described in [6]. We denote the Weyl-ordered symbol of an operator A(t) = A(ẑ, t) by A(z, t) and define the expectation value forÂ(t) over the state Ψ( x, t) as As Ψ 2 does not depend on time, we have from (2.6), (2.7), and (2.8) We call (2.10) the Ehrenfest equation for the RGPE (2.6) as is common practice in quantum mechanics for the linear Schrödinger equation (κ = 0 in (2.1)).
Let z Ψ (t) = (z Ψl (t)) and ∆ Ψkl (t) denote the expectation values over Ψ( x, t) for the operatorŝ respectively. Here ∆ẑ l =ẑ l − (z Ψ ) l (t). We call z Ψ (t) the first moments and ∆ From (2.6), (2.7), (2.8), and (2.10) we immediately obtain a dynamical system in matrix notation: (2.11) We call (2.11) the Hamilton-Ehrenfest system (HES) of the second order for the RGPE (2.6) as (2.11) contain the first and second moments. For brevity, we use a shorthand notation for the total set of the first and second moments of Ψ( x, t): Ψ (t) . (2.12) The functions g = g Ψ (t) describe phase orbits in the phase space of system (2.11).
Then the Cauchy problem (2.4) for the RGPE (2.6) can be written equivalently aŝ ) is a concise form of the HES (2.11), and Γ(t, g Ψ (t)) designates the r.h.s. of (2.11). We call the reduced GPE (2.13) and the corresponding HES (2.15) the consistent system for the RGPE (2.6). The reduced GPE (2.13) can be assigned to the class of nearly linear equations [18], as the operator (2.14) of the RGPE (2.13) is a linear partial differential operator with coefficients depending only on the first and second moments g Ψ (t).
The consistent system (2.13), (2.15) allows us to reduce the Cauchy problem for the RGPE (2.13) to the Cauchy problem for a linear PDE, therefore the Cauchy problem (2.16) for HES (2.15) can be solved independently of equation (2.13). Let be the general solution of the HES (2.15) and C = (C 1 , C 2 , . . . , C N ) denote the set of integration constants. Consider a linear PDE with coefficients depending on the parameters C: The operatorĤ q (t, C) of (2.17) is obtained from (2.14) where the general solution g(t, C) of the HES (2.15) stands for the moments g Ψ (t). We call ( where the integration constants C have been replaced by the functionals C = C[ψ] determined from the algebraic conditions Then the solution of the Cauchy problem (2.13), (2.14) for the RGPE (see [6,33] for details) is and, hence, i.e., the functionals C[Ψ](t) are the integrals of (2.1). Also, we have Let us now turn to the construction of symmetry operators for the RGPE (2.6). By using an operator intertwining a pair of ALEs of the form (2.17). Analysis of the GPE of general form involves a great number of additional technical issues associated with the semiclassical approximation that requires a separate study. To illustrate the main ideas of the proposed approach, we restrict our discussion to the case of a quadratic operator for which equation (2.17) is integrable.

The intertwining operator and symmetry operators
In this section, we establish a relationship between the symmetry operators and the intertwining operator for the reduced Gross-Pitavevskii equation (2.6). A class of intertwining operators can be found as a set of products of the fundamental intertwining operator by the symmetry operators of the ALE (2.17).
According to definition (2.9), the nonlinear symmetry operatorÂ(t) maps any solution Ψ( x, t) of equation (2.13) into its another solution: Forâ =Â(t) t=s and ψ( x) given by (2.4), we can set ψ a ( x) =âψ( x) = Ψ A (t) t=s and use the notation g ψa for the first and second moments of ψ a ( x) similar to (2.12).
From the solution of the Cauchy problem for the HES (2.16) with the initial condition g Ψ (t) t=s = g ψa , analogously to (2.24), we have g(t, C[ψ a ]) = g ψa (t). (3.1) According to (2.21), the solutions Ψ( x, t) and Ψ A ( x, t) of the RGPE (2.13) are found as where Φ( x, t, C) and Φ( x, t, C ) are the solutions of two ALEs of the form (2.17) with two different sets of integration constants C and C , respectively, and the corresponding linear operatorsL(t, C ) andL(t, C).
To construct the symmetry operatorÂ(t) we relate the functions Φ( x, t, C [ψ a ]) and Φ( x, t, C[ψ]) by a linear operatorM (t, s, C , C) intertwining the operatorsL(t, C ) andL(t, C): Here the linear operatorR(t, s, C , C) is a Lagrangian multiplier, and the initial condition iŝ M (t, s, C , C)| t=s =â. From (3.3) we have that Φ( x, t, C ) =M (t, s, C , C)Φ( x, t, C) for two arbitrary sets of constants C and C, and this is especially true for Φ( x, t, C [ψ a ]) and Φ( x, t, C[ψ]) with the constants C [ψ a ] and C[ψ].
To find the operatorM (t, s, C , C), we consider a linear intertwining operator D(t, s, C , C) forL(t, C ) andL(t, C) satisfying the conditionŝ L(t, C )D(t, s, C , C) =D(t, s, C , C)L(t, C), and B is the family of linear symmetry operators of the ALE (3.6). Hence, given the operator D(t, s, C , C) of (3.4) and the family B of linear symmetry operators of the ALE (3.6) we can construct the family of nonlinear symmetry operators for the GPE (2.1).
Proof . In view of (3.8) and (3.10), equation (3.4) for the fundamental intertwining operator can be written aŝ Therefore, the operatorL 0 ( x, t) given by (3.10)  Here we used the notation The solution of the Cauchy problem (3.13) for the operator D(t, s, C , C) can be obtained with the standard methods (see, e.g., [3,21]) as Then the symmetry operatorÂ(t) for equation (2.13) (or, equivalently, for equation (2.6)) can be presented as (3.7), where the intertwining operator D(t, s, C , C) is defined by (3.12) and B(t, C) is the symmetry operator for the ALE (2.17).
Using the explicit form (3.12) of the intertwining operator D(t, s, C , C) and the operator K( x, t, s, C) from (3.8), we have Note that expression (3.14) for the symmetry operators is not simple and requires further analysis, but other forms of symmetry operators for GPEs are unknown.
To obtain simplier examples of symmetry operators in explicit form, we consider the 1D case of equations (2.6), (2.7), and (2.8).

Symmetry operators in the 1D case
Based on the results of the previous section, here we construct in explicit form the symmetry operators for the RGPE (2.6) in the one-dimensional case and obtain two countable sets of exact solutions to the one-dimensional GPE using the symmetry operators.
We introduce the notation and assume thatΩ 2 = σ 0 µ − ρ 2 > 0. Indeed, in this case, the general solution of system (4.3) is (4.5) and all solutions of system (4.3) are localized. Assume that the wave packets that describe the evolution of particles by equation (4.1) do not spread. This takes place if Ω 2 =σµ − ρ 2 > 0.
For system (4.4) we have and all solutions of system (4.4) are also localized. Here C = (C 1 , . . . , C 5 ) and C l , l = 1, 5, are arbitrary integration constants. The 1D associated linear equation (2.17) iŝ We can immediately verify that for the associated linear equation (4.7) we can construct the following set of symmetry operators linear in x andp: Here the functions B(t) and C(t) are solutions of the linear Hamiltonian systeṁ B = −ρB −σC, The Cauchy matrix X (t) for system (4.10) can easily be found as The set of solutions normalized by the condition [24] B(t)C * (t) − C(t)B * (t) = 2i (4.12) can be written as (4.13) Equation (4.12) results in the following commutation relations for the symmetry operators (4.8) and (4.9): For the function φ given by (3.8) in the 1D case, we obtain Φ(x, t, C) =K( x, t, s, C)φ( x, t), (4.14) where, according to (3.9), From (3.10) we find Then the symmetry operatorÂ(t) (3.7) for equation (4.1) can be presented as where B(t, C) is the symmetry operator of the associated linear equation (4.7). The intertwining operator D(t, C , C) presented, according to (3.12), as where, according to (3.11), The matrix X (t) is given by (4.11). The symmetry operatorÂ(t) of the nonlinear equation (4.1) involved into (4.17) has the structure of a linear pseudodifferential operator whose parameters are functionals of the function on which the operator acts. Therefore, the explicit form of the operatorÂ(t) is determined not only by the symmetry operator B(t, C) of the associated linear equation, but also by the function Ψ(x, t). Note that for some values of the parameters (more precisely, for the function Ψ(x, t) that defines them) the pseudodifferential operator becomes a differential one.
We set where the operatorâ + (t, C) is defined in (4.9).
The symmetry operatorsÂ ν (t) in (4.28) generalize those of the linear equations used in the Maslov complex germ theory [4,24], as in the limit κ → 0 (κ is the nonlinearity parameter in equation (4.2)), the operatorsÂ ν (t) become the creation operators of the Maslov complex germ theory. As in the linear case (κ = 0), the operatorsÂ ν (t) generate a countable set of exact solutions Ψ ν (x, t) to the nonlinear equation (4.2).

Discussion
Direct calculation of symmetry operators for a nonlinear equation is, as a rule, a severe problem because of the nonlinearity and complexity of the determining equations [26]. However, for nearly linear equations [18] a wide class of symmetry operators can be constructed by solving linear determining equations for operators of this type much as symmetry operators are found for linear PDEs. We have illustrated this situation with the example of the generalized multidimensional Gross-Pitaevskii equation (2.1). The formalism of semiclassical asymptotics leads to the semiclassically reduced GPE (2.6) (or (2.13)), which belongs to the class of nearly linear equations. Note that the solutions of GPE can be found in a special class of functions decreasing at infinity [6]. The reduced GPE is the quadratic one in the space coordinates and derivatives and contains a nonlocal term of special form. In constructing the symmetry operators for the reduced Gross-Pitaevskii equation (2.13), we use the fact that this equation can be associated with the linear equation (2.17). The symmetry operatorÂ(t) of the reduced GPE (2.13), which is a particular case of (3.7), has the structure of a linear pseudodifferential operator with coefficients C depending on the function Ψ on which the operator acts. The operatorÂ is determined in terms of the linear intertwining operator D and of the symmetry operators of the associated linear equation (2.17). The dependence of the coefficients C on Ψ arises from the algebraic condition (3.1), and therefore the operatorÂ(t) is nonlinear. This is the key point of the presented approach. The 1D examples considered show that for a special choice of the parameters C we can construct symmetry operators and generate the families of solutions to the nonlinear equation (4.1) written in explicit form.
The further development of the study of symmetry operators is seen as a generalization to the approach for integro-differential GPEs of more general form and to systems of equations of this type.