Exactly solvable Gross-Pitaevskii type equations

The Gross-Pitaevskii equation is a nonlinear equation and is difficult to solve. There are few exact solutions for the Gross-Pitaevskii type equation. In this paper, we present a method to construct exactly solvable Gross-Pitaevskii type equations. We find that every Gross-Pitaevskii type equation belongs to a certain family. Every family consists of infinite number of Gross-Pitaevskii type equations. Once an equation in a family is solved, the solutions of all other equations in the family can be obtained immediately by a transform. For example, the nonlinear Schr\"odinger equation, a special case of the Gross-Pitaevskii equation, belongs to a family, so the solutions of all the other Gross-Pitaevskii type equations in the family can be obtained from the solution of the nonlinear Schr\"odinger equation. That is, one can construct infinite number of exactly solvable Gross-Pitaevskii type equations from a Gross-Pitaevskii type equation whose solution is known. As examples, we also consider the family of the quintic Gross-Pitaevskii equation and the family of the cubic-quintic Gross-Pitaevskii equation. Moreover, we also show that for the three-dimensional Gross-Pitaevskii type equation, there also exist such kinds of families.


Introduction
The Gross-Pitaevskii (GP) equation has many applications in various branches of physics. In Bose-Einstein condensation, the GP equation is used to describe dilute Bose gases [1], the BEC in optical lattices [2,3], and the spinor BEC [4,5]. Moreover, the dynamics of BEC is studied by the time-dependent GP equation from first-principle [6] and the limit of GP equation's emergence is also discussed [7]. Some methods based on the GP equation, e.g., the truncated Wigner method [8], the positive-P method [9], the mean-field theory, and the Hartree-Fock-Bogoliubov method [10], are used to describe BEC. The GP equation is also used to describe the Josephson plasma oscillations [11]. In general relativity, the GP equation is used to study the gravastar of black hole physics [12] and the black hole in the anti-de Sitter space [13]. The GP equation is a nonlinear equation and is difficult to solve. Many studies are devoted to solving the GP equation, such as stationary solutions [14], numerical solutions [15][16][17], analytical solutions [18], and soliton solutions [19][20][21][22]. Some methods for solving the GP equation are developed, such as the inverse scattering method [23]. Some solutions of the GP equation with various external potentials are obtained, e.g., the harmonic-oscillator potential [24], the multi-well potential [25], the changed external trap [26], the nonlinear lattice pseudopotential [27], the external magnetic field [28], and a sort of parity-time-symmetric potentials [23,29]. Moreover, the problem of scattering is also studied [30]. In this paper, we suggest a method for constructing exactly solvable GP type equations.
The time-independent GP equation is where g is the coupling constant and U eff (r) = U (r) − µ with U (r) the external potential and µ the chemical potential. Here we take = 1 and the mass 2m = 1 for simplicity. The variable coefficient GP equation, also called the inhomogeneous GP equation, whose variable coefficients are space-dependent is a kind of important GP type equations [31][32][33].
In the following, we consider the time-independent GP type equation in a general form, which includes any power of the density |ψ (r)| and the coefficient is spatially inhomogeneous and depends on the coordinate. , some GP type equations are better models for describing BEC, e.g., the cubic-quintic GP equation which takes the three-particle interactions into account [34].
In this paper, we show that there exists a relation between the GP type equations. If two GP type equations are connected by the relation, their solutions will be connected by a transform. The GP type equations who are connected by the relation form a family. A family consists of infinite number of GP type equations, and every GP type equation belongs to a certain family. In a family, once an equation is solved, the solutions of all other equations in the family can be obtained immediately by the transform.
As examples, we consider three families of the GP type equation, the family of the nonlinear Schrödinger equation, the family of the quintic GP equation, and the family of the cubic-quintic GP equation. These three GP type equations belong to three families. Each family consists of infinite number of GP type equations. The nonlinear Schrödinger equation, the quintic GP equation, and the cubic-quintic GP equation have exact solutions. By the transform obtained in the present paper, we can solve all the GP type equations in these three families exactly.
In section 2, we show that there exist families of the GP type equation and take the nonlinear Schrödinger equation as an example. In sections 3 and 4, we discuss the family of the quintic GP equation and the family of the cubic-quintic GP equation. In section 5, we discuss the family of the three-dimensional spherically symmetric GP type equation. The conclusions are summarized in section 6.

The family of the Gross-Pitaevskii type equation
In this section, we show that there exists a relation among the GP type equations. If the GP type equations are related by such a relation, their solutions are related by a transform, i.e., their solutions can be transformed to each other by the transform. All the GP type equations who are related by the relation form a family. In a family, once a family member is solved, the solutions of all other family members can be obtained by the transform immediately.

The relation and the transform
For the GP type equations we have the following relation.
Two one-dimensional GP type equations their solutions are related by the transform: Here σ is a constant chosen arbitrarily. This result can be verified directly. Substituting the transforms (2.5) and (2.6) into the GP type equation (2.1) gives This is just the one-dimensional GP type equation (2.2) with

The family
In the relation ( In a family of the GP type equation, the family members are connected by a transform with a transform parameter σ. This implies that there exists an algebraic structure.

The family of the Gross-Pitaevskii equation
The GP equation is the most important special case of the GP type equation (1.2), which has only the |ψ (x)| 2 term and g 2 (x) = g, The family containing the GP equation (2.10), by the relations (2.3) and (2.4), consists of the following family members: (2.14) The GP type equations with the potential (2.12) and the coupling (2.13) forms a family. The specificity of this family is that the GP equation itself is a member of the family.

The family of the nonlinear Schrödinger equation
The stationary nonlinear Schrödinger equation [35] is a special case of the GP equation (2.10) with a vanishing external potential and the chemical potential µ replaced by the energy E.

The equation connected to the stationary nonlinear Schrödinger equation (2.15), by the relations (2.3) and (2.4), is a GP type equation
The GP type equations with the potential (2.17) and the coupling (2.18) form a family, a family with the nonlinear Schrödinger equation as one of its family members.
It can be checked that the nonlinear Schrödinger equation (2.15) has a solution Different choices of the parameter σ lead to different potentials in Eq. (2.16). For example, σ = 1/2 and E = α/4 give and σ = 2 and E = 4α give The family of the quintic Gross-Pitaevskii equation The quintic GP equation describes BEC when the interaction between the atoms is moderate or strong [34]. The quintic GP equation has a solution [34] ψ (x) = 3 −g 4 where g 4 is a negative constant.

Conclusion
We show that there exist families of the GP type equations. A family contains infinite number of GP type equations. The solutions of the GP type equations in a family are related by a transform. So long as one GP type equation in a family is solved, the solution of all other family members can be obtained immediately by the transform. That is, starting with a solved GP equation, we can construct a series of solvable GP equations.
As examples, we consider the families of the nonlinear Schrödinger equation, the quintic GP equation, and the cubic-quintic GP equation.
The GP type equation is difficult to solve. The method presented in the paper provides an approach to construct exactly solvable GP type equations.