Exactly solvable Gross–Pitaevskii type equations

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, 2], the BEC in optical lattices [3, 4], and the spinor BEC [5, 6]. Moreover, the dynamics of BEC is studied by the time-dependent GP equation from first-principle [7] and the limit of GP equation's emergence is also discussed [8]. Some methods based on the GP equation, e.g., the truncated Wigner method [9], the positive-P method [10], the mean-field theory, and the Hartree–Fock-Bogoliubov method [11] are used to describe BEC. The influence of the inter-particle interaction to BEC is an important problem [12, 13], and the GP type equation is suitable for describing the condensate of the interacting Bose gas. The GP equation is also used to describe the Josephson plasma oscillations [14]. In general relativity, the GP equation is used to study the gravastar of black hole physics [15] and the black hole in the anti-de Sitter space [16]. 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 [17], numerical solutions [18–20], analytical solutions [21], and soliton solutions [22–25]. Some methods for solving the GP equation are developed, such as the inverse scattering method [26]. Some solutions of the GP equation with various external potentials are obtained, e.g., the harmonic-oscillator potential [27], the multi-well potential [28], the changed external trap [29], the nonlinear lattice pseudopotential [30], the external magnetic field [31], and a sort of parity-time-symmetric potentials [26, 32]. The problem of scattering is also studied [33]. Moreover, there are studies on the one-dimensional GP equation and its applications [34–38]. In this paper, we suggest a method for constructing exactly solvable GP type equations, especially for variable-coefficient GP type equations. The GP equation is also important in studying the nonlinear solitary and periodic waves in the condensate of a superfluid [39] and the ultracold Bose-condensed atomic vapors in mesoscopic waveguide structures [40].

The time-independent GP equation is

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}\psi \left({\bf{r}}\right)-\left[{U}_{\mathrm{eff}}\left({\bf{r}}\right)+g{\left|\psi \left({\bf{r}}\right)\right|}^{2}\right]\psi \left({\bf{r}}\right)=0,\end{eqnarray} \tag{ 1.1 }$$

where g is the coupling constant and ${U}_{\mathrm{eff}}\left({\bf{r}}\right)=U\left({\bf{r}}\right)-\mu$ with $U\left({\bf{r}}\right)$ the external potential and μ the chemical potential. Here we take ℏ = 1 and the mass 2m = 1 for simplicity. The variable-coefficient GP equation, also called the inhomogeneous GP equation, whose coefficients are space-dependent is a kind of important GP type equations [41–43]. In the following, we consider the time-independent GP type equation in a general form,

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}\psi \left({\bf{r}}\right)-\left[{U}_{\mathrm{eff}}\left({\bf{r}}\right)+{g}_{1}\left({\bf{r}}\right)\left|\psi \left({\bf{r}}\right)\right|+{g}_{2}\left({\bf{r}}\right){\left|\psi \left({\bf{r}}\right)\right|}^{2}\ +\cdots +\ {g}_{n}\left({\bf{r}}\right){\left|\psi \left({\bf{r}}\right)\right|}^{n}\right]\psi \left({\bf{r}}\right)=0,\end{eqnarray} \tag{ 1.2 }$$

which includes any power of the density $\left|\psi \left({\bf{r}}\right)\right|$ and the coefficient depends on the spatial coordinate. The GP equation, the nonlinear Schrödinger equation, etc., are the special cases of equation (1.2). The GP type equation is used to describe the many-body interaction in BEC [44–51]. On the other hand, the variable-coefficient GP type equation has a wide application in nonlinear optics. The nonlinear Schrödinger equation performs numerical analysis of the pulse mechanism of laser [52–54]. The variable-coefficient nonlinear Schrödinger equation describes the pulse in inhomogeneous optical systems [55] and the interactions between periodic optical solitons [56]. The variable-coefficient cubic-quintic GP equation describes the brightlike and darklike solitary wave solutions [57].

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. In a family, once an equation is solved, the solutions of other equations in the family can be obtained by the transform.

As examples, we consider some families of the GP type equation, the family of the GP 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 GP type equations belong to different families. The solutions of these equations are known, so by the transform we can solve all the equations in their families.

In section 2, we consider the family of the GP type equation. In section 3, we discuss some exactly solvable families, including the nonlinear Schrödinger equation, the quintic GP equation, and the cubic-quintic GP equation. In section 4, we consider the family of the generalized Gross–Pitaevskii type equation. In section 5, we consider the family of the three-dimensional spherically symmetric GP type equation. The conclusions are summarized in section 6.

In this section, we show that there exists a relation between the GP type equations. All the GP type equations who are related by the relation form a family.

2.1. The relation

For the GP type equations we have the following relation.

Two one-dimensional GP type equations

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\psi \left(x\right)}{{{dx}}^{2}}-\left[{U}_{\mathrm{eff}}\left(x\right)+{g}_{1}\left(x\right)\left|\psi \left(x\right)\right|+{g}_{2}\left(x\right){\left|\psi \left(x\right)\right|}^{2}\ +\cdots +\ {g}_{n}\left(x\right){\left|\psi \left(x\right)\right|}^{n}\right]\psi \left(x\right)=0,\end{eqnarray} \tag{ 2.1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-\left[{V}_{\mathrm{eff}}\left(\xi \right)+{G}_{1}\left(\xi \right)\left|\phi \left(\xi \right)\right|+{G}_{2}\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{2}+\cdots +{G}_{n}\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{n}\right]\phi \left(\xi \right)=0,\end{eqnarray} \tag{ 2.2 }$$

if ${U}_{\mathrm{eff}}\left(x\right)$ and ${V}_{\mathrm{eff}}\left(\xi \right),{g}_{l}\left(x\right)$ and ${G}_{l}\left(\xi \right)$ (l = 1, ⋯ ,n) satisfy the relations

$$\begin{eqnarray}&&\sigma \left({x}^{2}{U}_{\mathrm{eff}}\left(x\right)+\displaystyle \frac{1}{4}\right)={\sigma }^{-1}\left({\xi }^{2}{V}_{\mathrm{eff}}\left(\xi \right)+\displaystyle \frac{1}{4}\right),\end{eqnarray} \tag{ 2.3 }$$

$$\begin{eqnarray}&&\sigma {x}^{\left(l+4\right)/2}{g}_{l}\left(x\right)={\sigma }^{-1}{\left|\xi \right|}^{\left(l+4\right)/2}{G}_{l}\left(\xi \right),\quad l=1,\cdots ,n,\end{eqnarray} \tag{ 2.4 }$$

their solutions are related by the transform:

$$\begin{eqnarray}&&x\leftrightarrow {\left|\xi \right|}^{\sigma },\end{eqnarray} \tag{ 2.5 }$$

$$\begin{eqnarray}&&\psi \left(x\right)\leftrightarrow {\left|\xi \right|}^{{}^{\left(\sigma -1\right)/2}}\phi \left(\xi \right).\end{eqnarray} \tag{ 2.6 }$$

Hereσ 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

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-{\sigma }^{2}\left[\displaystyle \frac{1-{\sigma }^{-2}}{4{\xi }^{2}}+{\left|\xi \right|}^{2\left(\sigma -1\right)}{U}_{\mathrm{eff}}\left({\left|\xi \right|}^{\sigma }\right)+{\left|\xi \right|}^{\left(5/2\right)\left(\sigma -1\right)}{g}_{1}\left({\left|\xi \right|}^{\sigma }\right)\left|\phi \left(\xi \right)\right|\right.\\ \quad \left.+\ {\left|\xi \right|}^{3\left(\sigma -1\right)}{g}_{2}\left({\left|\xi \right|}^{\sigma }\right){\left|\phi \left(\xi \right)\right|}^{2}\ +...+\ {\left|\xi \right|}^{\left(\sigma -1\right)\left(n+4\right)/2}{g}_{n}\left({\left|\xi \right|}^{\sigma }\right){\left|\phi \left(\xi \right)\right|}^{n}\right]\phi \left(\xi \right)=0.\end{array}\end{eqnarray} \tag{ 2.7 }$$

This is just the one-dimensional GP type equation (2.2) with

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}\left(\xi \right) & = & {\sigma }^{2}\left[\displaystyle \frac{1-{\sigma }^{-2}}{4{\xi }^{2}}+{\left|\xi \right|}^{2\left(\sigma -1\right)}{U}_{\mathrm{eff}}\left({\left|\xi \right|}^{\sigma }\right)\right],\\ {G}_{l}\left(\xi \right) & = & {\sigma }^{2}{\left|\xi \right|}^{\left(\sigma -1\right)\left(l+4\right)/2}{g}_{l}\left({\left|\xi \right|}^{\sigma }\right),\ l=1,\cdots ,n.\end{array}\end{eqnarray} \tag{ 2.8 }$$

This proves the relations (2.3)–(2.6).

2.2. The family

2.3. The fixed point

In a family, all family members are connected by a transform. This transform has two fixed points.

The transforms (2.5) and (2.6) give

$$\begin{eqnarray}&&\phi \left(\xi \right)={\left|\xi \right|}^{\left(1-\sigma \right)/2}\psi \left({\left|\xi \right|}^{\sigma }\right).\end{eqnarray} \tag{ 2.9 }$$

It can be seen directly that the points

$$\begin{eqnarray}&&\left(1,\psi \left(1\right)\right)\quad \mathrm{and}\quad \left(-1,\psi \left(1\right)\right)\end{eqnarray} \tag{ 2.10 }$$

are fixed points in the transform. That is, in a family, all family members pass through these two points.

2.4. 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 ${\left|\psi \left(x\right)\right|}^{2}$ term and a constant coefficient ${g}_{2}\left(x\right)=g$:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\psi \left(x\right)}{{{dx}}^{2}}-\left[{U}_{\mathrm{eff}}\left(x\right)+g{\left|\psi \left(x\right)\right|}^{2}\right]\psi \left(x\right)=0.\end{eqnarray} \tag{ 2.11 }$$

The family of the GP equation, in which GP equation is one of its family member, by the relations (2.3) and (2.4), consists of the following family members:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-\left[{V}_{\mathrm{eff}}\left(\xi \right)+G\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{2}\right]\phi \left(\xi \right)=0\end{eqnarray} \tag{ 2.12 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}\left(\xi \right) & = & {\sigma }^{2}\left[\displaystyle \frac{1-{\sigma }^{-2}}{4{\xi }^{2}}+{\left|\xi \right|}^{2\left(\sigma -1\right)}{U}_{\mathrm{eff}}\left({\left|\xi \right|}^{\sigma }\right)\right],\\ G\left(\xi \right) & = & g{\sigma }^{2}{\left|\xi \right|}^{3\left(\sigma -1\right)}.\end{array}\end{eqnarray} \tag{ 2.13 }$$

The family members are labeled by the parameter σ.

The solutions of family members are connected by the transforms (2.5) and (2.6):

$$\begin{eqnarray}&&\phi \left(\xi \right)={\left|\xi \right|}^{\left(1-\sigma \right)/2}\psi \left({\left|\xi \right|}^{\sigma }\right).\end{eqnarray} \tag{ 2.14 }$$

In the present paper, we consider the GP type equation in a general form. The general GP type equation contains any power of the wave function with space-dependent coefficients. Here we discuss the correspondence between the general GP type equation and the BEC system.

Under the frame of the mean field theory, the interaction between two particles is proportional to the particle number density ${\left|\psi \right|}^{2}$. Therefore, the term ${\left|\psi \right|}^{2}\psi$ describes the two-body interaction, the term ${\left|\psi \right|}^{4}\psi$ describes the three-body interaction, and so on [44–51] . That is, the even-power of $\left|\psi \right|$ describes the many-body interaction. Moreover, the odd-power term is also used to describe the effect beyond the mean field treatment [58, 59].

The coupling parameter g is proportional to the s-wave scattering length of inter-atomic scattering in BEC [60, 61] and the scattering length can be determined by the Feshbach resonance [47]. In the BEC experiment, the coupling parameter can be controlled by the Feshbach resonance which changes the scattering length [62, 63]. For example, the scattering length can be changed by the magnetic field [64, 65]. The spatial modulation leads to a space-dependent coupling parameter and the temporal modulation leads to a time-dependent coupling parameter [61, 62, 66]. In the present paper, we consider the GP type equations with space-dependent coupling parameter.

For two-body interactions, the coupling parameter of the term ${\left|\psi \right|}^{2}\psi$ is of the magnitude [45, 49]

$$\begin{eqnarray}&&g\sim \displaystyle \frac{4\pi {{\hslash }}^{2}}{m}{a}_{s},\end{eqnarray} \tag{ 3.1 }$$

where a_s is the s-wave scattering length. If only consider two-body interactions, it should satisfy $\sqrt{{{na}}_{s}^{3}}\ll 1$ with n the particle number density.

In the case of high densities, the three-body interaction becomes important and the two-body description is no longer effective [49]. Moreover, the coupling parameter of the three-body interaction, i.e., the coefficient of the term ${\left|\psi \right|}^{4}\psi$, is of the magnitude [49]

$$\begin{eqnarray}&&{g}_{3}=\displaystyle \frac{12\pi {{\hslash }}^{2}{a}_{s}^{4}}{m}\left({d}_{1}+{d}_{2}\right)\tan \left({s}_{0}\mathrm{ln}\left(\displaystyle \frac{{a}_{s}}{{a}_{0}}+\displaystyle \frac{\pi }{2}\right)\right)\end{eqnarray} \tag{ 3.2 }$$

which is proportional to a_s ⁴, so when the scattering length is large, the three-body interaction needs to be taken into account. The coupling parameter g₃ may be a complex number. For example, for the ⁸⁷Rb condensate the real part of g₃/ℏ is about 10⁻²⁶ ∼ 10⁻²⁷ cm⁶s⁻¹ [49, 67, 68] and the imaginary part is about 10⁻³⁰cm⁶s⁻¹ and is ignorable [69]. The other parameters in equation (3.2) can be determined numerically [67, 70].

In this section, we consider some exactly solvable families, the families of the nonlinear Schrödinger equation, the quintic GP equation, and the cubic-quintic GP equation.

3.1. The family of the nonlinear Schrödinger equation

The stationary nonlinear Schrödinger equation [71]

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\psi \left(x\right)}{{{dx}}^{2}}+\left[E-g{\left|\psi \left(x\right)\right|}^{2}\right]\psi \left(x\right)=0\end{eqnarray} \tag{ 3.3 }$$

is a special case of the GP equation (2.11) with a vanishing external potential and the chemical potential μ replaced by the energy E.

The family of the nonlinear Schrödinger equation, by the relations (2.3) and (2.4), consists of the following family members:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-\left[{V}_{\mathrm{eff}}\left(\xi \right)+G\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{2}\right]\phi \left(\xi \right)=0\end{eqnarray} \tag{ 3.4 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}\left(\xi \right) & = & {\sigma }^{2}\left[\displaystyle \frac{1-{\sigma }^{-2}}{4{\xi }^{2}}-{\left|\xi \right|}^{2\left(\sigma -1\right)}E\right],\\ G\left(\xi \right) & = & g{\sigma }^{2}{\left|\xi \right|}^{3\left(\sigma -1\right)},\end{array}\end{eqnarray} \tag{ 3.5 }$$

where ${V}_{\mathrm{eff}}\left(\xi \right)=V\left(\xi \right)-{ \mathcal E }$. The solution of equation (3.4) by the transforms (2.5) and (2.6) is

$$\begin{eqnarray}&&\phi \left(\xi \right)={\left|\xi \right|}^{\left(1-\sigma \right)/2}\psi \left({\left|\xi \right|}^{\sigma }\right).\end{eqnarray} \tag{ 3.6 }$$

It can be checked that the nonlinear Schrödinger equation (3.3) has a solution

$$\begin{eqnarray}&&\psi \left(x\right)=\sqrt{\displaystyle \frac{E}{g}}\tanh \left(\sqrt{\displaystyle \frac{E}{2}}\left(x+b\right)\right),\end{eqnarray} \tag{ 3.7 }$$

where b is a constant.

Then the solution of the family member, equation (3.4), by the transform (3.6) is

$$\begin{eqnarray}&&\phi \left(\xi \right)=\left|\sigma \right|{\left|\xi \right|}^{\sigma -1}\sqrt{\displaystyle \frac{E}{G\left(\xi \right)}}\tanh \left(\sqrt{\displaystyle \frac{E}{2}}\left({\left|\xi \right|}^{\sigma }+b\right)\right).\end{eqnarray} \tag{ 3.8 }$$

The family members of the nonlinear Schrödinger equation with various value of σ are shown in figure 1.

Zoom In Zoom Out Reset image size

For a special case of the stationary nonlinear Schrödinger equation (3.3)

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\psi \left(x\right)}{{{dx}}^{2}}+\left[2\mu +2{\left|\psi \left(x\right)\right|}^{2}\right]\psi \left(x\right)=0,\end{eqnarray} \tag{ 3.9 }$$

[72] provides an exact bright soliton solution:

$$\begin{eqnarray}&&\psi \left(x\right)=\sqrt{-2\mu }{\rm{sech}} \left[\sqrt{-2\mu }\left(x-{x}_{0}\right)\right].\end{eqnarray} \tag{ 3.10 }$$

The family of the nonlinear Schrödinger equation (3.9), by the relations (2.3) and (2.4), consists of the following family members:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-\left[{V}_{{eff}}\left(\xi \right)+{G}_{2}\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{2}\right]\phi \left(\xi \right)=0,\end{eqnarray} \tag{ 3.11 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{V}_{{eff}}\left(\xi \right) & = & {\sigma }^{2}\left[\displaystyle \frac{1-{\sigma }^{-2}}{4{\xi }^{2}}-2\mu {\left|\xi \right|}^{2\left(\sigma -1\right)}\right],\\ {G}_{2}\left(\xi \right) & = & -2{\sigma }^{2}{\left|\xi \right|}^{3\left(\sigma -1\right)}.\end{array}\end{eqnarray} \tag{ 3.12 }$$

The solution of the family members (3.11) is

$$\begin{eqnarray}\begin{array}{rcl}\phi \left(\xi \right) & = & {\left|\xi \right|}^{\left(1-\sigma \right)/2}\psi \left({\left|\xi \right|}^{\sigma }\right)\\ & = & {\left|\xi \right|}^{\left(1-\sigma \right)/2}\sqrt{-2\mu }{\rm{sech}} \left[\sqrt{-2\mu }\left({\left|\xi \right|}^{\sigma }-{\xi }_{0}\right)\right].\end{array}\end{eqnarray} \tag{ 3.13 }$$

3.2. The family of the quintic Gross–Pitaevskii equation

The quintic GP equation describes BEC when the interaction between the atoms is moderate or strong [50].

The quintic GP equation

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\psi \left(x\right)}{{{dx}}^{2}}-\left[1+{g}_{4}{\left|\psi \left(x\right)\right|}^{4}\right]\psi \left(x\right)=0\end{eqnarray} \tag{ 3.14 }$$

has a solution [50]

$$\begin{eqnarray}&&\psi \left(x\right)={\left(\displaystyle \frac{3}{-{g}_{4}}\right)}^{1/4}\displaystyle \frac{1}{\sqrt{\cosh \left(2x\right)}},\end{eqnarray} \tag{ 3.15 }$$

where g₄ is a negative constant.

The family of the quintic GP equation, by the relations (2.3) and (2.4), consists of the family members:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-\left[{V}_{\mathrm{eff}}\left(\xi \right)+{G}_{4}\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{4}\right]\phi \left(\xi \right)=0,\end{eqnarray} \tag{ 3.16 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}\left(\xi \right) & = & {\sigma }^{2}\left[\displaystyle \frac{1-{\sigma }^{-2}}{4{\xi }^{2}}+{\left|\xi \right|}^{2\left(\sigma -1\right)}\right],\\ {G}_{4}\left(\xi \right) & = & {\sigma }^{2}{\left|\xi \right|}^{4\left(\sigma -1\right)}{g}_{4}.\end{array}\end{eqnarray} \tag{ 3.17 }$$

The solution of equation (3.16) by the transforms (2.5) and (2.6) is

$$\begin{eqnarray}&&\phi \left(\xi \right)={\left|\sigma \right|}^{1/2}{\left|\xi \right|}^{\left(\sigma -1\right)/2}{\left(\displaystyle \frac{3}{-{G}_{4}\left(\xi \right)}\right)}^{1/4}\displaystyle \frac{1}{\sqrt{\cosh \left(2{\left|\xi \right|}^{\sigma }\right)}}.\end{eqnarray} \tag{ 3.18 }$$

3.3. The family of the cubic-quintic Gross–Pitaevskii equation

The cubic-quintic GP equation which describes BEC considers two-particle and three-particle interactions [50, 51].

The cubic-quintic GP equation

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\psi \left(x\right)}{{{dx}}^{2}}-\left[1+{g}_{2}{\left|\psi \left(x\right)\right|}^{2}+{g}_{4}{\left|\psi \left(x\right)\right|}^{4}\right]\psi \left(x\right)=0\end{eqnarray} \tag{ 3.19 }$$

has a solution [50]

$$\begin{eqnarray}&&\psi \left(x\right)=\displaystyle \frac{2}{\sqrt{\sqrt{{g}_{2}^{2}-\tfrac{16}{3}{g}_{4}}\cosh \left(2x\right)-{g}_{2}}},\end{eqnarray} \tag{ 3.20 }$$

where g₂ and g₄ are negative constants.

The family of the cubic-quintic GP equation, by the relations (2.3) and (2.4), consists of the family members:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-\left[{V}_{\mathrm{eff}}\left(\xi \right)+{G}_{2}\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{2}+{G}_{4}\left(\xi \right){\left|\phi \left(\xi \right)\right|}^{4}\right]\phi \left(\xi \right)=0\end{eqnarray} \tag{ 3.21 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}\left(\xi \right) & = & {\sigma }^{2}\left[\displaystyle \frac{1-{\sigma }^{-2}}{4{\xi }^{2}}+{\left|\xi \right|}^{2\left(\sigma -1\right)}\right],\\ {G}_{2}\left(\xi \right) & = & {\sigma }^{2}{\left|\xi \right|}^{3\left(\sigma -1\right)}{g}_{2},\\ {G}_{4}\left(\xi \right) & = & {\sigma }^{2}{\left|\xi \right|}^{4\left(\sigma -1\right)}{g}_{4}.\end{array}\end{eqnarray} \tag{ 3.22 }$$

The solution of equation (3.21) by the transforms (2.5) and (2.6) is

$$\begin{eqnarray}&&\phi \left(\xi \right)={\left|\sigma \right|}^{1/2}{\left|\xi \right|}^{\left(\sigma -1\right)/2}\displaystyle \frac{2}{\sqrt{\sqrt{{\left[{\left|\sigma \right|}^{-1}{\left|\xi \right|}^{1-\sigma }{G}_{2}\left(\xi \right)\right]}^{2}-\tfrac{16}{3}{G}_{4}\left(\xi \right)}\cosh \left(2{\left|\xi \right|}^{\sigma }\right)-{\left|\sigma \right|}^{-1}{\left|\xi \right|}^{1-\sigma }{G}_{2}\left(\xi \right)}}.\end{eqnarray} \tag{ 3.23 }$$

For academic interest, we consider a generalized Gross–Pitaevskii type equation and its family. By the generalized Gross–Pitaevskii type equation we mean that ${\left|\psi \left(x\right)\right|}^{n}\psi \left(x\right)$ in the Gross–Pitaevskii type equation is replaced by $\psi {\left(x\right)}^{n+1}$.

Two one-dimensional generalized GP type equations

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\psi \left(x\right)}{{{dx}}^{2}}-\left[{U}_{\mathrm{eff}}\left(x\right)+{g}_{1}\left(x\right)\psi \left(x\right)+{g}_{2}\left(x\right){\psi }^{2}\left(x\right)\ +\cdots +\ {g}_{n}\left(x\right){\psi }^{n}\left(x\right)\right]\psi \left(x\right)=0,\end{eqnarray} \tag{ 4.1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\phi \left(\xi \right)}{d{\xi }^{2}}-\left[{V}_{\mathrm{eff}}\left(\xi \right)+{G}_{1}\left(\xi \right)\psi \left(\xi \right)+{G}_{2}\left(\xi \right){\phi }^{2}\left(\xi \right)+\cdots +{G}_{n}\left(\xi \right){\phi }^{n}\left(\xi \right)\right]\phi \left(\xi \right)=0,\end{eqnarray} \tag{ 4.2 }$$

if ${U}_{\mathrm{eff}}\left(x\right)$ and ${V}_{\mathrm{eff}}\left(\xi \right),{g}_{l}\left(x\right)$ and ${G}_{l}\left(\xi \right)$ (l = 1, ⋯ ,n) satisfy the relations

$$\begin{eqnarray}\begin{array}{rcl}\sigma \left({x}^{2}{U}_{\mathrm{eff}}\left(x\right)+\displaystyle \frac{1}{4}\right) & = & {\sigma }^{-1}\left({\xi }^{2}{V}_{\mathrm{eff}}\left(\xi \right)+\displaystyle \frac{1}{4}\right),\\ \sigma {x}^{\left(l+4\right)/2}{g}_{l}\left(x\right) & = & {\sigma }^{-1}{\xi }^{\left(l+4\right)/2}{G}_{l}\left(\xi \right),\quad l=1,\cdots ,n,\end{array}\end{eqnarray} \tag{ 4.3 }$$

their solutions are related by the transform:

$$\begin{eqnarray}\begin{array}{rcl}x & \leftrightarrow & {\xi }^{\sigma },\\ \psi \left(x\right) & \leftrightarrow & {\xi }^{\left(\sigma -1\right)/2}\phi \left(\xi \right).\end{array}\end{eqnarray} \tag{ 4.4 }$$

Hereσ is a constant chosen arbitrarily.

This result can be verified directly by the same procedure as for the GP type equation.

The fixed points by the transform (4.3), i.e.,

$$\begin{eqnarray}&&\phi \left(\xi \right)={\xi }^{\left(1-\sigma \right)/2}\psi \left({\xi }^{\sigma }\right).\end{eqnarray} \tag{ 4.5 }$$

are

$$\begin{eqnarray}&&\left(1,\psi \left(1\right)\right)\quad \mathrm{and}\quad \left(-1,\psi \left(-1\right)\right).\end{eqnarray} \tag{ 4.6 }$$

In a family, all family members pass through these two points.

The generalized GP type equation (4.1) is a Liénard equation [73–75] with space-dependent coefficients. The result obtained here can be used to consider the family of the Liénard equation.

The three dimensional spherically symmetric GP type equation also has the similar relation.

Two three-dimensional radial GP type equations

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{\psi }_{l}\left(r\right)}{{{dr}}^{2}}-\left[{U}_{\mathrm{eff}}\left(r\right)+\displaystyle \frac{l\left(l+1\right)}{{r}^{2}}+g\left(r\right)\displaystyle \frac{1}{{r}^{2}}{\left|{\psi }_{l}\left(r\right)\right|}^{2}\right]{\psi }_{l}\left(r\right)=0,\end{eqnarray} \tag{ 5.1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{\phi }_{{\ell }}\left(\rho \right)}{d{\rho }^{2}}-\left[{V}_{\mathrm{eff}}\left(\rho \right)+\displaystyle \frac{{\ell }\left({\ell }+1\right)}{{\rho }^{2}}+G\left(\rho \right)\displaystyle \frac{1}{{\rho }^{2}}{\left|{\phi }_{{\ell }}\left(\rho \right)\right|}^{2}\right]{\phi }_{{\ell }}\left(\rho \right)=0,\end{eqnarray} \tag{ 5.2 }$$

where l and ℓ are angular quantum numbers, if ${U}_{\mathrm{eff}}\left(r\right)$ and ${V}_{\mathrm{eff}}\left(\rho \right),g\left(r\right)$ and $G\left(\rho \right)$ satisfy the relations

$$\begin{eqnarray}\begin{array}{rcl}\sigma {r}^{2}{U}_{\mathrm{eff}}\left(r\right) & = & {\sigma }^{-1}{\rho }^{2}{V}_{\mathrm{eff}}\left(\rho \right),\\ \sigma {rg}\left(r\right) & = & {\sigma }^{-1}\rho G\left(\rho \right),\end{array}\end{eqnarray} \tag{ 5.3 }$$

their solutions are related by the transform:

$$\begin{eqnarray}\begin{array}{rcl}r & \leftrightarrow & {\rho }^{\sigma },\\ {\psi }_{l}\left(r\right) & \leftrightarrow & {\rho }^{\left(\sigma -1\right)/2}{\phi }_{{\ell }}\left(\rho \right),\\ l+\displaystyle \frac{1}{2} & \leftrightarrow & {\sigma }^{-1}\left({\ell }+\displaystyle \frac{1}{2}\right).\end{array}\end{eqnarray} \tag{ 5.4 }$$

Hereσ is a constant chosen arbitrarily.

This result can be verified directly. Substituting the transform (5.4) into equation (5.1) gives

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{\phi }_{{\ell }}\left(\rho \right)}{d{\rho }^{2}}-\left[{\sigma }^{2}{\rho }^{2\left(\sigma -1\right)}{U}_{\mathrm{eff}}\left({\rho }^{\sigma }\right)+\displaystyle \frac{{\ell }\left({\ell }+1\right)}{{\rho }^{2}}+g\left({\rho }^{\sigma }\right){\sigma }^{2}{\rho }^{\sigma -1}\displaystyle \frac{1}{{\rho }^{2}}{\left|{\phi }_{{\ell }}\left(\rho \right)\right|}^{2}\right]{\phi }_{{\ell }}\left(\rho \right)=0,\end{eqnarray} \tag{ 5.5 }$$

This is just the three-dimensional radial GP type equation (5.2) with

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}\left(\rho \right) & = & {\sigma }^{2}{\rho }^{2\left(\sigma -1\right)}{U}_{\mathrm{eff}}\left({\rho }^{\sigma }\right),\\ G\left(\rho \right) & = & g\left({\rho }^{\sigma }\right){\sigma }^{2}{\rho }^{\sigma -1}.\end{array}\end{eqnarray} \tag{ 5.6 }$$

This proves the relations (5.3) and (5.4).

We show that there exist families of the GP type equations. The GP type equations in a family are related by a transform. In a family, so long as one family member is solved, all family members are solved by the transform. The GP type equation is difficult to solve. The method presented in the paper provides an approach to construct exactly solvable GP type equations.

As examples, we consider the family of some special GP type equations: the nonlinear Schrödinger equation, the quintic GP equation, and the cubic-quintic GP equation.

We also consider family of the generalized GP type equation. The result of the generalized GP type equation inspires us to consider the family of the Liénard equation in the future work.

For three-dimensional cases, we consider the family of the three-dimensional spherically symmetric GP type equation.

We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supported in part by Special Funds for Theoretical Physics Research Program of the National Natural Science Foundation of China under Grant No. 11 947 124 and NSF of China under Grant No. 11 575 125 and No. 11 675 119.