Lie Symmetry Group, Invariant Subspace, and Conservation Law for the Time-Fractional Derivative Nonlinear Schrödinger Equation

Abstract

In this paper, a time-fractional derivative nonlinear Schrödinger equation involving the Riemann–Liouville fractional derivative is investigated. We first perform a Lie symmetry analysis of this equation, and then derive the reduced equations under the admitted optimal-symmetry system. Moreover, with the invariant subspace method, several exact solutions for the equation and their figures are presented. Finally, the new conservation theorem is applied to construct the conservation laws of the equation.

1. Introduction

Fractional calculus is the theory of derivatives and integrals with arbitrary real or complex orders, and it unifies and generalizes the notion of integer-order differentiation and n-fold integration [1,2,3]. By the end of the nineteenth century, several different forms of fractional derivatives had been proposed, primarily by Liouville, Grünwald, Letnikov, and Riemann. It can provide a remarkable instrument for the description of various phenomena with memory and hereditary properties [4,5]. This advantage naturally leads to the investigation of the differential equations of fractional order and to the necessity of solving such equations. Although effective general methods for solving fractional differential equations (FDEs) cannot be found, there are several useful ways to construct explicit solutions in special forms, such as the differential transform method [6], finite difference method [7], sub-equation method [8], homotopy perturbation method [9], etc.

A Lie symmetry group, which is a valid and mature method for studying partial differential equations (PDEs), can be used to derive group-invariant solutions, conservation laws, and linearization of nonlinear PDEs [10,11]. As for the Lie group theory of FDEs, a dominant position was taken by Gazizov et al. in [12,13], where they proposed extended formulas and studied the nonlinear anomalous time-fractional diffusion equation involving Riemann–Liouville and Caputo fractional derivatives. In [14], the higher-order symmetries for a class of linear FDEs were discussed in the same manner as Lie point symmetries for this equation. Moreover, potential symmetries for time-fractional PDEs were presented by introducing potential systems with which several exact solutions were constructed [15]. Concerning the conservation law of FDEs, Lukashchuk [16] first achieved the construction of the Noether operator by employing the concept of nonlinear self-adjointness developed by Ibragimov [17]. One of the methods based on invariance principles is the invariant subspace method, which stems from the work of Galaktionov and his collaborator [18]. Recently, Gazizov et al. established the modification of the invariant subspace method for FDEs [19], where they also found that the jointness of invariant subspace and the Lie symmetry group is helpful for finding exact solutions of FDEs. Based on these established schemas, numerous works have focused on FDEs, including scalar-time FDEs [20,21], multidimensional-time FDEs [22,23], coupled-time FDEs [24,25,26], and space-time FDEs [27,28,29].

It is well known that the derivative nonlinear Schrödinger equation for $u(t,x)$

$$\begin{array}{c}\hfill u_t+\mathrm{i}{u}_{xx}\pm {\left(\right|u|}^2{u)}_x=0,\end{array}$$

(1)

originated from both nonlinear plasma physics and fiber optics, where $\left|u\right|={\left(u\overline{u}\right)}^{1/2}$ and $\overline{u}$ is the complex conjugate of u. In the field of plasma physics, Equation (1) models the long-wavelength dynamics of dispersive Alfvén waves propagating along an ambient magnetic field [30]. As in fiber optics, Equation (1) represents the propagation of nonlinear pulses in optical fibers as the effects of higher-order nonlinear perturbations [31]. The integrability of Equation (1) also stimulates the attention of mathematicians. In [32], the inverse scattering transform and the Lax pair for Equation (1) were investigated by Kaup and Newell. Moreover, the Hirota method [33] and Darboux transformation [34,35] were also used to solve Equation (1). In this paper, we will consider the time-fractional derivative nonlinear Schrödinger equation

$$\begin{array}{c}\hfill \mathrm{i}{\partial }_t^{\alpha }u=u_{xx}+{\mathrm{i}a\left|u\right|}^p{u}_x+{\mathrm{i}b\left(\right|u|}^p{)}_{x}u,0<\alpha <1,\end{array}$$

(2)

where a, b, and p are real constants, and ${\partial }_t^{\alpha }$ is the left-sided Riemann–Liouville fractional differential operator of order $\alpha$ with respect to t. To the best of the authors’ knowledge, the symmetry-related analysis of Equation (2) is still open, which motivates us to make a contribution to the study of the Lie symmetry group, invariant subspace, and conservation law for Equation (2). Furthermore, several exact solutions of Equation (2) are also constructed.

This paper is structured as follows. Section 2 provides some basic results related to fractional calculus. In Section 3, we investigate the point symmetry groups admitted by Equation (2) and the corresponding optimal system, with which Equation (2) is reduced to a fractional ordinary differential equation (FODE), and then derive the explicit solutions of Equation (2). Several separable solutions of Equation (2) are constructed through the study of the admitted invariant subspace in Section 4. Section 5 is devoted to deriving the conservation laws of Equation (2) from the obtained symmetry groups by applying the nonlinear self-adjoint method. Section 6 presents our conclusions.

2. Preliminaries

Here, we list the definition of the Riemann–Liouville fractional derivative and several related results arising from the theory of fractional calculus [2]. Consider the differentiable functions $f\left(t\right)$ and $g\left(t\right)$ on $0<t<T$. The left-side and right-side fractional integral operators of order $\mu >0$ are, respectively, defined by

$$\begin{array}{cc}\hfill & {}_0{I}_t^{\mu }f\left(t\right)=\frac{1}{\Gamma \left(\mu \right)}{\int }_0^t{(t-s)}^{\mu -1}f\left(s\right)ds,\hfill \end{array}$$

(3)

$$\begin{array}{c}\hfill {}_t{I}_T^{\mu }f\left(t\right)=\frac{1}{\Gamma \left(\mu \right)}{\int }_t^T{(s-t)}^{\mu -1}f\left(s\right)ds,\end{array}$$

(4)

where $\Gamma \left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt$, $(\Re (z)>0)$ represents the Gamma function. The left-side and right-side Riemann–Liouville fractional derivatives of order $\alpha >0$ are, respectively, defined by

$$\begin{array}{cc}\hfill & {\partial }_t^{\alpha }f\left(t\right)=\frac{d^n}{d{t}^n}\left({}_0{I}_t^{n-\alpha }f\left(t\right)\right)=\frac{1}{\Gamma (n-\alpha )}\frac{d^n}{d{t}^n}{\int }_0^t{(t-s)}^{n-\alpha -1}f\left(s\right)ds,\hfill \end{array}$$

(5)

$$\begin{array}{c}\hfill {}_t{\partial }_T^{\alpha }f\left(t\right)={(-1)}^n\frac{d^n}{d{t}^n}\left({}_t{I}_T^{n-\alpha }f\left(t\right)\right)=\frac{{(-1)}^n}{\Gamma (n-\alpha )}\frac{d^n}{d{t}^n}{\int }_t^T{(s-t)}^{n-\alpha -1}f\left(s\right)ds,\end{array}$$

(6)

where $0\le n-1<\alpha <n$ and $n\in {\mathbb{Z}}^{+}$. For brevity, the left-side Riemann–Liouville fractional derivative is usually said to be the Riemann–Liouville fractional derivative, which has several properties that are different from the ones for the integer-order derivative, such as

$${\partial }_t^{\alpha }k=\frac{k}{\Gamma (1-\alpha )}{t}^{-\alpha },\mathrm{for}\mathrm{constant}k,$$

(7)

$${\partial }_t^{\alpha }{t}^{\nu }=\left\{\begin{array}{cc}\frac{\Gamma (1+\nu )}{\Gamma (1-\alpha +\nu )}{t}^{\nu -\alpha },\hfill & \mathrm{for}\nu >\alpha -1;\hfill \\ 0,\hfill & \mathrm{for}\nu =\alpha -1.\hfill \end{array}\right.$$

(8)

It also has the linear property and generalized Leibniz rule, which are given by

$$\begin{array}{cc}\hfill & {\partial }_t^{\alpha }(af+bg)=a{\partial }_t^{\alpha }f+b{\partial }_t^{\alpha }g,\mathrm{for}\mathrm{constants}a,b,\hfill \end{array}$$

(9)

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }\left(fg\right)=\sum _{j=0}^{\infty }\left(\genfrac{}{}{0pt}{}{\alpha }{j}\right){\partial }_t^{j}f{\partial }_t^{\alpha -j}g,\end{array}$$

(10)

where

$$\left(\genfrac{}{}{0pt}{}{\alpha }{j}\right)=\frac{\Gamma (\alpha +1)}{\Gamma (j+1)\Gamma (\alpha +1-j)}.$$

(11)

In particular, if $j>\alpha$, we have $\left(\genfrac{}{}{0pt}{}{\alpha }{j}\right)=0$.

3. Symmetry Analysis

In this section, we determine the point symmetry groups admitted by Equation (2) and their corresponding one-dimensional optimal system. Then, we perform the symmetry reductions of Equation (2) under the obtained optimal system.

3.1. Lie Symmetry Group

Let us assume that Equation (2) is invariant under a one-parameter Lie group of transformation

$$\begin{array}{cc}\hfill & \tilde{t}=t+ϵ\tau (t,x,u,\overline{u})+O\left({ϵ}^2\right),\hfill \end{array}$$

(12a)

$$\begin{array}{cc}& \tilde{x}=x+ϵ\xi (t,x,u,\overline{u})+O\left({ϵ}^2\right),\hfill \end{array}$$

(12b)

$$\begin{array}{cc}& \tilde{u}=u+ϵ\eta (t,x,u,\overline{u})+O\left({ϵ}^2\right),\hfill \end{array}$$

(12c)

$$\begin{array}{cc}& \tilde{\overline{u}}=u+ϵ\zeta (t,x,u,\overline{u})+O\left({ϵ}^2\right),\hfill \end{array}$$

(12d)

where $ϵ$ is the parameter. The corresponding infinitesimal generator with $\tau =\tau (t,x,u,\overline{u})$, $\xi =\xi (t,x,u,\overline{u})$, $\eta =\eta (t,x,u,\overline{u})$, $\zeta =\zeta (t,x,u,\overline{u})$ is

$$V=\tau {\partial }_t+\xi {\partial }_x+\eta {\partial }_u+\zeta {\partial }_{\overline{u}},$$

(13)

and the $(\alpha ,2)$-th order prolongation ${\mathrm{Pr}}^{(\alpha ,2)}V$ of generator V is defined as follows:

$$\begin{array}{c}\hfill {\mathrm{Pr}}^{(\alpha ,2)}V=V+{\eta }^{\alpha ,t}{\partial }_{{\partial }_t^{\alpha }u}+{\zeta }^{\alpha ,t}{\partial }_{{\partial }_t^{\alpha }\overline{u}}+{\eta }^x{\partial }_{u_x}+{\zeta }^x{\partial }_{{\overline{u}}_x}+{\eta }^{xx}{\partial }_{u_{xx}}+{\zeta }^{xx}{\partial }_{{\overline{u}}_{xx}}.\end{array}$$

(14)

In (14), the integer-order extended infinitesimals [11] are given by

$$\begin{array}{cc}\hfill & {\eta }^x=D_x\eta -u_t{D}_x\tau -u_x{D}_x\xi ,\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & {\eta }^{xx}=D_x{\eta }^x-u_{xt}{D}_x\tau -u_{xx}{D}_x\xi ,\hfill \end{array}$$

(15b)

$$\begin{array}{cc}\hfill & {\zeta }^x=D_x\zeta -{\overline{u}}_t{D}_x\tau -{\overline{u}}_x{D}_x\xi ,\hfill \end{array}$$

(15c)

$$\begin{array}{cc}\hfill & {\zeta }^{xx}=D_x{\zeta }^x-{\overline{u}}_{xt}{D}_x\tau -{\overline{u}}_{xx}{D}_x\xi ,\hfill \end{array}$$

(15d)

and the $\alpha$-th order extended infinitesimals are of the form

$$\begin{array}{c}\hfill {\eta }^{\alpha ,t}={\partial }_t^{\alpha }\eta +({\eta }_u-\alpha D_t\tau ){\partial }_t^{\alpha }u-u{\partial }_t^{\alpha }{\eta }_u+{\eta }_{\overline{u}}{\partial }_t^{\alpha }\overline{u}-\overline{u}{\partial }_t^{\alpha }{\eta }_{\overline{u}}+{\mu }_1+{\mu }_2-\sum _{n=1}^{\infty }\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right)D_t^n\left(\xi \right){\partial }_t^{\alpha -n}\left(u_x\right)\hfill \\ \hfill +\sum _{n=1}^{\infty }\left(\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right){\partial }_t^n{\eta }_u-\left(\genfrac{}{}{0pt}{}{\alpha }{n+1}\right)D_t^{n+1}\tau \right){\partial }_t^{\alpha -n}u+\sum _{n=1}^{\infty }\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right){\partial }_t^n\left({\eta }_{\overline{u}}\right){\partial }_t^{\alpha -n}\left(\overline{u}\right),\hfill \end{array}$$

(16a)

$$\begin{array}{c}\hfill {\zeta }^{\alpha ,t}={\partial }_t^{\alpha }\zeta +({\zeta }_{\overline{u}}-\alpha D_t\tau ){\partial }_t^{\alpha }\overline{u}-\overline{u}{\partial }_t^{\alpha }{\zeta }_{\overline{u}}+{\zeta }_u{\partial }_t^{\alpha }u-u{\partial }_t^{\alpha }{\zeta }_u+{\mu }_3+{\mu }_4-\sum _{n=1}^{\infty }\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right)D_t^n\left(\xi \right){\partial }_t^{\alpha -n}\left({\overline{u}}_x\right)\hfill \\ \hfill +\sum _{n=1}^{\infty }\left(\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right){\partial }_t^n{\zeta }_{\overline{u}}-\left(\genfrac{}{}{0pt}{}{\alpha }{n+1}\right)D_t^{n+1}\tau \right){\partial }_t^{\alpha -n}\overline{u}+\sum _{n=1}^{\infty }\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right){\partial }_t^n\left({\zeta }_u\right){\partial }_t^{\alpha -n}\left(u\right),\hfill \end{array}$$

(16b)

where $D_t$ and $D_x$ denote the total derivatives with respect to t and x, respectively, and

$$\begin{array}{cc}\hfill & {\mu }_1=\sum _{n=2}^{\infty }\sum _{m=2}^n\sum _{k=2}^m\sum _{r=0}^{k-l}\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right)\left(\genfrac{}{}{0pt}{}{n}{m}\right)\left(\genfrac{}{}{0pt}{}{k}{r}\right)\frac{1}{k!}\frac{t^{n-\alpha }}{\Gamma (n+1-\alpha )}{(-u)}^r{\partial }_t^m{u}^{k-r}\frac{{\partial }^{n-m+k}\eta }{\partial t^{n-m}\partial u^k},\hfill \end{array}$$

(17a)

$$\begin{array}{cc}& {\mu }_2=\sum _{n=2}^{\infty }\sum _{m=2}^n\sum _{k=2}^m\sum _{r=0}^{k-l}\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right)\left(\genfrac{}{}{0pt}{}{n}{m}\right)\left(\genfrac{}{}{0pt}{}{k}{r}\right)\frac{1}{k!}\frac{t^{n-\alpha }}{\Gamma (n+1-\alpha )}{(-\overline{u})}^r{\partial }_t^m{\overline{u}}^{k-r}\frac{{\partial }^{n-m+k}\eta }{\partial t^{n-m}\partial {\overline{u}}^k},\hfill \end{array}$$

(17b)

$$\begin{array}{cc}& {\mu }_3=\sum _{n=2}^{\infty }\sum _{m=2}^n\sum _{k=2}^m\sum _{r=0}^{k-l}\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right)\left(\genfrac{}{}{0pt}{}{n}{m}\right)\left(\genfrac{}{}{0pt}{}{k}{r}\right)\frac{1}{k!}\frac{t^{n-\alpha }}{\Gamma (n+1-\alpha )}{(-u)}^r{\partial }_t^m{u}^{k-r}\frac{{\partial }^{n-m+k}\zeta }{\partial t^{n-m}\partial u^k},\hfill \end{array}$$

(17c)

$$\begin{array}{cc}& {\mu }_4=\sum _{n=2}^{\infty }\sum _{m=2}^n\sum _{k=2}^m\sum _{r=0}^{k-l}\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right)\left(\genfrac{}{}{0pt}{}{n}{m}\right)\left(\genfrac{}{}{0pt}{}{k}{r}\right)\frac{1}{k!}\frac{t^{n-\alpha }}{\Gamma (n+1-\alpha )}{(-\overline{u})}^r{\partial }_t^m{\overline{u}}^{k-r}\frac{{\partial }^{n-m+k}\zeta }{\partial t^{n-m}\partial {\overline{u}}^k}.\hfill \end{array}$$

(17d)

The infinitesimal invariance criterion for Equation (2) with respect to the operator (13) demands

$${\tau |}_{t=0}=0,$$

(18)

and

$$P{r}^{(\alpha ,2)}V\left(\mathrm{i}{\partial }_t^{\alpha }u-u_{xx}-{\mathrm{i}a\left|u\right|}^p{u}_x-{\mathrm{i}b\left(\right|u|}^p{)}_{x}u\right){|}_{\left(1.2\right)}=0.$$

(19)

Substituting the prolongations (15) and (16) and setting the coefficients of linearly independent derivatives $u_t$, ${\overline{u}}_t$, $u_x$, ${\overline{u}}_x$, $u_{xx}$, ${\overline{u}}_{xx}$,…, ${\partial }_t^{\alpha -n}u$, ${\partial }_t^{\alpha -n}\overline{u}$, ${\partial }_t^{\alpha -n}{u}_x$, ${\partial }_t^{\alpha -n}{\overline{u}}_x$ to zero in (19), the determining equations, which are the overdetermined system for the infinitesimals $\tau$, $\xi$, $\eta$, and $\zeta$, can be obtained as follows:

$$\begin{array}{cc}\hfill & {\tau }_x={\tau }_u={\tau }_{\overline{u}}={\xi }_u={\xi }_{\overline{u}}={\eta }_{\overline{u}}={\zeta }_u={\eta }_{xu}={\zeta }_{x\overline{u}}={\eta }_{uu}={\zeta }_{\overline{u}\overline{u}}=0,\hfill \end{array}$$

(20a)

$$\begin{array}{cc}& 2{\xi }_x-\alpha {\tau }_t=p(\overline{u}\eta +u\zeta )+2u\overline{u}{\xi }_x=u\overline{u}({\eta }_u-{\zeta }_{\overline{u}})-\overline{u}\eta +u\zeta =0,\hfill \end{array}$$

(20b)

$$\begin{array}{cc}& D_t^n\xi =\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right){\partial }_t^n{\eta }_u-\left(\genfrac{}{}{0pt}{}{\alpha }{n+1}\right)D_t^{n+1}\tau =\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right){\partial }_t^n{\zeta }_{\overline{u}}-\left(\genfrac{}{}{0pt}{}{\alpha }{n+1}\right)D_t^{n+1}\tau =0.\hfill \end{array}$$

(20c)

Solving determining Equations (20) together with condition (18) yields

$$\begin{array}{cc}\hfill & \tau =c_{1}t,\xi =\frac{c_1\alpha }{2}x+c_2,\eta =\left(-\frac{\alpha }{2p}{c}_1+\mathrm{i}{c}_3\right)u,\zeta =\left(-\frac{\alpha }{2p}{c}_1-\mathrm{i}{c}_3\right)\overline{u},\hfill \end{array}$$

(21)

where $c_i$, $(i=1,2,3)$ are arbitrary real constants. Thus, we have the following assertion.

Theorem 1. 

The point symmetry group of Equation (2) is spanned by

$$\begin{array}{c}{V}_1=\mathrm{i}u{\partial }_u-\mathrm{i}\overline{u}{\partial }_{\overline{u}},\hfill \end{array}$$

(22a)

$$\begin{array}{c}{V}_2={\partial }_x,\hfill \end{array}$$

(22b)

$$\begin{array}{c}{V}_3=2pt{\partial }_t+\alpha px{\partial }_x-\alpha u{\partial }_u-\alpha \overline{u}{\partial }_{\overline{u}}.\hfill \end{array}$$

(22c)

Note that symmetries $V_1$ and $V_2$ are phase-rotation and space-translation transformations, respectively, while symmetry $V_3$ is a scaling transformation on $(t,x,u,\overline{u})$. We list the Lie algebra commutator structure of generator (22) in Table 1.

Table 1. The commutator structure of (22).

	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$
$V_1$	0	0	$p\alpha V_1$
$V_2$	0	0	0
$V_3$	$-p\alpha V_1$	0	0

Next, the optimal system of Equation (2) will be constructed to find all group-invariant solutions. The optimal system is the maximal set of one-dimensional point symmetry subgroups that are conjugacy-inequivalent in the full Lie group of point symmetries admitted by Equation (2). The method for finding such a system consists of taking a general operator of Lie algebra and subjecting it to various adjoint transformations to simplify it as much as possible [10]. The adjoint action is given by

$$\mathrm{Ad}\left(\exp \left(ϵV_i\right)\right)V_j=V_j-ϵ[V_i,V_j]+\frac{{ϵ}^2}{2}[V_i,[V_i,V_j]]-\cdots ,$$

(23)

where $[V_i,V_j]$ denotes the commutator of the Lie algebra and $ϵ$ is a parameter. Thus, a direct calculation leads to the corresponding adjoint actions of generators (22), which are shown in Table 2, where the $(i,j)$-th entry represents $\mathrm{Ad}\left(\exp \left(ϵV_i\right)\right)V_j$.

Table 2. The adjoint action of (22).

Ad	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$
$V_1$	$V_1$	$V_2$	$V_3-ϵp\alpha V_1$
$V_2$	$V_1$	$V_2$	$V_3$
$V_3$	$e^{ϵp\alpha }{V}_1$	$V_2$	$V_3$

With the help of the commutator structures in Table 1 and the adjoint representations in Table 2, the one-dimensional optimal subgroups can be determined with a systematic calculation. The result is summarized as follows.

Theorem 2. 

The one-dimensional optimal system of Equation (2) consists of

$$V_1,V_2+\mu V_1,V_3+\nu V_1,$$

(24)

with parameters $-\infty <\mu ,\nu <\infty$.

3.2. Symmetry Reduction

Now, let us run the reductions under the optimal symmetry generators shown in (24).

3.2.1. Reduction under $V_2+\mu V_1$

Solving the corresponding characteristic equation

$$\frac{dt}{0}=\frac{dx}{1}=\frac{du}{\mathrm{i}\mu u}=\frac{d\overline{u}}{-\mathrm{i}\mu \overline{u}},$$

(25)

we get the invariants $z=t$, $U=e^{-\mathrm{i}\mu x}u$, and $\overline{U}=e^{\mathrm{i}\mu x}\overline{u}$. Then, Equation (2) is reduced to the following FODE:

$$\begin{array}{c}\hfill \mathrm{i}{\partial }_z^{\alpha }U+{\mu }^{2}U+\mu a{\left|U\right|}^{p}U=0.\end{array}$$

(26)

To solve Equation (26), we distinguish two cases: $a\ne 0$ and $a=0$.

Case 1.$a\ne 0$. In this case, we find that the FODE (26) has the symmetry group of phase rotation

$${\mathrm{Y}}_1=\mathrm{i}U{\partial }_U-\mathrm{i}\overline{U}{\partial }_{\overline{U}},$$

(27)

with which no explicit solution of Equation (26) can be derived.

Case 2.$a=0$. In this case, the FODE (26) possesses a solution

$$U=c_1{z}^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }\left(\mathrm{i}{\mu }^2{z}^{\alpha }\right),$$

(28)

where $c_1$ is an arbitrary complex constant and ${\mathrm{E}}_{\alpha ,\beta }\left(z\right)$ is the Mittag–Leffler function [3] defined by

$${\mathrm{E}}_{\alpha ,\beta }\left(z\right)=\sum _{j=0}^{\infty }\frac{z^j}{\Gamma (\alpha j+\beta )}.$$

(29)

Thus, Equation (2) has a solution

$$u=c_1{e}^{\mathrm{i}\mu x}{t}^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }\left(\mathrm{i}{\mu }^2{t}^{\alpha }\right),\mathrm{for}a=0,$$

(30)

which is a space-periodic separable solution.

3.2.2. Reduction under $V_3+\nu V_1$

The characteristic equation of the generator $V_3+\nu V_1$ is given by

$$\frac{dt}{2pt}=\frac{dx}{p\alpha x}=\frac{du}{-(\alpha -\mathrm{i}\nu )u}=\frac{d\overline{u}}{-(\alpha +\mathrm{i}\nu )\overline{u}},$$

(31)

which leads to invariants $z=x{t}^{-\alpha /2}$, $U=u{t}^{\frac{\alpha -\mathrm{i}\nu }{2p}}$, and $\overline{U}=\overline{u}{t}^{\frac{\alpha +\mathrm{i}\nu }{2p}}$. Now, let us insert these invariants into the definition of the Riemann–Liouville fractional derivative (5), yielding

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }u& =\frac{{\partial }^n}{\partial t^n}\left(\frac{1}{\Gamma (n-\alpha )}{\int }_0^t{(t-s)}^{n-\alpha -1}{s}^{\frac{-\alpha +\mathrm{i}\nu }{2p}}U\left(x{s}^{-\frac{\alpha }{2}}\right)ds\right)\hfill \\ \hfill & =\frac{{\partial }^n}{\partial t^n}\left(\frac{t^{n-\alpha -1}}{\Gamma (n-\alpha )}{\int }_0^t{(1-s/t)}^{n-\alpha -1}{s}^{\frac{-\alpha +\mathrm{i}\nu }{2p}}U\left(z{(s/t)}^{-\frac{\alpha }{2}}\right)ds\right)\hfill \end{array}$$

(32)

The change in variable $y=t/s$ transforms Equation (32) into

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }u& =\frac{{\partial }^n}{\partial t^n}\left(\frac{t^{n-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p}}}{\Gamma (n-\alpha )}{\int }_1^{\infty }{(y-1)}^{n-\alpha -1}{y}^{-\left(n-\alpha +1+\frac{-\alpha +\mathrm{i}\nu }{2p}\right)}U\left(z{y}^{\frac{\alpha }{2}}\right)ds\right)\hfill \\ & =\frac{{\partial }^n}{\partial t^n}\left(t^{n-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p}}\left({\mathrm{K}}_{2/\alpha }^{1+\frac{-\alpha +\mathrm{i}\nu }{2p},n-\alpha }U\right)\left(z\right)\right)\hfill \\ & =\frac{{\partial }^{n-1}}{\partial t^{n-1}}\left(t^{n-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p}-1}\left(n-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p}-\frac{\alpha }{2}z\frac{d}{dz}\right)\left({\mathrm{K}}_{2/\alpha }^{1+\frac{-\alpha +\mathrm{i}\nu }{2p},n-\alpha }U\right)\left(z\right)\right).\hfill \end{array}$$

(33)

Differentiating $n-1$ times, we get

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }u& =t^{-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p}}{\displaystyle \prod _{j=0}^{l-1}}\left(1-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p}+j-\frac{\alpha }{2}z\frac{d}{dz}\right)\left({\mathrm{K}}_{2/\alpha }^{1+\frac{-\alpha +\mathrm{i}\nu }{2p},n-\alpha }U\right)\left(z\right)\hfill \\ \hfill & =t^{-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p}}\left({\mathrm{P}}_{2/\alpha }^{1-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p},\alpha }U\right)\left(z\right)\hfill \end{array}$$

(34)

Therefore, Equation (2) is reduced to the following FODE:

$$\begin{array}{c}\hfill \mathrm{i}\left({\mathrm{P}}_{2/\alpha }^{1-\alpha +\frac{-\alpha +\mathrm{i}\nu }{2p},\alpha }U\right)\left(z\right)=U^{″}+{\mathrm{i}(a+bp/2)\left|U\right|}^p{U}^{\prime }+\mathrm{i}(bp/2){\left|U\right|}^{p}U{\overline{U}}^{\prime }.\end{array}$$

(35)

In (35), $\left({\mathrm{P}}_{\delta }^{\beta ,\alpha }v\right)\left(z\right)$ is the Erdélyi–Kober fractional differential operator, which is defined by

$$\left({\mathrm{P}}_{\delta }^{\beta ,\alpha }v\right)\left(z\right)={\displaystyle \prod _{j=0}^{l-1}}\left(\beta +j-\frac{1}{\delta }z\frac{d}{dz}\right)\left({\mathrm{K}}_{\delta }^{\beta +\alpha ,l-\alpha }v\right)\left(z\right),\mathrm{with}l=\left\{\begin{array}{cc}\left[\alpha \right]+1,\hfill & \alpha \notin \mathbb{N};\hfill \\ \alpha ,\hfill & \alpha \in \mathbb{N},\hfill \end{array}\right.$$

(36)

where

$$\begin{array}{c}\hfill \left({\mathrm{K}}_{\delta }^{\beta ,\alpha }v\right)\left(z\right)=\left\{\begin{array}{cc}\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }{(y-1)}^{\alpha -1}{y}^{-\beta -\alpha }v\left(z{y}^{1/\delta }\right)dy,\hfill & \alpha >0;\hfill \\ v\left(z\right),\hfill & \alpha =0.\hfill \end{array}\right.\end{array}$$

(37)

4. Invariant Subspace

This section is devoted to the invariant subspace analysis for Equation (2). The maximal dimension of the invariant subspace is determined, and then the reduction of (2) to the FODE is described in terms of the admitted invariant subspaces. Moreover, the symmetry group theory is used to solve the reduced FODEs so that several solutions of Equation (2) can be explicitly derived.

Consider the time-fractional derivative nonlinear Schrödinger Equation (2); we introduce

$$F\left[u\right]=-\mathrm{i}{u}_{xx}+{a\left|u\right|}^p{u}_x+{b\left(\right|u|}^p{)}_{x}u.$$

(38)

Let $f_1\left(x\right),\dots ,f_n\left(x\right)$ be linearly independent functions that generate a linear subspace

$$W_n=\mathfrak{L}\{f_1\left(x\right),\dots ,f_n\left(x\right)\}=\left\{\sum _{j=1}^n{\phi }_j\left(t\right)f_j\left(x\right)\right\}.$$

(39)

Suppose that $W_n$ is invariant under the operator F. Then, there exist functions ${\Psi }_j$, $(j=1,\dots ,n)$ depending on ${\phi }_j={\phi }_j\left(t\right)$, $(j=1,\dots ,n)$ that satisfy

$$F\left[\sum _{j=1}^n{\phi }_j{f}_j\left(x\right)\right]=\sum _{j=1}^n{\Psi }_j({\phi }_1,\dots ,{\phi }_n)f_j\left(x\right).$$

(40)

This implies that Equation (2) has a generalized separable solution

$$u(t,x)=\sum _{j=1}^n{\phi }_j\left(t\right)f_j\left(x\right),$$

(41)

where the coefficients ${\phi }_j\left(t\right)$, $(j=1,\dots ,n)$ satisfy a system of FODEs

$${\partial }_t^{\alpha }{\phi }_j={\Psi }_j({\phi }_1,\dots ,{\phi }_n).$$

(42)

Note that the space $W_n=\mathfrak{L}\{f_1\left(x\right),\dots ,f_n\left(x\right)\}$ can be determined by the solution space of a linear equation

$$L\left[y\right]\equiv y^{\left(n\right)}+a_{n-1}\left(x\right)y^{(n-1)}+\cdots +a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=0,$$

(43)

where $y^{\left(l\right)}=d^{l}y/d^{l}x$, $l\in \mathbb{Z}$. If the operator F in (38) admits the invariant subspace $W_n$ governed by Equation (43), then the invariance condition is given by

$$L\left[F\right]{|}_E\equiv 0,$$

(44)

where E denotes the solution set of equation $L\left[u\right]=0$ and its differential consequences with respect to x. To construct solutions of FDEs, we need to know the maximal dimension of the invariant subspaces. Indeed, the maximal dimension for a nonlinear ordinary differential operator of order k is bounded by $2k+1$ (see [18,36]). It is easily seen that all possible dimensions of the invariant subspace admitted by operator F in (38) are $n=1,2,3,4,5$. For each dimension n, the invariance condition (44) with operator L shown in (43) leads to an overdetermined system. Solving the resulting system, we find that operator F in (38) admits no three-, four-, or five-dimensional invariant subspace, where the case of $a=b=0$ is out of our consideration. The results for one- and two-dimensional invariant subspaces are listed in the Table 3 and Table 4, where $k_j$, $(j=1,2,3)$ are arbitrary real constants.

Table 3. One-dimensional invariant subspaces $W_1$ of (38).

${\mathit{a}}_0\left(\mathit{x}\right)$	Invariant Subspace ${\mathit{W}}_1=\mathfrak{L}\left\{{\mathit{f}}_1\left(\mathit{x}\right)\right\}$	Conditions
0	$\mathfrak{L}\left\{k_1\right\}$
$\mathrm{i}{k}_1$	$\mathfrak{L}\left\{e^{-\mathrm{i}{k}_{1}x}\right\}$	$k_1\ne 0$
$-1/(x+k_1)$	$\mathfrak{L}\{-x+k_1\}$	$(p-1)(a+bp)=0$
$k_1\tan \left(k_1(x+k_2)\right)$	$\mathfrak{L}\left\{\cos \left(k_1(x+k_2)\right)\right\}$	$a=-bp,k_1>0$
$-k_1\tanh \left(k_1(x+k_2)\right)$	$\mathfrak{L}\left\{\cosh \left(k_1(x+k_2)\right)\right\}$	$a=-bp,k_1>0$
$\frac{\mathrm{i}{k}_2}{1-\mathrm{i}{k}_2(x+k_3)}$	$\mathfrak{L}\{1-\mathrm{i}{k}_2(x+k_3)\}$	$a=-2b,p=2,k_2\ne 0$
$\begin{array}{cc}& \frac{k_1^2\left(4k_1^2+k_2^2-4k_1^4{k}_3^2{e}^{\pm 4k_{1}x}\right)+8\mathrm{i}{k}_1^4{k}_3{e}^{\pm 2k_{1}x}}{4k_1^2+{\left(2k_1^2{k}_3{e}^{\pm 2k_{1}x}-k_2\right)}^2}\hfill \end{array}$	$\mathfrak{L}\{(2k_1+\mathrm{i}{k}_2)e^{\mp k_{1}x}-2\mathrm{i}{k}_1^2{k}_3{e}^{\pm k_{1}x}\}$	$a=-2b,p=2,k_1>0$

Table 4. Two-dimensional invariant subspaces $W_2$ of (38).

${\mathit{a}}_1\left(\mathit{x}\right)$	${\mathit{a}}_0\left(\mathit{x}\right)$	Invariant Subspace ${\mathit{W}}_2=\mathfrak{L}\{{\mathit{f}}_1\left(\mathit{x}\right),{\mathit{f}}_2\left(\mathit{x}\right)\}$	Conditions
0	$k_1^2$	$\mathfrak{L}\{\sin k_{1}x,\cos k_{1}x\}$	$a=-2b,p=2,k_1>0$
0	$-k_1^2$	$\mathfrak{L}\{e^{k_{1}x},e^{-k_{1}x}\}$	$a=-2b,p=2,k_1>0$

Next, we will apply the obtained invariant subspaces in Table 3 and Table 4 to reduce Equation (2) to a system of FODEs. The symmetry groups admitted by the reduced FODEs are then derived in order to construct the solutions of Equation (2).

Example 1. 

The invariant subspace $W_1=\mathfrak{L}\left\{1\right\}$ suggests that Equation (2) has a time-dependent solution

$$u(t,x)={\phi }_1\left(t\right),$$

(45)

where ${\phi }_1\left(t\right)$ is an unknown complex function to be determined. Substituting (45) into Equation (2) leads to an FODE for ${\phi }_1={\phi }_1\left(t\right)$

$${\partial }_t^{\alpha }{\phi }_1=0,$$

(46)

which has the solution

$${\phi }_1=t^{\alpha -1}.$$

(47)

Thus, the exact solution of Equation (2) is given by

$$u(t,x)=t^{\alpha -1},$$

(48)

which is dispersive, as illustrated in Figure 1a.

Figure 1. Behaviors of (a) solution (48) and (b) solution (56) with $\alpha =1/2$.

Example 2. 

The invariant subspace $W_1=\mathfrak{L}\left\{e^{-\mathrm{i}x}\right\}$ suggests that Equation (2) possesses an exact solution of the form

$$u(t,x)={\phi }_1\left(t\right)e^{-\mathrm{i}x},$$

(49)

where ${\phi }_1={\phi }_1\left(t\right)$ satisfies

$${\partial }_t^{\alpha }{\phi }_1=\mathrm{i}{\phi }_1\left(1-a|{\phi }_1{|}^p\right).$$

(50)

It is easy to see that, when $a=0$, Equation (50) has the solution

$${\phi }_1=t^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }\left(\mathrm{i}t\right).$$

(51)

Then, we obtain a solution of Equation (2)

$$u(t,x)=e^{-\mathrm{i}x}{t}^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }\left(\mathrm{i}t\right),\mathrm{for}a=0,$$

(52)

which is (30) with $c_1=1$ and $\mu =-1$. When $a\ne 0$, Equation (50) admits a point symmetry

$${\mathrm{Y}}_1=\mathrm{i}{\phi }_1{\partial }_{{\phi }_1}-\mathrm{i}{\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},$$

(53)

with which no invariant solution can be determined.

Example 3. 

The invariant subspace $W_1=\mathfrak{L}\left\{x\right\}$ implies that the solution of Equation (2) with $(p-1)(a+bp)=0$ is of the form

$$u(t,x)={\phi }_1\left(t\right)x,$$

(54)

where ${\phi }_1={\phi }_1\left(t\right)$ is determined by the FODE

$${\partial }_t^{\alpha }{\phi }_1=(a+bp)\left|{\phi }_1\right|{\phi }_1.$$

(55)

It is easily seen that, when $a+bp=0$ Equation (55) has solution (47), it yields an exact solution for Equation (2)

$$u(t,x)=x{t}^{\alpha -1},\mathrm{for}a+bp=0.$$

(56)

Solution (56) is the same as solution (48) up to a fixed x, so they have the same analytic behavior with respect to the variable t (see Figure 1a). We depict the perspective view of solution (56) with $\alpha =1/2$ in 3D in Figure 1b.

For the case of $p=1$ and $a\ne -b$, Equation (55) admits point symmetries

$$\begin{array}{cc}\hfill {\mathrm{Y}}_1=& \mathrm{i}{\phi }_1{\partial }_{{\phi }_1}-\mathrm{i}{\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_2=& t{\partial }_t-\alpha {\phi }_1{\partial }_{{\phi }_1}-\alpha {\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_3=& 3t^2{\partial }_t-2t{\phi }_1{\partial }_{{\phi }_1}-2t{\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},\mathrm{for}\alpha =1/3.\hfill \end{array}$$

(59)

To proceed, we use the symmetry (58) to find the invariant solution. The corresponding invariant is ${\phi }_1\left(t\right)=c{t}^{-\alpha }$, where c is a complex constant. The substitution of this invariant into the Equation (2) gives $\left|c\right|=\Gamma (1-\alpha )/\left(\right(a+b)\Gamma (1-2\alpha \left)\right)$ with $0<\alpha <1/2$ and $\alpha \ne 1/3$. Thus, Equation (2) has a solution

$$u(t,x)=\frac{\Gamma (1-\alpha )\exp \left(\mathrm{i}{c}_1\right)}{(a+b)\Gamma (1-2\alpha )}x{t}^{-\alpha },\mathrm{for}p=1,0<\alpha <1/2,\alpha \ne 1/3,$$

(60)

where $c_j$, $(j=1,2,\cdots )$ are arbitrary real constants. Solution (60) is blown up at $t=0$ and dispersive. Note that the modulo of solution (60) has similar behavior to that of solution (56) with respect to variables t and x (see Figure 1b). Considering the transformation corresponding to symmetry ${\mathrm{Y}}_3$, one can obtain a family of solutions from (60) given by

$$u(t,x)=\frac{\Gamma (2/3)\exp \left(\mathrm{i}{c}_1\right)}{(a+b)\Gamma (1/3)}x{t}^{-1/3}{(1+c_{2}t)}^{-1/3},\mathrm{for}p=1,\alpha =1/3.$$

(61)

Solution (61) not only has the analytic features of solution (60), but also blows up at $t=-1/c_2$.

The behaviors of the modulo of solution (61) when $c_2>0$ are shown in Figure 1a and Figure 2a. For the case of $c_2<0$, we use 2D and 3D plots against variable t (see Figure 2b,c).

Figure 2. Behaviors of $\left|u\right|$ in (61) (a) with $x=2$ and $c_2=1$, (b) with $x=2$ and $c_2=-1$, (c) with $c_2=-1$.

Example 4. 

The invariant subspace $W_1=\mathfrak{L}\left\{\cos x\right\}$ implies that Equation (2) with $a+bp=0$ can be solved by

$$u(t,x)={\phi }_1\left(t\right)\cos x.$$

(62)

Here, the complex function ${\phi }_1={\phi }_1\left(t\right)$ satisfies

$${\partial }_t^{\alpha }{\phi }_1=\mathrm{i}{\phi }_1,$$

(63)

whose solution is (51). Hence, we can derive one solution of Equation (2), given by

$$u(t,x)=t^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }\left(\mathrm{i}t\right)\cos x,\mathrm{for}a=-bp,$$

(64)

which is a space-periodic solution.

Example 5. 

The invariant subspace $W_1=\mathfrak{L}\left\{\cosh x\right\}$ implies that Equation (2) with $a+bp=0$ has a solution

$$u(t,x)={\phi }_1\left(t\right)\cosh x,$$

(65)

with which Equation (2) can be reduced to an FODE for ${\phi }_1={\phi }_1\left(t\right)$

$${\partial }_t^{\alpha }{\phi }_1=-\mathrm{i}{\phi }_1.$$

(66)

Equation (66) can be solved by ${\phi }_1\left(t\right)=t^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }(-\mathrm{i}t)$. Hence, we can derive one solution of Equation (2), given by

$$u(t,x)=t^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }(-\mathrm{i}t)\cosh x,\mathrm{for}a=-bp.$$

(67)

Example 6. 

The invariant subspace $W_1=\mathfrak{L}\{1+\mathrm{i}x\}$ leads to a solution of Equation (2) with $a=-2b$ and $p=2$, that is,

$$u(t,x)={\phi }_1\left(t\right)(1+\mathrm{i}x),$$

(68)

where the complex function ${\phi }_1={\phi }_1\left(t\right)$ satisfies

$${\partial }_t^{\alpha }{\phi }_1=-2\mathrm{i}b\left|{\phi }_1\right|{\phi }_1.$$

(69)

After a straightforward computation, we find that Equation (69) admits point symmetries

$$\begin{array}{cc}\hfill {\mathrm{Y}}_1=& \mathrm{i}{\phi }_1{\partial }_{{\phi }_1}-\mathrm{i}{\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_2=& 2t{\partial }_t-\alpha {\phi }_1{\partial }_{{\phi }_1}-\alpha {\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_3=& 2t^2{\partial }_t-t{\phi }_1{\partial }_{{\phi }_1}-t{\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},\mathrm{for}\alpha =1/2.\hfill \end{array}$$

(72)

The invariant of symmetry (71) is ${\phi }_1\left(t\right)=c{t}^{-\alpha /2}$, where c is a complex constant. The substitution of this invariant into the Equation (2) gives $c=0$. The same result is obtained if we consider the reduction of symmetry (72). Therefore, in this case, no invariant solution can be derived, except for $u(t,x)=0$.

Example 7. 

The invariant subspace $W_1=\mathfrak{L}\{2e^x-2\mathrm{i}{e}^{-x}\}$ implies that Equation (2) with $a=-2b$ and $p=2$ can be solved by

$$u(t,x)={\phi }_1\left(t\right)(2e^{-x}-2\mathrm{i}{e}^x),$$

(73)

where ${\phi }_1={\phi }_1\left(t\right)$ is the solution for the FODE

$${\partial }_t^{\alpha }{\phi }_1=-\mathrm{i}\left(16b|{\phi }_1|+1\right){\phi }_1.$$

(74)

Equation (74) admits one point symmetry

$$\begin{array}{cc}\hfill {\mathrm{Y}}_1=& \mathrm{i}{\phi }_1{\partial }_{{\phi }_1}-\mathrm{i}{\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1},\hfill \end{array}$$

(75)

which cannot be used to derive the invariant solution of Equation (74).

Example 8. 

The invariant subspace $W_2=\mathfrak{L}\{\sin x,\cos x\}$ yields an exact solution of Equation (2) with $a=-2b$ and $p=2$:

$$u(t,x)={\phi }_1\left(t\right)\sin x+{\phi }_2\left(t\right)\cos x.$$

(76)

In (76), the complex functions ${\phi }_j={\phi }_j\left(t\right)$, $(j=1,2)$ satisfy a system of FODEs

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }{\phi }_1=& \mathrm{i}{\phi }_1+b{\phi }_1\left({\overline{\phi }}_1{\phi }_2-{\phi }_1{\overline{\phi }}_2\right),\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }{\phi }_2=& \mathrm{i}{\phi }_2+b{\phi }_2\left({\overline{\phi }}_1{\phi }_2-{\phi }_1{\overline{\phi }}_2\right),\hfill \end{array}$$

(78)

which admits point symmetries

$$\begin{array}{cc}\hfill {\mathrm{Y}}_1=& {\phi }_1{\partial }_{{\phi }_2}+{\overline{\phi }}_1{\partial }_{{\overline{\phi }}_2},\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_2=& {\phi }_2{\partial }_{{\phi }_1}+{\overline{\phi }}_2{\partial }_{{\overline{\phi }}_1},\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_3=& {\phi }_1{\partial }_{{\phi }_1}-{\overline{\phi }}_2{\partial }_{{\overline{\phi }}_2},\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_4=& {\overline{\phi }}_1{\partial }_{{\overline{\phi }}_1}-{\phi }_2{\partial }_{{\phi }_2}.\hfill \end{array}$$

(82)

System (77) and (78) cannot be reduced under symmetries (79)–(82).

Example 9. 

The invariant subspace $W_2=\mathfrak{L}\{e^x,e^{-x}\}$ suggests that Equation (2) with $a=-2b$ and $p=2$ possesses a solution of the form

$$u(t,x)={\phi }_1\left(t\right)e^x+{\phi }_2\left(t\right)e^{-x},$$

(83)

where ${\phi }_j={\phi }_j\left(t\right)$, $(j=1,2)$ are unknown complex functions to be determined. Substituting (83) into Equation (2) leads to

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }{\phi }_1=& -\mathrm{i}{\phi }_1+2b{\phi }_1\left({\overline{\phi }}_1{\phi }_2-{\phi }_1{\overline{\phi }}_2\right),\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }{\phi }_2=& -\mathrm{i}{\phi }_2+2b{\phi }_2\left({\overline{\phi }}_1{\phi }_2-{\phi }_1{\overline{\phi }}_2\right).\hfill \end{array}$$

(85)

We find that the system of (84) and (85) also admits point symmetries (79)–(82), with which the reduction cannot be carried out.

5. Conservation Law

In this section, the nonlinear self-adjointness method is applied to construct the conservation laws of Equation (2). A conservation law of Equation (2) is a divergence expression $D_t\left(C^t\right)+D_x\left(C^x\right)=0$, which vanishes for all solutions of Equation (2). Here, the conserved quantities $C^t$ and $C^x$ are functions of $t,x,u,\overline{u}$ and all derivatives of the dependent variables u and $\overline{u}$. Note that, in view of the definition of the Riemann–Liouville fractional derivative, when $a=b$, Equation (2) has an apparent conservation law $D_t\left(C_0^t\right)+D_x\left(C_0^x\right)=0$ with

$$\begin{array}{c}\hfill C_0^t=\mathrm{i}{}_0{I}_t^{1-\alpha }u-\mathrm{i}{}_0{I}_t^{1-\alpha }\overline{u},C_0^x=-u_x-{\overline{u}}_x-{\mathrm{i}a\left|u\right|}^{p}u+\mathrm{i}a{\left|u\right|}^p\overline{u}.\end{array}$$

(86)

Next, let us apply the nonlinear self-adjointness method to find extra conservation laws. We assume that the formal Lagrangian of Equation (2) takes the form

$$\begin{array}{c}\hfill \mathcal{L}=f(t,x)G_1+g(t,x)G_2,\end{array}$$

(87)

where $G_1={\partial }_t^{\alpha }u+\mathrm{i}{u}_{xx}-{a\left|u\right|}^p{u}_x-{b\left(\right|u|}^p{)}_{x}u$, $G_2={\partial }_t^{\alpha }\overline{u}-\mathrm{i}{\overline{u}}_{xx}-{a\left|u\right|}^p{\overline{u}}_x-{b\left(\right|u|}^p{)}_x\overline{u}$, and $f(t,x)$ and $g(t,x)$ are newly introduced dependent variables. The adjoint equations of Equation (2) are defined by

$$\frac{\delta \mathcal{L}}{\delta u}=0,\frac{\delta \mathcal{L}}{\delta \overline{u}}=0,$$

(88)

where $\frac{\delta }{\delta u}$ and $\frac{\delta }{\delta \overline{u}}$ are the Euler–Lagrange operators with respect to variables u and $\overline{u}$:

$$\begin{array}{cc}\hfill & \frac{\delta }{\delta u}=\frac{\partial }{\partial u}+{}_t^C{\partial }_T^{\alpha }\frac{\partial }{\partial \left({\partial }_t^{\alpha }u\right)}-D_x\frac{\partial }{\partial u_x}+D_x^2\frac{\partial }{\partial u_{xx}},\hfill \end{array}$$

(89)

$$\begin{array}{cc}& \frac{\delta }{\delta \overline{u}}=\frac{\partial }{\partial \overline{u}}+{}_t^C{\partial }_T^{\alpha }\frac{\partial }{\partial \left({\partial }_t^{\alpha }\overline{u}\right)}-D_x\frac{\partial }{\partial {\overline{u}}_x}+D_x^2\frac{\partial }{\partial {\overline{u}}_{xx}}.\hfill \end{array}$$

(90)

Here, ${}_t^C{\partial }_T^{\alpha }$ is the Caputo fractional derivative operator of order $\alpha$, defined by

$${}_t^C{\partial }_T^{\alpha }f\left(t\right)={}_t{I}_T^{n-\alpha }\left(D_t^{n}f\left(t\right)\right)=\frac{1}{\Gamma (n-\alpha )}{\int }_t^T{(t-s)}^{n-\alpha -1}{f}^{\left(n\right)}\left(s\right)ds,$$

(91)

where $0\le n-1<\alpha <n$ and $n\in {\mathbb{Z}}^{+}$. Equation (2) is said to be nonlinear self-adjoint if the adjoint Equation (88) upon a substitution $(f(t,x),g(t,x))=(\varphi (t,x,u,\overline{u}),\psi (t,x,u,\overline{u}))$ satisfy

$$\begin{array}{c}\hfill \frac{\delta \mathcal{L}}{\delta u}={\lambda }_1{G}_1+{\lambda }_2{G}_2,\frac{\delta \mathcal{L}}{\delta \overline{u}}={\lambda }_3{G}_1+{\lambda }_4{G}_2,\end{array}$$

(92)

where ${\lambda }_j$, $(j=1,2,3,4)$ are unknown coefficients to be determined. Solving system (92), we get $\varphi =c_1$ and $\psi =c_2$ for $a=b$. Thus, for a given symmetry $V=\xi {\partial }_x+\tau {\partial }_t+\eta {\partial }_u+\zeta {\partial }_v$, the conserved quantities $C^t$ and $C^x$ can be expressed explicitly as follows [16,17]:

$$\begin{array}{cc}\hfill & C^t=D_t^{\alpha -1}\left(W^u\right)\frac{\partial \mathcal{L}}{\partial \left({\partial }_t^{\alpha }u\right)}+D_t^{\alpha -1}\left(W^{\overline{u}}\right)\frac{\partial \mathcal{L}}{\partial \left({\partial }_t^{\alpha }\overline{u}\right)},\hfill \end{array}$$

(93a)

$$\begin{array}{c}\hfill C^x=W^u\left(\frac{\partial \mathcal{L}}{\partial u_x}-D_x\frac{\partial \mathcal{L}}{\partial u_{xx}}\right)+W^{\overline{u}}\left(\frac{\partial \mathcal{L}}{\partial {\overline{u}}_x}-D_x\frac{\partial \mathcal{L}}{\partial {\overline{u}}_{xx}}\right)+D_x\left(W^u\right)\frac{\partial \mathcal{L}}{\partial u_{xx}}+D_x\left(W^{\overline{u}}\right)\frac{\partial \mathcal{L}}{\partial {\overline{u}}_{xx}},\end{array}$$

(93b)

where $W^u=\eta -\xi u_x-\tau u_t$ and $W^{\overline{u}}=\zeta -\xi {\overline{u}}_x-\tau {\overline{u}}_t$.

Now, the formulas (93) are used to construct the conservation laws. For symmetry $V_1$ with characteristic functions $W^u=\mathrm{i}u$ and $W^{\overline{u}}=-\mathrm{i}\overline{u}$, the corresponding conserved quantities are given by (86). For symmetry $V_2$ with characteristic functions $W^u=-u_x$ and $W^{\overline{u}}=-{\overline{u}}_x$, the corresponding conserved quantities are

$$\begin{array}{cc}\hfill & C_2^t=-{}_0{I}_t^{1-\alpha }{u}_x-{}_0{I}_t^{1-\alpha }{\overline{u}}_x,\hfill \end{array}$$

(94a)

$$\begin{array}{cc}& C_2^x=-\mathrm{i}{u}_{xx}+\mathrm{i}{\overline{u}}_{xx}+{a(1+p/2)\left|u\right|}^p(u_x+{\overline{u}}_x)+(ap/2){\left|u\right|}^{p-2}(u^2{\overline{u}}_x+{\overline{u}}^2{u}_x),\hfill \end{array}$$

(94b)

which imply $D_t\left(C_2^t\right)+D_x\left(C_2^x\right)=D_x(-G_1-G_2)$. Then, $D_t\left(C_2^t\right)+D_x(C_2^x+G_1+G_2)=0$ holds identically. Thus, the conserved quantities (94) yield a trivial conservation law. For symmetry $V_3$ with characteristic functions $W^u=-\alpha u-2pt{u}_t-\alpha px{u}_x$ and $W^{\overline{u}}=-\alpha \overline{u}-2pt{\overline{u}}_t-\alpha px{\overline{u}}_x$, the corresponding conserved quantities are

$$\begin{array}{cc}\hfill C_3^t=& -{}_0{I}_t^{1-\alpha }\left(\alpha u+\alpha \overline{u}+p\alpha x{u}_x+p\alpha x{\overline{u}}_x+2pt{u}_t+2pt{\overline{u}}_t\right),\hfill \end{array}$$

(95a)

$$\begin{array}{cc}\hfill C_3^x=& \mathrm{i}(\alpha {\overline{u}}_x+2pt{\overline{u}}_{tx}+p\alpha {\overline{u}}_x+p\alpha x{\overline{u}}_{xx})-\mathrm{i}(\alpha u_x+2pt{u}_{tx}+p\alpha u_x+p\alpha x{u}_{xx})\hfill \\ & +(\alpha u+2pt{u}_t+p\alpha x{u}_x)\left({a(1+p/2)\left|u\right|}^p+(ap/2){\left|u\right|}^{p-2}{\overline{u}}^2\right)\hfill \\ & +(\alpha \overline{u}+2pt{\overline{u}}_t+p\alpha x{\overline{u}}_x)\left({a(1+p/2)\left|u\right|}^p+(ap/2){\left|u\right|}^{p-2}{u}^2\right).\hfill \end{array}$$

(95b)

6. Conclusions

The aim of this paper was to study the Lie symmetry groups, invariant subspaces, and conservation laws for the time-fractional derivative nonlinear Schrödinger Equation (2). Through a systematic calculation, we found that Equation (2) admits three Lie symmetry groups, namely, phase-rotation transformation, space-translation transformation, and scaling transformation. Then, the corresponding optimal sub-algebras were utilized to transform Equation (2) into FODEs. Moreover, the invariant subspace method was applied in combination with the symmetry group method to compute separable solutions of Equation (2). By using the new conservation theorem, we obtained three conserved currents generated from the three admitted symmetry groups, one of which yielded a new nontrivial conservation law. The analysis in this paper implies that the proposed methods are profoundly beneficial for the study of FDEs, which prompts us to do more work on FDEs arising from real applied fields.