1 Introduction

The (2+1)-dimensional modified Bogoyavlenskii–Schiff equations are the important mathematical physical equations to describe the (2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. The equations are firstly derived by Bogoyavlenskii [1] and Schiff [2], which are given by

$$\begin{aligned} \left\{ \begin{array}{llll} u_{t}+u_{xxy}-4u^2u_y-2u_{x}v=0, \\ v_x=2uu_y. \\ \end{array}\right. \end{aligned}$$
(1)

Since then, Eq. (1) have been studied by many scholars with different methods in [3,4,5,6,7,8]. Among them, in [8], Kumar and Manju studied the integral order (2+1)-dimensional modified Bogoyavlenskii–Schiff equations by using Lie symmetry analysis method. They obtained the one-dimensional optimal subalgebras, optimal reductions and analytical solutions for Eq. (1).

As a generalization of the classical calculus, fractional calculus can be traced back to the letter written by L’Hôspital to Leibniz in 1695. Since then, it has gradually gained the attention of mathematicians. Especially in recent decades, it has developed rapidly and been successfully applied in many fields of science and technology [9,10,11,12]. Therefore, it is very important to find the solution of fractional differential equation. So far, there have been some numerical and analytical methods, such as Adomian decomposition method [13], finite difference method [14], homotopy perturbation method [15], the sub-equation method [16], the variational iteration method [17], Lie symmetry analysis method [18], invariant subspace method [19] and so on. Among them, Lie symmetry analysis method has received an increasing attention.

Lie symmetry analysis method was founded by Norwegian mathematician Sophus Lie at the end of the nineteenth century and then further developed by some other mathematicians, such as Ovsiannikov [20], Olver [21], Ibragimov [22,23,24] and so on. As a modern method among many analytic techniques, Lie symmetry analysis has been extended to fractional differential equations (FDEs) by Gazizov et al. [18] in 2007. It was then effectively applied to various models of the FDEs occurring in different areas of applied science (see [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]).

In this paper, Lie symmetry analysis method is extended to the following (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations:

$$\begin{aligned} \left\{ \begin{array}{llll} D^{\alpha }_{t}u+u_{xxy}-4u^2u_y-2u_{x}v=0, \\ v_x=2uu_y, \\ \end{array}\right. \end{aligned}$$
(2)

with \(0<\alpha <1\). When \(y=x\), the above equations become an (1+1)-dimensional time fractional mKdV equation, i.e.,

$$\begin{aligned} D^{\alpha }_{t}u-6u^2u_x+u_{xxx}=0, \end{aligned}$$
(3)

which is also researched by several scholars. As we all know, there are many types of definitions for fractional derivative, such as Riemann–Liouville type, Caputo type, Weyl type and so on. This paper adopts Riemann–Liouville fractional derivative defined by

$$\begin{aligned} _{a}D_{t}^\alpha f(t,x) =D_{t}^{n}\ _{a}I_{t}^{n-\alpha } f(t,x)= \left\{ \begin{array}{ll} \frac{1}{\Gamma (n-\alpha )}\frac{d^n}{dt^n}\int _a^t {\frac{f(s,x)}{(t-s)^{ \alpha -n+1 }} }\textrm{d}s,\ \ &{} n-1<\alpha <n, n\in \mathbb {N} \\ \\ D_{t}^{n}f(t,x),\ \ &{} \alpha =n\in \mathbb {N} \\ \end{array}\right. \end{aligned}$$

for \(t>a\). We denote the operator \(_{0}D^{\alpha }_{t}\) as \(D^{\alpha }_{t}\) throughout this paper.

The aim of this paper is to find all Lie symmetries for Eq. (2) by using Lie symmetry analysis method and reduce Eq. (2) to (1+1)-dimensional time fractional partial differential equations or time fractional ordinary differential equations. Moreover, we will derive explicit power series solutions by the power series method for the reduced equations and construct the conservation laws by the new conservation theorem and the generalization of Noether operators.

This paper is organized as follows. In Sect. 2, Lie symmetry analysis of Eqs. (2) is presented. In Sect. 3, the reduced equations, power series solutions and some trivial solutions for Eq. (2) are obtained. The conserved vectors for the symmetries admitted by Eq. (2) are constructed in Sects. 4, and the conclusion is given in the last section.

2 Lie Symmetry Analysis of Eq. (2)

2.1 Lie Symmetries

Consider the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations (2), which are assumed to be invariant under the one-parameter (\(\epsilon\)) Lie group of continuous point transformations, i.e.

$$\begin{aligned} t^{*}= & {} t+\epsilon \tau (t,x,y,u,v)+o(\epsilon ),\ \ \ \ x^{*}=x+\epsilon \xi (t,x,y,u,v)+o(\epsilon ), \nonumber \\ y^{*}= & {} y+\epsilon \theta (t,x,y,u,v)+o(\epsilon ),\ \ \ \ u^{*}=u+\epsilon \eta (t,x,y,u,v)+o(\epsilon ), \nonumber \\ v^{*}= & {} v+\epsilon \zeta (t,x,y,u,v)+o(\epsilon ),\ \ \ \ D^{\alpha }_{t^{*}}u^{*}=D^{\alpha }_{t}u+\epsilon \eta ^{\alpha ,t}+o(\epsilon ),\ \ \ \ \nonumber \\ D_{x^{*}}u^{*}= & {} D_{x}u+\epsilon \eta ^{x}+o(\epsilon ),\ \ \ \ D_{x^{*}}v^{*}=D_{x}v+\epsilon \zeta ^{x}+o(\epsilon ), \nonumber \\ D_{y^{*}}u^{*}= & {} D_{y}u+\epsilon \eta ^{y}+o(\epsilon ),\ \ \ \ D_{y^{*}}v^{*}=D_{y}v+\epsilon \zeta ^{y}+o(\epsilon ), \nonumber \\ \ \ \ \ \ \ \ D^{2}_{x^{*}}u^{*}= & {} D^{2}_{x}u+\epsilon \eta ^{xx}+o(\epsilon ),\ \ \ \ D_{x^{*}x^{*}y^{*}}u^{*}=D_{xxy}u+\epsilon \eta ^{xxy}+o(\epsilon ), \end{aligned}$$
(4)

where \(\tau\), \(\xi\), \(\theta\), \(\eta\) and \(\zeta\) are infinitesimals and \(\eta ^{\alpha ,t}\), \(\eta ^{x}\), \(\zeta ^{x}\), \(\eta ^{y}\), \(\zeta ^{y}\), \(\eta ^{xx}\), \(\eta ^{xxy}\) are the corresponding prolongations of orders \(\alpha\), 1, 2 and 3, respectively.

The corresponding group generator is defined by

$$\begin{aligned} X=\tau \frac{\partial }{\partial t }+\xi \frac{\partial }{\partial x }+\theta \frac{\partial }{\partial y }+\eta \frac{\partial }{\partial u }+\zeta \frac{\partial }{\partial v }. \end{aligned}$$
(5)

So the prolongation of the above group generator X has the form

$$\begin{aligned} prX=X+\eta ^{\alpha ,t}\frac{\partial }{\partial u^{\alpha }_{t}}+\eta ^{x}\frac{\partial }{\partial u_{x}}+\zeta ^{x}\frac{\partial }{\partial v_{x}}+\eta ^{y}\frac{\partial }{\partial u_{y}}+\eta ^{xxy}\frac{\partial }{\partial u_{xxy}}, \end{aligned}$$
(6)

where

$$\begin{aligned} \eta ^{\alpha ,t}&=D^{\alpha }_{t}(\eta )-D^{\alpha +1}_t(\tau u)+D^\alpha _t[D_t(\tau )u]+\tau D^{\alpha +1}_t(u)+\xi D^{\alpha }_{t}(u_{x})-D^{\alpha }_{t}(\xi u_{x}) \nonumber \\&\quad +\theta D^{\alpha }_{t}(u_{y})-D^{\alpha }_{t}(\theta u_{y}) \nonumber \\&=\frac{\partial ^\alpha \eta }{\partial {t^\alpha }}+(\eta _{u}-\alpha D_{t}(\tau ))\frac{\partial ^\alpha u }{\partial {t^\alpha }}-u\frac{\partial ^\alpha \eta _{u} }{\partial {t^\alpha }}+(\eta _{v}\frac{\partial ^\alpha v }{\partial {t^\alpha }}-v\frac{\partial ^\alpha \eta _{v} }{\partial {t^\alpha }})+\mu _1+\mu _2 \nonumber \\&\quad +\sum _{n=1}^{\infty } [\left( {\begin{array}{c}\alpha \\ n\end{array}}\right) \frac{\partial ^n \eta _{u} }{\partial {t^n}}-\left( {\begin{array}{c}\alpha \\ n+1\end{array}}\right) D^{n+1}_{t}(\tau )]D^{\alpha -n}_{t}(u)+\sum _{n=1}^{\infty }\left( {\begin{array}{c}\alpha \\ n\end{array}}\right) \frac{\partial ^n \eta _{v} }{\partial {t^n}}D^{\alpha -n}_{t}(v) \nonumber \\&\quad -\sum _{n=1}^{\infty }\left( {\begin{array}{c}\alpha \\ n\end{array}}\right) D^{n}_{t}(\xi )D^{\alpha -n}_{t}(u_{x})-\sum _{n=1}^{\infty }\left( {\begin{array}{c}\alpha \\ n\end{array}}\right) D^{n}_{t}(\theta )D^{\alpha -n}_{t}(u_{y}),\ \ \end{aligned}$$
(7)

with

$$\begin{aligned} \mu _1= & {} \sum _{n=2}^{\infty }\sum _{m=2}^{n}\sum _{k=2}^{m}\sum _{r=0}^{k-1 }\left( {\begin{array}{c}\alpha \\ n\end{array}}\right) \left( {\begin{array}{c}n \\ m\end{array}}\right) \left( {\begin{array}{c}k \\ r\end{array}}\right) \frac{t^{n-\alpha }(-u)^r}{k!\Gamma (n+1-\alpha )}\frac{\partial ^m u^{k-r}}{\partial t^m}\frac{\partial ^{n-m+k} \eta }{\partial t^{n-m}\partial u^{k}}, \\ \mu _2= & {} \sum _{n=2}^{\infty }\sum _{m=2}^{n}\sum _{k=2}^{m}\sum _{r=0}^{k-1 }\left( {\begin{array}{c}\alpha \\ n\end{array}}\right) \left( {\begin{array}{c}n \\ m\end{array}}\right) \left( {\begin{array}{c}k \\ r\end{array}}\right) \frac{t^{n-\alpha }(-v)^r}{k!\Gamma (n+1-\alpha )}\frac{\partial ^m v^{k-r}}{\partial t^m}\frac{\partial ^{n-m+k} \eta }{\partial t^{n-m}\partial v^{k}}, \end{aligned}$$

and

$$\begin{aligned} \eta ^{x}= & {} D_{x}(\eta )-u_{t}D_{x}(\tau )-u_{x}D_{x}(\xi )-u_{y}D_{x}(\theta ), \end{aligned}$$
(8)
$$\begin{aligned} \eta ^{y}= & {} D_{y}(\eta )-u_{t}D_{y}(\tau )-u_{x}D_{y}(\xi )-u_{y}D_{y}(\theta ), \end{aligned}$$
(9)
$$\begin{aligned} \zeta ^{x}= & {} D_{x}(\zeta )-v_{t}D_{x}(\tau )-v_{x}D_{x}(\xi )-v_{y}D_{x}(\theta ), \end{aligned}$$
(10)
$$\begin{aligned} \eta ^{xx}= & {} D_{x}(\eta ^{x})-u_{xt}D_{x}(\tau )-u_{xx}D_{x}(\xi )-u_{xy}D_{x}(\theta ), \end{aligned}$$
(11)
$$\begin{aligned} \eta ^{xxy}= & {} D_{y}(\eta ^{xx})-u_{xxt}D_{y}(\tau )-u_{xxx}D_{y}(\xi )-u_{xxy}D_{y}(\theta ), \end{aligned}$$
(12)

where \(D_{t}\), \(D_{x}\) and \(D_{y}\) are the total derivative with respect to t, x and y, respectively.

Remark 1

The infinitesimal transformations (4) should conserve the structure of the Riemann–Liouville fractional derivative operator, of which the lower limit in the integral is fixed. Therefore, the manifold \(t=0\) should be invariant with respect to such transformations. The invariance condition arrives at

$$\begin{aligned} \tau (t,x,y,u,v)|_{t=0}=0. \end{aligned}$$
(13)

Remark 2

From the expressions of \(\mu _1\) and \(\mu _2\), if the infinitesimals \(\eta\) and \(\zeta\) be linear with respect to the variables u and v, then \(\mu _1=\mu _2=0\), that is,

$$\begin{aligned} \frac{\partial ^{2} \eta }{\partial u^{2}}=\frac{\partial ^{2} \eta }{\partial v^{2}}=0. \end{aligned}$$
(14)

The one-parameter Lie symmetry transformations (4) are admitted by the system (2), if the following invariance criterion holds:

$$\begin{aligned} \left\{ \begin{array}{llll} prX\big (D^{\alpha }_{t}u+u_{xxy}-4u^2u_y-2u_{x}v\big )|_{(1.2)}=0, \\ prX(v_x-2uu_y)|_{(1.2)}=0, \\ \end{array}\right. \end{aligned}$$
(15)

which can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{llll} \big (\eta ^{\alpha ,t}+\eta ^{xxy}-4u^2\eta ^{y}-2v\eta ^{x}-8uu_{y}\eta -2u_x\zeta \big )|_{(1.2)}=0,\\ \big (\zeta ^{x}-2u\eta ^{y}-2u_{y}\eta \big )|_{(1.2)}=0. \\ \end{array}\right. \end{aligned}$$
(16)

Putting \(\eta ^{\alpha ,t}\), \(\eta ^{x}\), \(\zeta ^{x}\), \(\eta ^{y}\) and \(\eta ^{xxy}\) into (16) and letting coefficients of various derivatives of u and v to be zero, we can obtain the over-determined system of differential equations as follows:

$$\begin{aligned} \tau _{x}= & {} \tau _{y}=\tau _{u}=\tau _{v}=0,\ \ \theta _{t}=\theta _{x}=\theta _{u}=\theta _{v}=0, \end{aligned}$$
(17)
$$\begin{aligned} \xi _{t}= & {} \xi _{u}=\xi _{v}=\eta _{v}=\zeta _{u}=0,\ \ \eta _{uu}=\zeta _{uu}=\zeta _{uv}=\zeta _{vv}=0, \end{aligned}$$
(18)
$$\begin{aligned}{} & {} \left( {\begin{array}{c}\alpha \\ n\end{array}}\right) \frac{\partial ^n \eta _{u} }{\partial {t^n}}-\left( {\begin{array}{c}\alpha \\ n+1\end{array}}\right) D^{n+1}_{t}(\tau )=0,\ \ n\in N, \end{aligned}$$
(19)
$$\begin{aligned}{} & {} 2\xi _{x}+\theta _{y}-\alpha \tau _t=0, \end{aligned}$$
(20)
$$\begin{aligned}{} & {} v(\xi _x-\alpha \tau _t)-\zeta =0, \end{aligned}$$
(21)
$$\begin{aligned}{} & {} u(\theta _{y}-\alpha \tau _t)-2\eta =0, \end{aligned}$$
(22)
$$\begin{aligned}{} & {} u(\zeta _{v}+\theta _{y}-\eta _{u}-\xi _{x})-\eta =0, \end{aligned}$$
(23)
$$\begin{aligned}{} & {} \frac{\partial ^\alpha \eta }{\partial {t^\alpha }}-u\frac{\partial ^\alpha \eta _{u} }{\partial {t^\alpha }}-v\frac{\partial ^\alpha \eta _{v} }{\partial {t^\alpha }}+\eta _{xxy}-4u^2\eta _y-2v\eta _x=0. \end{aligned}$$
(24)

Solving these equations altogether, with the conditions (13) and (14), we can obtain infinitesimals as follows:

$$\begin{aligned} \tau =c_1t,\ \ \xi =c_2x+c_3,\ \ \theta =(\alpha c_1-2c_2)y+c_4,\ \ \eta =-c_2u,\ \ \zeta =(c_2-\alpha c_1)v, \end{aligned}$$
(25)

where \(c_1\), \(c_2\), \(c_3\) and \(c_4\) are arbitrary constants.

So the system (2) admitted the four-dimension Lie algebra spanned by

$$\begin{aligned} X_1=t\frac{\partial }{\partial t }+\alpha y\frac{\partial }{\partial y }-\alpha v\frac{\partial }{\partial v },\ \ X_2=x\frac{\partial }{\partial x }-2y\frac{\partial }{\partial y }-u\frac{\partial }{\partial u }+v\frac{\partial }{\partial v },\ \ X_3=\frac{\partial }{\partial x },\ \ X_4=\frac{\partial }{\partial y }. \end{aligned}$$
(26)

2.2 One-Dimensional Optimal System

It is easy to check that the group generators in (26) are closed under the Lie bracket defined by

$$\begin{aligned}{}[X_i,X_j]=X_iX_j-X_jX_i,\ \ (i,j=1,2,3,4). \end{aligned}$$
(27)

The commutation relationships of these group generators can be seen in Table 1.

Table 1 The commutation table of Lie algebra
Full size table

Then we consider the action of the adjoint operator which is given by the Lie series

$$\begin{aligned} Ad(exp(\varepsilon X_i))X_j=X_j-\varepsilon [X_i,X_j]+\frac{\varepsilon ^2 }{2}[X_i,[X_i,X_j]]-\cdots , \end{aligned}$$
(28)

where \(\varepsilon\) is an arbitrary parameter. According to (28), we calculate the adjoint action of the group generators in (26) which is listed in Table 2.

Table 2 The adjoint representation of Lie algebra
Full size table

From Tables 1 and 2, by Olver’s method [21], the optimal system for the one-dimensional subalgebras of Eq. (2) can be obtained as the following forms:

$$\begin{aligned} X_1,\ \ X_2,\ \ X_3,\ \ X_4,\ \ X_1+\beta X_2,\ \ X_3+\gamma X_4. \end{aligned}$$
(29)

3 Similarity reductions and invariant solutions of Eq. (2)

In this section, the aimed equations (2) can be reduced to (1+1)-dimensional time fractional partial differential equations with the left-hand Erdélyi–Kober fractional derivative. Then the reduced equations can be solved by the power series method and Laplace transform method to construct the invariant solutions for Eq. (2). In what follows, we consider the following cases.

Case 1: \(X_1+\beta X_2\)

For the convenience of calculation, assuming \(\beta =\frac{\alpha }{2}\), we can obtain the following group generator:

$$\begin{aligned} X_1+\frac{\alpha }{2} X_2=t\frac{\partial }{\partial t }+\frac{\alpha }{2}x\frac{\partial }{\partial x }-\frac{\alpha }{2}u\frac{\partial }{\partial u }-\frac{\alpha }{2}v\frac{\partial }{\partial v }, \end{aligned}$$

which can also be obtained by making \(c_2=\frac{\alpha }{2} c_1\) in (25). The characteristic equation corresponding to the group generator \(X_1+\frac{\alpha }{2} X_2\) is

$$\begin{aligned} \frac{\textrm{d} t }{t }=\frac{2\textrm{d}x }{\alpha x }=\frac{\textrm{d} y }{0 }=\frac{2\textrm{d}u }{-\alpha u }=\frac{2\textrm{d}v }{-\alpha v }, \end{aligned}$$
(30)

from which, we obtain the similarity variables \(xt^{-\frac{\alpha }{2}}\), y, \(ut^{\frac{\alpha }{2}}\) and \(vt^{\frac{\alpha }{2}}\). So we get the invariant solutions of the system (2) as follows:

$$\begin{aligned} u(t,x,y)=t^{-\frac{\alpha }{2}}f(\omega _1,\omega _2),\ \ v(t,x,y)=t^{-\frac{\alpha }{2}}g(\omega _1,\omega _2), \end{aligned}$$
(31)

with \(\omega _1=xt^{-\frac{\alpha }{2}}\), \(\omega _2=y\).

Theorem 2.1

The similarity transformations \(u(t,x,y)=t^{-\frac{\alpha }{2}}f(\omega _1,\omega _2)\), \(v(t,x,y)=t^{-\frac{\alpha }{2}}g(\omega _1,\omega _2)\) with the similarity variables \(\omega _1=xt^{-\frac{\alpha }{2}}\), \(\omega _2=y\) reduce the system (2) to the system of (1+1)-dimensional fractional differential equations given by

$$\begin{aligned} \left\{ \begin{array}{llll} (\mathcal {P}^{1-\frac{3\alpha }{2},\alpha }_{\frac{2 }{\alpha },\infty }f)(\omega _1,\omega _2)+f_{\omega _1\omega _1\omega _2}-4f^2f_{\omega _2}-2f_{\omega _1}g=0, \\ \\ g_{\omega _1}-2ff_{\omega _2}=0, \\ \end{array}\right. \end{aligned}$$
(32)

where \((\mathcal {P}^{\iota ,\kappa }_{\delta _1,\delta _2})\) is the left-hand Erdélyi–Kober fractional differential operator defined by

$$\begin{aligned}{} & {} (\mathcal {P}^{\iota ,\kappa }_{\delta _1,\delta _2}\psi )(\omega _1,\omega _2):=\prod _{j=0}^{m-1}\left( \iota +j-\frac{1 }{\delta _1}\omega _1\frac{\partial }{\partial \omega _1 }-\frac{1 }{\delta _2}\omega _2\frac{\partial }{\partial \omega _2 })(\mathcal {K}^{\iota +\kappa ,m-\kappa }_{\delta _1,\delta _2}\psi \right) (\omega _1,\omega _2),\ \kappa >0, \nonumber \\{} & {} \quad m=\left\{ \begin{array}{lll} [\kappa ]+1,\ \ {} &{}if \ \kappa \notin \mathbb {N}, \\ \kappa ,\ \ {} &{}if \ \kappa \in \mathbb {N}, \\ \end{array}\right. \end{aligned}$$
(33)

where

$$\begin{aligned} (\mathcal {K}^{\iota ,\kappa }_{\delta _1,\delta _2}\psi )(\omega _1,\omega _2):=\left\{ \begin{array}{lll} \frac{1 }{\Gamma (\kappa ) }\int _1^\infty (s-1)^{\kappa -1}s^{-(\iota +\kappa )}\psi (\omega _1 s^{\frac{1 }{\delta _1 }},\omega _2 s^{\frac{1 }{\delta _2 }})\textrm{d}s,\ \ {} &{}\kappa >0, \\ \psi (\omega _1,\omega _2),\ \ {} &{}\kappa =0, \\ \end{array}\right. \end{aligned}$$
(34)

is the left-hand Erdélyi–Kober fractional integral operator.

Proof

For \(0<\alpha <1\), the Riemann–Liouville time fractional derivative of u(t, x) can be obtained as follows:

$$\begin{aligned} \frac{\partial ^\alpha u}{\partial t^\alpha }=\frac{\partial ^\alpha }{\partial t^\alpha }(t^{-\frac{\alpha }{2}}f(\omega _1,\omega _2))=\frac{\partial }{\partial t }\left[ \frac{1}{\Gamma (1-\alpha )}\int _0^t (t-s)^{-\alpha }s^{\frac{-\alpha }{2}}f(xs^{-\frac{\alpha }{2}},y )\textrm{d}s\right] . \end{aligned}$$

Assuming \(r=\frac{t }{s}\), we have

$$\begin{aligned} \frac{\partial ^\alpha u}{\partial t^\alpha }=\frac{\partial }{\partial t }\left[ \frac{t^{1-\frac{3\alpha }{2}}}{\Gamma (1-\alpha )}\int _1^\infty (r-1)^{-\alpha }r^{\frac{3\alpha }{2}-2}f(\omega _1 r^{\frac{\alpha }{2}},\omega _2 )\textrm{d}r\right] =\frac{\partial }{\partial t }\left[ t^{1-\frac{3\alpha }{2}}(\mathcal {K}^{1-\frac{\alpha }{2},1-\alpha }_{\frac{2 }{\alpha },\infty }f)(\omega _1,\omega _2)\right] . \end{aligned}$$

Because of \(\omega _1=xt^{-\frac{\alpha }{2}}\) and \(\omega _2=y\), the following relation holds:

$$\begin{aligned} t\frac{\partial }{\partial t }\psi (\omega _1,\omega _2)=tx\left( -\frac{\alpha }{2}\right) t^{-\frac{\alpha }{2}-1}\psi _{\omega _1}=-\frac{\alpha }{2}\omega _1\frac{\partial }{\partial \omega _1 }\psi (\omega _1,\omega _2). \end{aligned}$$

Hence, we arrive at

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial t }\left[ t^{1-\frac{3\alpha }{2}}\left( \mathcal {K}^{1-\frac{\alpha }{2},1-\alpha }_{\frac{2 }{\alpha },\infty }f\right) (\omega _1,\omega _2)\right] =&t^{-\frac{3\alpha }{2}}\left[ \left( 1-\frac{3\alpha }{2}-\frac{\alpha }{2}\omega _1\frac{\partial }{\partial \omega _1 }\right) \left( \mathcal {K}^{1-\frac{\alpha }{2},1-\alpha }_{\frac{2 }{\alpha },\infty }f\right) (\omega _1,\omega _2)\right] \\ =&t^{-\frac{3\alpha }{2}}\left( \mathcal {P}^{1-\frac{3\alpha }{2},\alpha }_{\frac{2 }{\alpha },\infty }f\right) (\omega _1,\omega _2).\ \ \end{aligned} \end{aligned}$$

Meanwhile,

$$\begin{aligned}{} & {} u_{xxy}-4u^2u_{y}-2u_{x}v=t^{-\frac{3\alpha }{2}}(f_{\omega _1\omega _1\omega _2}-4f^2f_{\omega _2}-2f_{\omega _1}g), \\{} & {} \quad v_{x}-2uu_{y}=t^{-\alpha }(g_{\omega _1}-2ff_{\omega _2}). \end{aligned}$$

This completes the proof. \(\square\)

Next we use the power series method introduced in [42, 43] to derive the power series solutions of the reduced equations (32). Let us introduce the variable \(\omega =\omega _1+a\omega _2\) and assume

$$\begin{aligned} f(\omega _1,\omega _2)= & {} f(\omega )=\sum _{k=0}^{\infty }a_k (\omega _1+a\omega _2)^k=\sum _{k=0}^{\infty }a_k \omega ^k, \end{aligned}$$
(35)
$$\begin{aligned} g(\omega _1,\omega _2)= & {} g(\omega )=\sum _{k=0}^{\infty }b_k (\omega _1+a\omega _2)^k=\sum _{k=0}^{\infty }b_k \omega ^k, \end{aligned}$$
(36)

where \(a_k\) and \(b_k\) are constants to be determined later. Substituting (35) and (36) into the reduced equations (32) and using the derivations in [43] yields that

$$\begin{aligned}{} & {} \sum _{k=0}^{\infty }\frac{\Gamma \left( 1-\frac{(k+1)\alpha }{2}\right) }{\Gamma \left( 1-\frac{(k+3)\alpha }{2} \right) }a_k \omega ^k+a\sum _{k=0}^{\infty }(k+3)(k+2)(k+1)a_{k+3}\omega ^k\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\{} & {} \quad -4a\sum _{k=0}^{\infty }\sum _{l+m+n=k}^{}(n+1)a_{l}a_{m}a_{n+1}\omega ^k-2\sum _{k=0}^{\infty }\sum _{m+n=k}^{}(n+1)b_{m}a_{n+1}\omega ^k=0,\ \ \end{aligned}$$
(37)
$$\begin{aligned}{} & {} \quad \sum _{k=0}^{\infty }(k+1)b_{k+1}\omega ^k-2a\sum _{k=0}^{\infty }\sum _{m+n=k}^{}(n+1)a_{m}a_{n+1}\omega ^k=0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(38)

Equating the coefficients of different powers of \(\omega\) arrives at the following system:

$$\begin{aligned} \left\{ \begin{array}{llll} \frac{\Gamma (1-\frac{(k+1)\alpha }{2})}{\Gamma (1-\frac{(k+3)\alpha }{2} )}a_k-4a\sum _{l+m+n=k}^{}(n+1)a_{l}a_{m}a_{n+1}-2\sum _{m+n=k}^{}(n+1)b_{m}a_{n+1}\\ +a(k+3)(k+2)(k+1)a_{k+3}=0,\\ (k+1)b_{k+1}-2a\sum _{m+n=k}^{}(n+1)a_{m}a_{n+1}=0,\\ \end{array}\right. \end{aligned}$$
(39)

from which, we can obtain the explicit expressions of \(a_k\) and \(b_k\). For \(k=0\), we have

$$\begin{aligned} \left\{ \begin{array}{llll} b_1=2aa_0a_1, \\ a_3=-\frac{1}{6a}\Big [\frac{\Gamma (1-\frac{\alpha }{2})}{\Gamma (1-\frac{3\alpha }{2})}a_0-4aa^2_0a_1-2b_0a_1\Big ]. \\ \end{array}\right. \end{aligned}$$
(40)

For \(k=1\), we have

$$\begin{aligned} \left\{ \begin{array}{llll} b_2=a(a_1^2+2a_0a_2), \\ a_4=-\frac{1}{24a}\Big [\frac{\Gamma (1-\alpha )}{\Gamma (1-2\alpha ) }a_1-8a(a_0a_1^2+a_0^2a_2)-2(a_1b_1+2b_0a_2)\Big ]. \\ \end{array}\right. \end{aligned}$$
(41)

For \(k=2\), we have

$$\begin{aligned} \left\{ \begin{array}{llll} b_3=2a(a_1a_2+a_0a_3), \\ a_5=-\frac{1}{60a}\Big [\frac{\Gamma (1-\frac{3\alpha }{2})}{\Gamma (1-\frac{5\alpha }{2}) }a_2-4a(a_1^3+3a_0^2a_3+6a_0a_1a_2)-2(a_1b_2+2b_1a_2+3b_0a_3)\Big ]. \\ \end{array}\right. \end{aligned}$$
(42)

For \(k>2\), we have

$$\begin{aligned} \left\{ \begin{array}{llll} b_{k+1}=\frac{2a}{(k+1)}\sum _{m+n=k}^{}(n+1)a_{m}a_{n+1}, \\ a_{k+3}=\frac{-1}{a(k+3)(k+2)(k+1)}\Big [\frac{\Gamma (1-\frac{(k+1)\alpha }{2})}{\Gamma (1-\frac{(k+3)\alpha }{2} )}a_k-4a\sum _{l+m+n=k}^{}(n+1)a_{l}a_{m}a_{n+1}\\ \ \ \ \ \ \ \ -2\sum _{m+n=k}^{}(n+1)b_{m}a_{n+1}\Big ]. \\ \end{array}\right. \end{aligned}$$
(43)

In what follows, applying the method in [44, 45], the convergence analysis of the power series solutions for the system (32) will be presented. From (40)–(43), we get

$$\begin{aligned} \left\{ \begin{array}{llll} |b_{k+1}|\le 2a\sum _{m+n=k}^{}|a_{m}||a_{n+1}|, \\ |a_{k+3}|\le M\Big [|a_k|+\sum _{l+m+n=k}^{}|a_{l}||a_{m}||a_{n+1}|+\sum _{m+n=k}^{}|b_{m}||a_{n+1}|\Big ], \\ \end{array}\right. \end{aligned}$$
(44)

with \(M=\max \left( \frac{\Gamma \left( 1-\frac{(k+1)\alpha }{2}\right) }{6a\Gamma \left( 1-\frac{(k+3)\alpha }{2} \right) },\frac{2}{3},\frac{1}{3a}\right)\). Assuming

$$\begin{aligned} C(\theta )=\sum _{k=0}^{\infty }c_k \theta ^k,\ \ D(\theta )=\sum _{k=0}^{\infty }d_k \theta ^k, \end{aligned}$$
(45)

with \(c_0=|a_0|\), \(d_0=|b_0|\), \(c_1=|a_1|\), \(c_2=|a_2|\) and

$$\begin{aligned} \left\{ \begin{array}{llll} d_{k+1}=2a\sum _{m+n=k}^{}c_{m}c_{n+1},\ k=0,1,2\ldots ,\\ c_{k+3}=M(c_k+\sum _{l+m+n=k}^{}c_{l}c_{m}c_{n+1}+\sum _{m+n=k}^{}d_{m}c_{n+1}),\ k=0,1,2\ldots ,\\ \end{array}\right. \end{aligned}$$
(46)

that is, \(|a_k|\le c_k\), \(|b_k|\le d_k\), \(k=0,1,2\ldots\), we can get

$$\begin{aligned} C(\theta )&=c_0+c_1\theta +c_2\theta ^2+\sum _{k=0}^{\infty }c_{k+3} \theta ^{k+3}\ \ \ \ \ \nonumber \\&=c_0+c_1\theta +c_2\theta ^2+M\sum _{k=0}^{\infty }\left( c_k+\sum _{l+m+n=k}^{}c_{l}c_{m}c_{n+1}+\sum _{m+n=k}^{}d_{m}c_{n+1}\right) \theta ^{k+3}\ \ \ \ \nonumber \\&=c_0+c_1\theta +c_2\theta ^2+MC(\theta )\theta ^3+M\sum _{k=0}^{\infty }\left[ \sum _{l+m+n=k}^{}c_{l}c_{m}c_{n+1}+\sum _{m+n=k}^{}d_{m}c_{n+1}\right] \theta ^{k+3}, \ \ \end{aligned}$$
(47)
$$\begin{aligned} D(\theta&)=d_0+\sum _{k=0}^{\infty }d_{k+1} \theta ^{k+1}=d_0+2a\sum _{k=0}^{\infty }\sum _{m+n=k}^{}c_{m}c_{n+1}\theta ^{k+1}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(48)

Consider the implicit functional system with respect to the independent variable \(\theta\),

$$\begin{aligned}&H(\theta ,C,D)=C-c_0-c_1\theta -c_2\theta ^2-MC\theta ^3-M\sum _{k=0}^{\infty }\Big [\sum _{l+m+n=k}^{}c_{l}c_{m}c_{n+1}\nonumber \\&\quad +\sum _{m+n=k}^{}d_{m}c_{n+1}\Big ]\theta ^{k+3}, \end{aligned}$$
(49)
$$\begin{aligned}&\quad I(\theta ,C,D)=D-d_0-2a\sum _{k=0}^{\infty }\sum _{m+n=k}^{}c_{m}c_{n+1}\theta ^{k+1},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(50)

from which, H and I are analytic in the neighborhood of point \((0,c_0,d_0)\) and

$$\begin{aligned} H(0,c_0,d_0)=0,\ \ I(0,c_0,d_0)=0. \end{aligned}$$
(51)

The Jacobian determinant is

$$\begin{aligned} J=\frac{\partial (H,I)}{\partial (C,D)}\ne 0. \end{aligned}$$
(52)

Then the two series \(C=C(\theta )\) and \(D=D(\theta )\) are analytic in the neighborhood of \((0,c_0,d_0)\) with positive radius by implicit function theorem, that is, the series \(f(\omega )\) and \(g(\omega )\) are convergent in the neighborhood of \((0,c_0,d_0)\).

Therefore, the power series solutions of (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations (2) have the form

$$\begin{aligned} u(t,x,y)=t^{-\frac{\alpha }{2}}f(\omega )=t^{-\frac{\alpha }{2}}\sum _{k=0}^{\infty }a_k(xt^{-\frac{\alpha }{2}}+ay)^k=a_0t^{-\frac{\alpha }{2}}+a_1(xt^{-\frac{\alpha }{2}}+ay)t^{-\frac{\alpha }{2}}\ \ \ \ \ \nonumber \\ +a_2(xt^{-\frac{\alpha }{2}}+ay)^2t^{-\frac{\alpha }{2}}+\sum _{k=0}^{\infty }\frac{-t^{-\frac{\alpha }{2}}}{a(k+3)(k+2)(k+1)}\Big [\frac{\Gamma \left( 1-\frac{(k+1)\alpha }{2}\right) }{\Gamma \left( 1-\frac{(k+3)\alpha }{2} \right) }a_k \nonumber \\ -4a\sum _{l+m+n=k}^{}(n+1)a_{l}a_{m}a_{n+1}-2\sum _{m+n=k}^{}(n+1)b_{m}a_{n+1}\Big ](xt^{-\frac{\alpha }{2}}+ay)^{k+3}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(53a)
$$\begin{aligned} v(t,x,y)=t^{-\frac{\alpha }{2}}g(\omega )=t^{-\frac{\alpha }{2}}\sum _{k=0}^{\infty }b_k(xt^{-\frac{\alpha }{2}}+ay)^k\ \ \ \ \ \nonumber \\ =b_0t^{-\frac{\alpha }{2}}+\sum _{k=0}^{\infty }\frac{2at^{-\frac{\alpha }{2}}}{(k+1)}\sum _{m+n=k}^{}(n+1)a_{m}a_{n+1}(xt^{-\frac{\alpha }{2}}+ay)^{k+1},\ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(53b)

where \(a_k\) and \(b_k\) are defined by (40)–(43) with arbitrary initial conditions \(a_0=f(0,0)\), \(b_0=g(0,0)\), \(a_1=f'(0,0)\) and \(a_2=\frac{1}{2}f''(0,0)\).

Case 2: \(X_1\)

The characteristic equation corresponding to the group generator \(X_1\) is

$$\begin{aligned} \frac{\textrm{d} t }{t }=\frac{\textrm{d}x }{0 }=\frac{\textrm{d} y }{\alpha y }=\frac{\textrm{d}u }{0 }=\frac{\textrm{d}v }{-\alpha v }, \end{aligned}$$
(54)

from which, we obtain the similarity variables x, \(yt^{-\alpha }\), u and \(vt^{\alpha }\). So we get the invariant solutions of the system (2) as follows:

$$\begin{aligned} u(t,x,y)=f(\omega _1,\omega _2),\ \ v(t,x,y)=t^{-\alpha }g(\omega _1,\omega _2), \end{aligned}$$
(55)

with \(\omega _1=x\), \(\omega _2=yt^{-\alpha }\).

Theorem 2.2

The similarity transformations \(u(t,x,y)=f(\omega _1,\omega _2)\), \(v(t,x,y)=t^{-\alpha }g(\omega _1,\omega _2)\) with the similarity variables \(\omega _1=x\), \(\omega _2=yt^{-\alpha }\) reduce the system (2) to the system of (1+1)-dimensional fractional differential equations given by

$$\begin{aligned} \left\{ \begin{array}{llll} (\mathcal {P}^{1-\alpha ,\alpha }_{\infty ,\frac{1 }{\alpha }}f)(\omega _1,\omega _2)+f_{\omega _1\omega _1\omega _2}-4f^2f_{\omega _2}-2f_{\omega _1}g=0, \\ \\ g_{\omega _1}-2ff_{\omega _2}=0. \\ \end{array}\right. \end{aligned}$$
(56)

The proof of Theorem 2.2 is similar to that of Theorem 2.1. Meanwhile, assuming \(\omega =\omega _2+a\omega _1\) and using the procedure in Case 1, we can obtain the power series solution of Eqs. (2) as follows:

$$\begin{aligned} u(t,x,y) &= f(\omega )=\sum _{k=0}^{\infty }a_k(yt^{-\alpha }+ax)^k=a_0+a_1(yt^{-\alpha }+ax) \nonumber \\{} & +a_2(yt^{-\alpha }+ax)^2+\sum _{k=0}^{\infty }\frac{-1}{(k+3)(k+2)(k+1)a^2}\Big [\frac{\Gamma (1-k\alpha )}{\Gamma (1-(k+1)\alpha )}a_k \nonumber \\ & -4\sum _{l+m+n=k}^{}(n+1)a_{l}a_{m}a_{n+1}-2a\sum _{m+n=k}^{}(n+1)b_{m}a_{n+1}\Big ](yt^{-\alpha }+ax)^{k+3}, \end{aligned}$$
(57a)
$$\begin{aligned} v(t,x,y) & = t^{-\alpha }g(\omega )=t^{-\alpha }\sum _{k=0}^{\infty }b_k(yt^{-\alpha }+ax)^k\ \ \ \ \ \nonumber \\ &= b_0t^{-\alpha }+\sum _{k=0}^{\infty }\frac{2t^{-\alpha }}{(k+1)a}\sum _{m+n=k}^{}(n+1)a_{m}a_{n+1}(yt^{-\alpha }+ax)^{k+1},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(57b)

with arbitrary initial conditions \(a_0=f(0,0)\), \(b_0=g(0,0)\), \(a_1=f'(0,0)\) and \(a_2=\frac{1}{2}f''(0,0)\).

Case 3: \(X_2\)

The characteristic equation corresponding to the group generator \(X_2\) is

$$\begin{aligned} \frac{\textrm{d} t }{0}=\frac{\textrm{d}x }{x }=\frac{\textrm{d} y }{-2y }=\frac{\textrm{d}u }{-u }=\frac{\textrm{d}v }{v }, \end{aligned}$$
(58)

from which, we obtain the similarity variables t, \(yx^2\), ux and \(vx^{-1}\). So we get the invariant solutions of the system (2)

$$\begin{aligned} u=x^{-1}f(t,X),\ \ v=xg(t,X),\ \ X=yx^2, \end{aligned}$$
(59)

and the reduced equations

$$\begin{aligned} \left\{ \begin{array}{llll} D^{\alpha }_{t}f+4X^2f_{XXX}+6Xf_{XX}-4f^2f_{X}-4Xgf_{X}+2fg=0, \\ 2ff_{X}-2Xg_X-g=0, \\ \end{array}\right. \end{aligned}$$
(60)

which is (1+1)-dimensional time fractional partial differential equations. For Eqs. (60), we can once again use the Lie symmetry analysis method to further reduce them to fractional ordinary differential equations, and obtain the following generators:

$$\begin{aligned} \varLambda _1=\frac{\partial }{\partial X },\ \ \varLambda _2=t\frac{\partial }{\partial t }+\alpha X\frac{\partial }{\partial X }-\alpha g\frac{\partial }{\partial g }. \end{aligned}$$
(61)

For \(\varLambda _1=\frac{\partial }{\partial X }\), Eqs. (60) have the following invarant solutions:

$$\begin{aligned} f(t,X)=F(t),\ \ g(t,X)=G(t). \end{aligned}$$
(62)

Substituting (62) into Eqs. (60), we get the following solutions:

$$\begin{aligned} F(t)=\frac{C_0 }{\Gamma (\alpha )}t^{\alpha -1},\ \ G(t)=0, \end{aligned}$$
(63)

That is, Eqs. (2) have one trivial solution, i.e.,

$$\begin{aligned} u=\frac{C_0 }{\Gamma (\alpha )}x^{-1}t^{\alpha -1},\ \ v=0. \end{aligned}$$
(64)

For \(\varLambda _2=t\frac{\partial }{\partial t }+\alpha X\frac{\partial }{\partial X }-\alpha g\frac{\partial }{\partial g }\), Eqs. (60) have the following invarant solutions:

$$\begin{aligned} f(t,X)=F(\sigma ),\ \ g(t,X)=t^{-\alpha }G(\sigma ),\ \ \sigma =Xt^{-\alpha }. \end{aligned}$$
(65)

Substituting (65) into Eqs. (60), we can obtain the following fractional ordinary differential equations:

$$\begin{aligned} \left\{ \begin{array}{llll} (\mathcal {P}^{1-\alpha ,\alpha }_{\frac{1 }{\alpha }}F)(\sigma )+4\sigma ^2F_{\sigma \sigma \sigma }+6\sigma F_{\sigma \sigma }-4F^2F_{\sigma }-4\sigma GF_{\sigma }-2FG=0, \\ 2FF_{\sigma }-2\sigma G_{\sigma }-G=0. \\ \end{array}\right. \end{aligned}$$
(66)

The power series solution of (66) can be obtained as

$$\begin{aligned} F(\sigma )=\sum _{k=0}^{\infty }a_k\sigma ^k, \ \ G(\sigma )=\sum _{k=0}^{\infty }b_k\sigma ^k, \end{aligned}$$
(67)

where \(a_k\) and \(b_k\) are defined by \(a_0=f(0)\), \(a_1=\frac{1}{4a_0\Gamma (1-\alpha )}-\frac{1}{2}\), \(b_0=2a_0a_1\), \(a_2=\Big [\frac{\Gamma (1-\alpha )}{\Gamma (1-2\alpha )}a_1-8a_0a_1^2+4b_0a_1-4a_0a_1\Big ]/(8a_0^2-12)\) and

$$\begin{aligned} \left\{ \begin{array}{llll} b_{k+1}=\frac{2}{2k+3}\sum _{m+n=k+1}^{}(n+1)a_{m}a_{n+1}, \\ \\ a_{k+3}=\frac{\frac{\Gamma (1-(k+2)\alpha )}{\Gamma (1-(k+3)\alpha )}a_{k+2}-4\sum _{l+m+n=k+2}^{n\ne k+2}(n+1)a_{l}a_{m}a_{n+1}-4\sum _{m+n=k+1}^{}(n+1)b_{m}a_{n+1}-2\sum _{m+n=k+2}^{}a_{m}a_{n}}{4(k+3)a_0^2-4(k+3)(k+2)(k+1)-6(k+3)(k+2)}. \\ \end{array}\right. \end{aligned}$$
(68)

That is, Eqs. (2) have the following power series solution:

$$\begin{aligned} u(t,x,y)= & {} x^{-1}f(t,X)=x^{-1}F(\sigma )=a_0x^{-1}+\left( \frac{1}{4a_0\Gamma (1-\alpha )}-\frac{1}{2}\right) xyt^{-\alpha } \nonumber \\{} & {} +\left[ \left( \frac{\Gamma (1-\alpha )}{\Gamma (1-2\alpha )}a_1-8a_0a_1^2+4b_0a_1-4a_0a_1\right) /(8a_0^2-12)\right] x^3y^2t^{-2\alpha }\nonumber \\{} & {} +\sum _{k=0}^{\infty }\left[ \frac{\frac{\Gamma (1-(k+2)\alpha )}{\Gamma (1-(k+3)\alpha )}a_{k+2}-4\sum _{l+m+n=k+2}^{n\ne k+2}(n+1)a_{l}a_{m}a_{n+1}}{4(k+3)a_0^2-4(k+3)(k+2)(k+1)-6(k+3)(k+2)} \right. \nonumber \\{} & {} -\left. \frac{4\sum _{m+n=k+1}^{}(n+1)b_{m}a_{n+1}+2\sum _{m+n=k+2}^{}a_{m}a_{n}}{4(k+3)a_0^2-4(k+3)(k+2)(k+1)-6(k+3)(k+2)}\right] x^{2k+5}y^{k+3}t^{-(k+3)\alpha }, \end{aligned}$$
(69a)
$$\begin{aligned} v(t,x,y)= & {} xg(t,X)=xt^{-\alpha }G(\sigma )=2a_0a_1xt^{-\alpha } \nonumber \\{} & {} +\sum _{k=0}^{\infty }\frac{2}{2k+3}\sum _{m+n=k+1}^{}(n+1)a_{m}a_{n+1}x^{2k+3}y^{k+1}t^{-(k+2)\alpha }. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(69b)

Case 4: \(X_3\)

The characteristic equation corresponding to the group generator \(X_3\) is

$$\begin{aligned} \frac{\textrm{d} t }{0}=\frac{\textrm{d}x }{1 }=\frac{\textrm{d} y }{0 }=\frac{\textrm{d}u }{0 }=\frac{\textrm{d}v }{0 }, \end{aligned}$$
(70)

from which, we obtain the similarity variables t, y, u and v. So we get the invariant solutions of the system (2) as follows:

$$\begin{aligned} u=f(t,y),\ \ v=g(t,y). \end{aligned}$$
(71)

Substituting (71) into Eqs. (2), we have the following reduced equations:

$$\begin{aligned} \left\{ \begin{array}{llll} D^{\alpha }_{t}f-4f^2f_y=0, \\ 2ff_y=0, \\ \end{array}\right. \end{aligned}$$
(72)

which is rewritten as

$$\begin{aligned} D^{\alpha }_{t}f(t)=0. \end{aligned}$$
(73)

We can easily use the Laplace transform of Riemann–Liouville fractional derivative, i.e.,

$$\begin{aligned} \mathcal {L}\{D_{t}^\alpha f(t)\}=s^\alpha F(s)-\sum _{k=0}^{n-1 }s^kf^{(\alpha -k-1)}(0),\ \ n-1<\alpha \le n, n\in N \end{aligned}$$
(74)

to obtain the exact solutions of (73) as

$$\begin{aligned} f(t)=\frac{k_1}{\Gamma (\alpha )}t^{\alpha -1}, \end{aligned}$$
(75)

where \(k_1\) is a constant determined by initial condition, i.e., \(k_1=f^{(\alpha -1)}(0)\).

In this case, we obtain the following trivial solutions:

$$\begin{aligned} u=\frac{k_1}{\Gamma (\alpha )}t^{\alpha -1},\ \ v=g(t,y), \end{aligned}$$
(76)

where g(t, y) is an arbitrary function.

Case 5: \(X_4\)

The characteristic equation corresponding to the group generator \(X_4\) is

$$\begin{aligned} \frac{\textrm{d} t }{0}=\frac{\textrm{d}x }{0 }=\frac{\textrm{d} y }{1 }=\frac{\textrm{d}u }{0 }=\frac{\textrm{d}v }{0 }, \end{aligned}$$
(77)

from which, we obtain the similarity variables t, x, u and v. So we get the invariant solutions of the system (2)

$$\begin{aligned} u=f(t,x),\ \ v=g(t,x). \end{aligned}$$
(78)

and the reduced equations

$$\begin{aligned} \left\{ \begin{array}{llll} D^{\alpha }_{t}f-2f_xg=0, \\ g_x=0. \\ \end{array}\right. \end{aligned}$$
(79)

Now we use invariant subspace method for (79) and find that they admit the following invariant subspace:

$$\begin{aligned} W_2^1\times W_2^2=\{1,x\}\times \{1,x\}, \end{aligned}$$

from which, the solutions of (79) have the form \(f=f_1(t)+f_2(t)x\), \(g=g(t)\). So (79) can be rewritten as

$$\begin{aligned} D^{\alpha }_{t}f_1+xD^{\alpha }_{t}f_2-2f_2g=0, \end{aligned}$$
(80)

from which, we can get \(f_2(t)=\frac{k_2}{\Gamma (\alpha )}t^{\alpha -1}\) with \(k_2=f_2^{(\alpha -1)}(0)\) and

$$\begin{aligned} D^{\alpha }_{t}f_1-\frac{2k_2}{\Gamma (\alpha )}t^{\alpha -1}g=0. \end{aligned}$$
(81)

Then solving the equation (81), we obtain

$$\begin{aligned} f_1(t)=k_1t^\beta ,\ \ g(t)=\frac{k_1\Gamma (\alpha )\Gamma (\beta +1)}{2k_2\Gamma (\beta -\alpha +1)}t^{\beta -2\alpha +1}, \end{aligned}$$
(82)

where \(k_1\) and \(\beta\) are arbitrary constants.

In this case, we obtain the following trivial solutions:

$$\begin{aligned} u=k_1t^\beta +\frac{k_2}{\Gamma (\alpha )}t^{\alpha -1}x,\ \ v=\frac{k_1\Gamma (\alpha )\Gamma (\beta +1)}{2k_2\Gamma (\beta -\alpha +1)}t^{\beta -2\alpha +1}. \end{aligned}$$
(83)

Case 6: \(X_3+\gamma X_4\)

The characteristic equation corresponding to the group generator \(X_3+\gamma X_4\) is

$$\begin{aligned} \frac{\textrm{d} t }{0}=\frac{\textrm{d}x }{1 }=\frac{\textrm{d} y }{\gamma }=\frac{\textrm{d}u }{0 }=\frac{\textrm{d}v }{0 }, \end{aligned}$$
(84)

from which, we obtain the similarity variables t, \(y-\gamma x\), u and v. So we get the invariant solutions of the system (2)

$$\begin{aligned} u=f(t,X),\ \ v=g(t,X),\ \ X=y-\gamma x, \end{aligned}$$
(85)

and the reduced equations

$$\begin{aligned} \left\{ \begin{array}{llll} D^{\alpha }_{t}f+\gamma ^2f_{XXX}-4f^2f_{X}+2\gamma f_{X}g=0, \\ 2ff_{X}+\gamma g_X=0, \\ \end{array}\right. \end{aligned}$$
(86)

which can be further reduced to fractional ordinary differential equations by using Lie symmetry analysis method once again. The following generators are admitted by (86):

$$\begin{aligned} \varLambda _1=\frac{\partial }{\partial X },\ \ \varLambda _2=t\frac{\partial }{\partial t }+\frac{\alpha }{3 } X\frac{\partial }{\partial X }-\frac{\alpha }{3 } f\frac{\partial }{\partial f }-\frac{2\alpha }{3 } g\frac{\partial }{\partial g }. \end{aligned}$$
(87)

For \(\varLambda _1=\frac{\partial }{\partial X }\), similarly to Case 3, we can get another one trivial solution, i.e.,

$$\begin{aligned} u=\frac{C_0 }{\Gamma (\alpha )}t^{\alpha -1},\ \ v=0. \end{aligned}$$
(88)

For \(\varLambda _2=t\frac{\partial }{\partial t }+\frac{\alpha }{3 } X\frac{\partial }{\partial X }-\frac{\alpha }{3 } f\frac{\partial }{\partial f }-\frac{2\alpha }{3 } g\frac{\partial }{\partial g }\), Eqs. (86) have the following invarant solutions:

$$\begin{aligned} f(t,X)=t^{-\frac{\alpha }{3 }}F(\sigma ),\ \ g(t,X)=t^{-\frac{2\alpha }{3 }}G(\sigma ),\ \ \sigma =Xt^{-\frac{\alpha }{3 }}. \end{aligned}$$
(89)

Substituting (89) into Eqs. (86), we can obtain the following fractional ordinary differential equations:

$$\begin{aligned} \left\{ \begin{array}{llll} (\mathcal {P}^{1-\frac{4\alpha }{3 },\alpha }_{\frac{3 }{\alpha }}F)(\sigma )+\gamma ^2F_{\sigma \sigma \sigma }-4F^2F_{\sigma }+2\gamma GF_{\sigma }=0, \\ 2FF_{\sigma }+\gamma G_{\sigma }=0. \\ \end{array}\right. \end{aligned}$$
(90)

The power series solution of (3.61) can be obtained as

$$\begin{aligned} F(\sigma )=\sum _{k=0}^{\infty }a_k\sigma ^k, \ \ G(\sigma )=\sum _{k=0}^{\infty }b_k\sigma ^k, \end{aligned}$$
(91)

where \(a_k\) and \(b_k\) are defined by \(a_0=f(0)\), \(b_0=g(0)\), \(a_1=f'(0)\), \(a_2=\frac{1}{2}f''(0)\) and

$$\begin{aligned} \left\{ \begin{array}{llll} b_{k+1}=-\frac{2}{(k+1)\gamma }\sum _{m+n=k}^{}(n+1)a_{m}a_{n+1}, \\ a_{k+3}=\frac{-1}{(k+3)(k+2)(k+1)\gamma ^2}\Big [\frac{\Gamma (1-\frac{(k+1)\alpha }{3})}{\Gamma (1-\frac{(k+4)\alpha }{3} )}a_k-4\sum _{l+m+n=k}^{}(n+1)a_{l}a_{m}a_{n+1}\\ \ \ \ \ \ \ \ +2\gamma \sum _{m+n=k}^{}(n+1)b_{m}a_{n+1}\Big ]. \\ \end{array}\right. \end{aligned}$$
(92)

That is, Eqs. (2) have the following power series solution:

$$\begin{aligned} u(t,x,y)= & {} f(t,X)=t^{-\frac{\alpha }{3 }}F(\sigma )=a_0t^{-\frac{\alpha }{3 }}+a_1(y-\gamma x)t^{-\frac{2\alpha }{3 }}+a_2(y-\gamma x)^2t^{-\alpha } \nonumber \\{} & {} +\sum _{k=0}^{\infty }\frac{-1}{(k+3)(k+2)(k+1)\gamma ^2}\Big [\frac{\Gamma (1-\frac{(k+1)\alpha }{3})}{\Gamma (1-\frac{(k+4)\alpha }{3} )}a_k-4\sum _{l+m+n=k}^{}(n+1) \ \ \ \ \nonumber \\{} & {} a_{l}a_{m}a_{n+1}+2\gamma \sum _{m+n=k}^{}(n+1)b_{m}a_{n+1}\Big ](y-\gamma x)^{k+3}t^{-\frac{(k+4)\alpha }{3 }}, \end{aligned}$$
(93a)
$$\begin{aligned} v(t,x,y)= & {} g(t,X)=t^{-\frac{2\alpha }{3 }}G(\sigma ) \nonumber \\= & {} b_0t^{-\frac{2\alpha }{3 }}+\sum _{k=0}^{\infty }\frac{-2}{(k+1)\gamma }\sum _{m+n=k}^{}(n+1)a_{m}a_{n+1}(y-\gamma x)^{k+1}t^{-\frac{(k+3)\alpha }{3 }}. \ \ \ \ \ \ \end{aligned}$$
(93b)

4 Conservation Laws of Eq. (2)

In this section, we will construct conservation laws of Eq. (2) by using the generalization of the Noether operators and the new conservation theorem [46, 47].

The system (2) is denoted as

$$\begin{aligned} \left\{ \begin{array}{llll} F_1=D^{\alpha }_{t}u+u_{xxy}-4u^2u_y-2u_{x}v=0, \\ F_2=v_x-2uu_y=0, \\ \end{array}\right. \end{aligned}$$
(94)

of which the formal Lagrangian is given by

$$\begin{aligned} \begin{aligned} \mathcal {L}&=p(t,x,y)F_1+q(t,x,y)F_2\\&=p(t,x,y)\big (D^{\alpha }_{t}u+u_{xxy}-4u^2u_y-2u_{x}v\big )+q(t,x,y)\big (v_x-2uu_y\big ), \ \ \end{aligned} \end{aligned}$$
(95)

where p(t, x, y) and q(t, x, y) are new dependent variables. The Euler–Lagrange operators are

$$\begin{aligned}{} & {} \frac{\delta }{\delta u}=\frac{\partial }{\partial u }+(D^{\alpha }_{t})^*\frac{\partial }{\partial (D^{\alpha }_{t}u) }-D_{x}\frac{\partial }{\partial u_{x} }-D_{y}\frac{\partial }{\partial u_{y} }- D^2_{x}D_{y}\frac{\partial }{\partial u_{xxy} }, \end{aligned}$$
(96)
$$\begin{aligned}{} & {} \quad \frac{\delta }{\delta v}=\frac{\partial }{\partial v }-D_{x}\frac{\partial }{\partial v_{x} }, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(97)

where \((D^{\alpha }_{t})^*\) is the adjoint operator of \(D^{\alpha }_{t}\). It is defined by the right-sided of Caputo fractional derivative, i.e.

$$\begin{aligned} (D^{\alpha }_{t})^*f(t,x)\equiv \ _{t}^{c}D_{T}^{\alpha }f(t,x)= \left\{ \begin{array}{lll} \frac{1}{\Gamma (n-\alpha )}\int _t^T {\frac{1}{(t-s)^{ \alpha -n+1 }} }\frac{\partial ^n }{\partial {s^n}}f(s,x)\textrm{d}s,&{} n-1<\alpha <n, n\in \mathbb {N} \\ D_{t}^{n}f(t,x),&{} \alpha =n\in \mathbb {N}. \\ \end{array}\right. \end{aligned}$$

The system of adjoint equations to (94) is given by

$$\begin{aligned} \left\{ \begin{array}{llll} F_1^*=\frac{\delta \mathcal {L} }{\delta u}=(D^{\alpha }_{t})^*p+2vp_x+2v_xp+4u^2p_y+2uq_y-p_{xxy}=0, \\ F_2^*=\frac{\delta \mathcal {L} }{\delta v}=-2u_xp-q_x=0. \\ \end{array}\right. \end{aligned}$$
(98)

Next we will use the above adjoint equations and the new conservation theorem to construct conservation laws of Eq. (2). From the classical definition of the conservation laws, a vector \(C=(C^t,C^x,C^y)\) is called the conserved vector for the governing equation if it satisfies the conservation equation \([D_tC^t+D_xC^x+D_yC^y]_{F_1,F_2=0}=0\). We can obtain the components of the conserved vector by using the generalization of the Noether operators.

Firstly, from the fundamental operator identity, i.e.

$$\begin{aligned} prX+D_t\tau \cdot \mathcal {I}+D_x\xi \cdot \mathcal {I}+D_y\theta \cdot \mathcal {I}=W^u\cdot \frac{\delta }{\delta u}+W^v\cdot \frac{\delta }{\delta v}+D_t\mathcal {N}^t+D_x\mathcal {N}^x+D_y\mathcal {N}^y, \end{aligned}$$
(99)

where prX is mentioned in (6), \(\mathcal {I}\) is the identity operator and \(W^u=\eta -\tau u_t-\xi u_x-\theta u_y\), \(W^v=\zeta -\tau v_t-\xi v_x-\theta v_y\) are the characteristics for group generator X, we can get the Noether operators as follows:

$$\begin{aligned} \mathcal {N}^t= & {} \tau \mathcal {I}+\sum _{k=0}^{n-1 }(-1)^kD_{t}^{\alpha -1-k}(W^u)D_{t}^k\frac{\partial }{\partial (D^{\alpha }_{t}u) }-(-1)^nJ\left( W^u,D_t^n\frac{\partial }{\partial (D^{\alpha }_{t}u) }\right) ,\ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(100)
$$\begin{aligned} \mathcal {N}^x= & {} \xi \mathcal {I}+W^u\left( \frac{\partial }{\partial u_x }+D_{x}D_{y}\frac{\partial }{\partial u_{xxy}}\right) +W^v\frac{\partial }{\partial v_x }-D_{x}(W^u)D_{y}\frac{\partial }{\partial u_{xxy}}+D_{x}D_{y}(W^u)\frac{\partial }{\partial u_{xxy} }, \end{aligned}$$
(101)
$$\begin{aligned} \mathcal {N}^y= & {} \theta \mathcal {I}+W^u\left( \frac{\partial }{\partial u_y }+D^2_{x}\frac{\partial }{\partial u_{yxx}}\right) -D_{x}(W^u)D_{x}\frac{\partial }{\partial u_{yxx}}+D^2_{x}(W^u)\frac{\partial }{\partial u_{yxx} },\ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(102)

where \(n=[\alpha ]+1\) and J is given by

$$\begin{aligned} J(f,g)=\frac{1 }{\Gamma (n-\alpha ) }\int _0^t\int _t^T\frac{f(\tau ,x,y)g(\theta ,x,y) }{(\theta -\tau )^{\alpha +1-n} }\textrm{d}\theta \textrm{d}\tau . \end{aligned}$$
(103)

The components of conserved vector are defined by

$$\begin{aligned} C^t=\mathcal {N}^t\mathcal {L},\ \ C^x=\mathcal {N}^x\mathcal {L},\ \ C^y=\mathcal {N}^y\mathcal {L}. \end{aligned}$$
(104)

Case 1: \(X_1=t\frac{\partial }{\partial t }+\alpha y\frac{\partial }{\partial y }-\alpha v\frac{\partial }{\partial v }\)

The characteristics of \(X_1\) are

$$\begin{aligned} W^u=-tu_t-\alpha yu_y,\ \ W^v=-\alpha v-tv_t-\alpha yv_y. \end{aligned}$$
(105)

Therefore, for \(0<\alpha <1\),

$$\begin{aligned} C^t&=pD_{t}^{\alpha -1}(W^u)+J(W^u,p_t)\nonumber \\&=-pD_{t}^{\alpha -1}(tu_t+\alpha yu_y)-J(tu_t+\alpha yu_y,p_t), \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(106)
$$\begin{aligned} C^x&=(p_{xy}-2vp)W^u+qW^v-p_yD_x(W^u)+pD_xD_y(W^u)\nonumber \\&=(2vp-p_{xy})(tu_t+\alpha yu_y)-q(\alpha v+tv_t+\alpha yv_y)\nonumber \\&\quad +p_y(tu_{xt}+\alpha yu_{xy})-p(tu_{xyt}+\alpha yu_{xyy}), \end{aligned}$$
(107)
$$\begin{aligned} C^y&=(p_{xx}-4u^2p-2uq)W^u-p_xD_x(W^u)+pD^2_x(W^u)\nonumber \\&=(4u^2p+2uq-p_{xy})(tu_t+\alpha yu_y)+p_x(tu_{xt}+\alpha yu_{xy}) \nonumber \\&\quad -p(tu_{xxt}+\alpha yu_{xxy}). \end{aligned}$$
(108)

Case 2: \(X_2=x\frac{\partial }{\partial x }-2y\frac{\partial }{\partial y }-u\frac{\partial }{\partial u }+v\frac{\partial }{\partial v }\)

The characteristics of \(X_2\) are

$$\begin{aligned} W^u=-u-xu_x+2yu_y,\ \ W^v=v-xv_x+2yv_y. \end{aligned}$$
(109)

Therefore, for \(0<\alpha <1\),

$$\begin{aligned} C^t&=pD_{t}^{\alpha -1}(W^u)+J(W^u,p_t)\nonumber \\ &=-pD_{t}^{\alpha -1}(u+xu_x-2yu_y)-J(u+xu_x-2yu_y,p_t), \end{aligned}$$
(110)
$$\begin{aligned} C^x& = (p_{xy}-2vp)W^u+qW^v-p_yD_x(W^u)+pD_xD_y(W^u)\nonumber \\ & = (2vp-p_{xy})(u+xu_x-2yu_y)+q(v-xv_x+2yv_y)\nonumber \\&+p_y(2u_x+xu_{xx}-2yu_{xy})-p(xu_{xxy}-2yu_{xyy}), \ \ \end{aligned}$$
(111)
$$\begin{aligned} C^y &=(p_{xx}-4u^2p-2uq)W^u-p_xD_x(W^u)+pD^2_x(W^u)\nonumber \\ &= (4u^2p+2uq-p_{xy})(u+xu_x-2yu_y)+p_x(2u_x \nonumber \\&+xu_{xx}-2yu_{xy})-p(3u_{xx}+xu_{xxx}-2yu_{xxy}). \end{aligned}$$
(112)

Case 3: \(X_3=\frac{\partial }{\partial x }\)

The characteristics of \(X_3\) are

$$\begin{aligned} W^u=-u_x,\ \ W^v=-v_x. \end{aligned}$$
(113)

Therefore, for \(0<\alpha <1\),

$$\begin{aligned} C^t&=pD_{t}^{\alpha -1}(W^u)+J(W^u,p_t)=-pD_{t}^{\alpha -1}u_x-J(u_x,p_t), \end{aligned}$$
(114)
$$\begin{aligned} C^x&=(p_{xy}-2vp)W^u+qW^v-p_yD_x(W^u)+pD_xD_y(W^u)\nonumber \\&=(2vp-p_{xy})u_x-qv_x+p_yu_{xx}-pu_{xxy}, \ \ \end{aligned}$$
(115)
$$\begin{aligned} C^y&=(p_{xx}-4u^2p-2uq)W^u-p_xD_x(W^u)+pD^2_x(W^u) \ \ \ \ \nonumber \\&=(4u^2p+2uq-p_{xy})u_x+p_xu_{xx}-pu_{xxx}. \end{aligned}$$
(116)

Case 4: \(X_4=\frac{\partial }{\partial y }\)

The characteristics of \(X_4\) are

$$\begin{aligned} W^u=-u_y,\ \ W^v=-v_y. \end{aligned}$$
(117)

Therefore, for \(0<\alpha <1\),

$$\begin{aligned} C^t&=pD_{t}^{\alpha -1}(W^u)+J(W^u,p_t)=-pD_{t}^{\alpha -1}u_y-J(u_y,p_t), \end{aligned}$$
(118)
$$\begin{aligned} C^x&=(p_{xy}-2vp)W^u+qW^v-p_yD_x(W^u)+pD_xD_y(W^u)\nonumber \\&=(2vp-p_{xy})u_y-qv_y+p_yu_{xy}-pu_{xyy}, \ \ \end{aligned}$$
(119)
$$\begin{aligned} C^y&=(p_{xx}-4u^2p-2uq)W^u-p_xD_x(W^u)+pD^2_x(W^u) \ \ \ \ \nonumber \\&=(4u^2p+2uq-p_{xy})u_y+p_xu_{xy}-pu_{xxy}. \end{aligned}$$
(120)

5 Conclusion

This paper repeatedly uses the Lie symmetry analysis method to reduce the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations to some (1+1)- and (0+1)- dimensional fractional differential equations. For the reduced fractional differential equations with Riemann–Liouville fractional derivative, we use the Laplace transform method and the invariant subspace method to obtain some analytical solutions. For the reduced fractional differential equations with Erdélyi–Kober fractional derivative, we use the power series method to obtain some power series solutions. The main results of this study are summarized in Table 3. This paper demonstrates that the Lie symmetry analysis method can effectively reduce the dimensionality of high-dimensional fractional differential equations until they are reduced to solvable equations. In addition, the nonlocal symmetries have gradually been applied to treat integer order differential equations [48,49,50]. Inspired by this, our next step is to study nonlocal symmetries for fractional differential equations.

Table 3 The reduced equations and exact solutions of Eq. (2)
Full size table