1. Introduction
Nonlinear partial differential equations (NPDEs) usually play an indispensable role in revealing numerous physical phenomena in nature [1–9]. In the last few decades, researchers have paid much attention to the study of NPDEs, including their dynamic properties and solutions. A number of methods were put forward to acquire the solutions of NPDEs, for example, the homogeneous balance method [10–13], inverse scattering transform [14–18], Darboux transformation [19–24], Hirota's bilinear method [25–33], Riemann-Hilbert approach [34–43], nonlocal symmetry method and so on [44–49]. However, for most NPDEs, the solutions are difficult to obtain due to their complex expressions.
As a generalization of classical calculus, fractional calculus is proposed by many researchers. Fractional derivative is an important concept of fractional calculus. The order of a fractional derivative can be an integer or a fraction, whereas the order of a classical derivative can only be an integer. The fractional derivatives used widely are Riemann-Liouville, Caputo and Grünwald-Letnikov types. For many applied disciplines, especially in fluid mechanics and materials science, compared with integer-order NPDEs, nonlinear fractional partial differential equations (NFPDEs) can be better used to explain natural processes and phenomena. This is because the systems corresponding to NFPDEs possess good time memory and global correlation, which can better reflect the historically dependent process of the development of system functions. Thus, the performance of the systems is improved.
The Lie symmetry analysis approach [50–59] is mainly used to find the similarity reductions and solutions of NPDEs. This method was originally utilized to investigate the continuous transformation groups in solving differential equations. This theory was proposed by Sophus Lie and has achieved fruitful development over the last few decades. Because this method can be used to find the group-invariant solutions of NPDEs, it has become a prominent approach to studying the properties of solutions of NPDEs. Since the Lie symmetry analysis approach is mainly used to study integer-order NPDEs, it has an important theoretical research value to extend the method to NFPDEs. Although this method is not widely used in NFPDEs, there are still many achievements in this respect [60–62].
This paper aimed to investigate Lie point symmetries, similarity reductions, conservation laws and traveling wave solutions for the time-fractional Davey-Stewartson (DS) equation. The DS equation is [63]
$\begin{eqnarray}\left\{\begin{array}{l}{\rm{i}}{u}_{t}+{u}_{{xx}}+{u}_{{yy}}-| u{| }^{2}u+{uv}=0,\\ {v}_{{xx}}+{v}_{{yy}}=-{\left(| u{| }^{2}\right)}_{{xx}},\end{array}\right.\end{eqnarray}$
where x and y are spatial coordinates; t is a temporal coordinate; u = u(x, y, t) is a complex analytic function that represents the amplitude of wave; and v = v(x, y, t) is a real analytic function that can be treated as a forcing effect on the wave as it propagates.The DS equation was proposed by Davey and Stewartson. It is a two-dimensional extension of the famous (1+1)-dimensional nonlinear Schrödinger (NLS) equation. In fluid dynamics, the DS equation describes the evolution of wave envelope in water waves and has wide applications.
If the Riemann-Liouville derivative operator ${D}_{t}^{\gamma }$ is introduced, the time-fractional DS equation reads as
$\begin{eqnarray}\left\{\begin{array}{l}{\rm{i}}{D}_{t}^{\gamma }u+{u}_{{xx}}+{u}_{{yy}}-| u{| }^{2}u+{uv}=0,\\ {v}_{{xx}}+{v}_{{yy}}=-{\left(| u{| }^{2}\right)}_{{xx}},\end{array}\right.\end{eqnarray}$
where 0 < γ < 1 is the order of the Riemann-Liouville derivative.This paper is organized as follows. In section 2 , the definition and some properties of the Riemann-Liouville fractional derivative are described. In section 3 , based on the Lie symmetry analysis method, the Lie point symmetries of the time-fractional DS equation are presented. In section 4 , the similarity reductions of the time-fractional DS equation are obtained. Meanwhile, the original equation is converted to a system of fractional partial differential equations and two systems of fractional ordinary differential equations. In section 5 , the conserved vectors and the conservation laws of the time-fractional DS equation are derived. In section 6 , the traveling wave solutions are depicted and plotted. Finally, some main conclusions are presented.
2. Riemann-Liouville fractional derivative
Take $x,y$ as spatial variables and t as a temporal variable, respectively, and $n\in {\mathbb{N}}$, $\gamma \in {{\mathbb{R}}}^{+}$, then for function $f=f(x,y,t)$, its γth-order Riemann-Liouville fractional partial derivative is
$\begin{eqnarray}\begin{array}{l}{D}_{t}^{\gamma }f(x,y,t)=\displaystyle \frac{{\partial }^{\gamma }f}{\partial {t}^{\gamma }}\\ \,=\,\left\{\begin{array}{ll}\displaystyle \frac{\partial }{\partial {t}^{n}}{I}^{n-\gamma }f(x,y,t), & 0\leqslant n-1\lt \gamma \lt n,\\ \displaystyle \frac{{\partial }^{n}f}{\partial {t}^{n}}, & \gamma =n,\end{array}\right.\end{array}\end{eqnarray}$
where ${I}^{\gamma }f(x,y,t)$ is the γth-order Riemann-Liouville fractional integral, and it is defined as $\begin{eqnarray}{I}^{\gamma }f(x,y,t)=\left\{\begin{array}{ll}\displaystyle \frac{1}{{\rm{\Gamma }}(\gamma )}{\int }_{0}^{t}{\left(t-\tau \right)}^{\gamma -1}f(x,y,\tau ){\rm{d}}\tau , & \gamma \gt 0,\\ f(x,y,t), & \gamma =0,\end{array}\right.\end{eqnarray}$
where ${\rm{\Gamma }}(\gamma )$ is the gamma function.The properties of the Riemann-Liouville fractional derivative are
where f(t), f1(t) and f2(t) are functions of [a, b], n − 1 < γ ≤ n.
| (a)${D}_{t}^{\gamma }[{\left(t-a\right)}^{\mu }]=\tfrac{{\rm{\Gamma }}(1+\mu )}{{\rm{\Gamma }}(1-\gamma +\mu )}{\left(t-a\right)}^{\mu -\gamma }$, (t > a), particularly, if f(t) ≡ C, then ${D}_{t}^{\gamma }[f(t)]\,={D}_{t}^{\gamma }C=\tfrac{C}{{\rm{\Gamma }}(1-\gamma )}{\left(t-a\right)}^{-\gamma }$, | |
| (b)Linear rule: ${D}_{t}^{\gamma }[{C}_{1}{f}_{1}(t)+{C}_{2}{f}_{2}(t)]={C}_{1}{D}_{t}^{\gamma }[{f}_{1}(t)]+{C}_{2}{D}_{t}^{\gamma }[{f}_{2}(t)]$, | |
| (c)Leibnitz rule: ${D}_{t}^{\gamma }[{f}_{1}(t)\cdot {f}_{2}(t)]={\sum }_{i\,=\,0}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right){D}_{t}^{\gamma -i}[{f}_{1}(t)]\cdot {f}_{2}^{(i)}(t)$, where $\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right)=\tfrac{{\left(-1\right)}^{i-1}\gamma {\rm{\Gamma }}(i-\gamma )}{{\rm{\Gamma }}(1-\gamma ){\rm{\Gamma }}(i+1)}$. | |
| (d)${D}_{t}^{-\gamma }[{D}_{t}^{\gamma }f(t)]=f(t)-{\sum }_{i=1}^{n}{\left[{D}_{t}^{\gamma -i}f(t)\right]}_{t=a}\displaystyle \frac{{\left(t-a\right)}^{\gamma -i}}{{\rm{\Gamma }}(\gamma -i+1)}$, |
3. Lie point symmetries of the time-fractional DS equation
From equation (2 ), it can be seen that the time-fractional DS equation is a complex coupled system. To obtain the Lie point symmetries of the system, equation (2 ) needs to be transformed into a real coupled system.
First, we substitute the transformation u(x, y, t) = p(x, y, t) + iq(x, y, t) into equation (2 ). Consequently, equation (2 ) is transformed to the system as follows:
$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Delta }}}_{1}={D}_{t}^{\gamma }p+{q}_{{xx}}+{q}_{{yy}}-({p}^{2}q+{q}^{3})+{qv}=0,\\ {{\rm{\Delta }}}_{2}=-{D}_{t}^{\gamma }q+{p}_{{xx}}+{p}_{{yy}}-({p}^{3}+{{pq}}^{2})+{pv}=0,\\ {{\rm{\Delta }}}_{3}={v}_{{xx}}+{v}_{{yy}}+2({p}_{x}^{2}+{{pp}}_{{xx}}+{q}_{x}^{2}+{{qq}}_{{xx}})=0,\end{array}\right.\end{eqnarray}$
where p(x, y, t) and q(x, y, t) are real functions.The general form of equation (5 ) (system) is in the following:
$\begin{eqnarray}\left\{\begin{array}{l}{D}_{t}^{\gamma }p={F}_{1}(x,y,t,p,q,v,{p}_{x},{q}_{x},{v}_{x},{p}_{y},{q}_{y},{v}_{y},{p}_{{xx}},{q}_{{xx}},{v}_{{xx}},{p}_{{yy}},{q}_{{yy}},{v}_{{yy}},\cdot \cdot \cdot ),\\ {D}_{t}^{\gamma }q={F}_{2}(x,y,t,p,q,v,{p}_{x},{q}_{x},{v}_{x},{p}_{y},{q}_{y},{v}_{y},{p}_{{xx}},{q}_{{xx}},{v}_{{xx}},{p}_{{yy}},{q}_{{yy}},{v}_{{yy}},\cdot \cdot \cdot ),\\ {F}_{3}(x,y,t,p,q,v,{p}_{x},{q}_{x},{v}_{x},{p}_{y},{q}_{y},{v}_{y},{p}_{{xx}},{q}_{{xx}},{v}_{{xx}},{p}_{{yy}},{q}_{{yy}},{v}_{{yy}},\cdot \cdot \cdot )=0,\end{array}\right.\end{eqnarray}$
where 0 < γ < 1.The one-parameter Lie group of transformations for equation (5 ) has the following form:
$\begin{eqnarray}\left\{\begin{array}{l}{x}^{* }=x+{\varepsilon }{\lambda }_{1}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {y}^{* }=y+{\varepsilon }{\lambda }_{2}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {t}^{* }=t+{\varepsilon }\tau (x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {p}^{* }=p+{\varepsilon }{\phi }_{1}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {q}^{* }=q+{\varepsilon }{\phi }_{2}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {v}^{* }=v+{\varepsilon }{\phi }_{3}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {D}_{{t}^{* }}^{\gamma }{p}^{* }={D}_{t}^{\gamma }p+{\varepsilon }{\phi }_{1}^{\gamma }(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {D}_{{t}^{* }}^{\gamma }{q}^{* }={D}_{t}^{\gamma }q+{\varepsilon }{\phi }_{2}^{\gamma }(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {p}_{{x}^{* }}^{* }\,=\,{p}_{x}+{\varepsilon }{\phi }_{1}^{x}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {q}_{{x}^{* }}^{* }={q}_{x}+{\varepsilon }{\phi }_{2}^{x}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {v}_{{x}^{* }}^{* }={v}_{x}+{\varepsilon }{\phi }_{3}^{x}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {p}_{{y}^{* }}^{* }={p}_{y}+{\varepsilon }{\phi }_{1}^{y}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {q}_{{y}^{* }}^{* }={q}_{y}+{\varepsilon }{\phi }_{2}^{y}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {v}_{{y}^{* }}^{* }={v}_{y}+{\varepsilon }{\phi }_{3}^{y}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {p}_{{x}^{* }{x}^{* }}^{* }\,=\,{p}_{{xx}}+{\varepsilon }{\phi }_{1}^{{xx}}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {q}_{{x}^{* }{x}^{* }}^{* }={q}_{{xx}}+{\varepsilon }{\phi }_{2}^{{xx}}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {v}_{{x}^{* }{x}^{* }}^{* }={v}_{{xx}}+{\varepsilon }{\phi }_{3}^{{xx}}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {p}_{{y}^{* }{y}^{* }}^{* }\,=\,{p}_{{yy}}+{\varepsilon }{\phi }_{1}^{{yy}}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {q}_{{y}^{* }{y}^{* }}^{* }={q}_{{yy}}+{\varepsilon }{\phi }_{2}^{{yy}}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\\ {v}_{{y}^{* }{y}^{* }}^{* }={v}_{{yy}}+{\varepsilon }{\phi }_{3}^{{yy}}(x,y,t,p,q,v)+O({{\varepsilon }}^{2}),\end{array}\right.\end{eqnarray}$
where ε ≪ 1 is a parameter; λ1, λ2, λ3, $\tau$, φ1, φ2 and φ3 are infinitesimal functions; and ${\phi }_{1}^{\gamma },{\phi }_{2}^{\gamma },{\phi }_{1}^{x},{\phi }_{2}^{x},{\phi }_{3}^{x},{\phi }_{1}^{y},{\phi }_{2}^{y},{\phi }_{3}^{y},{\phi }_{1}^{{xx}},{\phi }_{2}^{{xx}},{\phi }_{3}^{{xx}},{\phi }_{1}^{{yy}},{\phi }_{2}^{{yy}}$ and ${\phi }_{3}^{{yy}}$ can be determined later.Taking vector field as follows:13 ) can be rewritten as15 ) into equation (14 ), one achieves12 ) and (16 ) into equation (11 ) and equating the coefficients of the partial derivatives and powers of the functions p, q and v with respect to the independent variables x, y and t be zero, then a system of determining equations is derived. Solving the determining equations, one has
$\begin{eqnarray}\begin{array}{rcl}V & = & {\lambda }_{1}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial x}+{\lambda }_{2}(x,y,t,p,q,v)\\ & & \times \displaystyle \frac{\partial }{\partial y}+\tau (x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial t}\\ & & +{\phi }_{1}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial p}+{\phi }_{2}(x,y,t,p,q,v)\\ & & \times \displaystyle \frac{\partial }{\partial q}+{\phi }_{3}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial v},\end{array}\end{eqnarray}$
therefore, the second prolongation of vector field V can be $\begin{eqnarray}\begin{array}{l}{{pr}}^{(\gamma ,2)}V={\lambda }_{1}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial x}+{\lambda }_{2}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial y}\\ \quad +\ \tau (x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial t}\\ \quad +{\phi }_{1}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial p}+{\phi }_{2}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial q}\\ \quad +{\phi }_{3}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial v}\\ \quad +{\phi }_{1}^{\gamma }(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {p}_{t}^{\gamma }}+{\phi }_{2}^{\gamma }(x,y,t,p,q,v)\\ \quad \times \displaystyle \frac{\partial }{\partial {q}_{t}^{\gamma }}\\ \quad +{\phi }_{1}^{x}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {p}_{x}}\\ \quad +{\phi }_{2}^{x}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {q}_{x}}\\ \quad +{\phi }_{1}^{{xx}}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {p}_{{xx}}}+{\phi }_{2}^{{xx}}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {q}_{{xx}}}\\ \quad +{\phi }_{3}^{{xx}}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {v}_{{xx}}}\\ \quad +{\phi }_{1}^{{yy}}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {p}_{{yy}}}+{\phi }_{2}^{{yy}}(x,y,t,p,q,v)\\ \quad \times \displaystyle \frac{\partial }{\partial {q}_{{yy}}}+{\phi }_{3}^{{yy}}(x,y,t,p,q,v)\displaystyle \frac{\partial }{\partial {v}_{{yy}}},\end{array}\end{eqnarray}$
where ${p}_{t}^{\gamma }={D}_{t}^{\gamma }p,{q}_{t}^{\gamma }={D}_{t}^{\gamma }q$, and the subscripts are denoted as partial derivatives. According to the Lie symmetry theory, pr(γ,2) should satisfy the invariance criterion, i.e. $\begin{eqnarray}\left\{\begin{array}{l}{{pr}}^{(\gamma ,2)}V({{\rm{\Delta }}}_{1}){| }_{{{\rm{\Delta }}}_{1}=0,{{\rm{\Delta }}}_{2}=0,{{\rm{\Delta }}}_{3}=0},\\ {{pr}}^{(\gamma ,2)}V({{\rm{\Delta }}}_{2}){| }_{{{\rm{\Delta }}}_{1}=0,{{\rm{\Delta }}}_{2}=0,{{\rm{\Delta }}}_{3}=0},\\ {{pr}}^{(\gamma ,2)}V({{\rm{\Delta }}}_{3}){| }_{{{\rm{\Delta }}}_{1}=0,{{\rm{\Delta }}}_{2}=0,{{\rm{\Delta }}}_{3}=0}.\end{array}\right.\end{eqnarray}$
Acting pr(γ,2)V on Δ1, Δ2 and Δ3, respectively, and acquiring $\begin{eqnarray}\left\{\begin{array}{l}{\phi }_{1}^{\gamma }+{\phi }_{2}^{{xx}}+{\phi }_{2}^{{yy}}-(2{pq}{\phi }_{1}+{p}^{2}{\phi }_{2}+3{q}^{2}{\phi }_{2})+v{\phi }_{2}+q{\phi }_{3}=0,\\ -{\phi }_{2}^{\gamma }+{\phi }_{1}^{{xx}}+{\phi }_{1}^{{yy}}-(3{p}^{2}{\phi }_{1}+{q}^{2}{\phi }_{1}+2{pq}{\phi }_{2})+v{\phi }_{1}+p{\phi }_{3}=0,\\ {\phi }_{3}^{{xx}}+{\phi }_{3}^{{yy}}+2({p}_{{xx}}{\phi }_{1}+2{p}_{x}{\phi }_{1}^{x}+p{\phi }_{1}^{{xx}}+{q}_{{xx}}{\phi }_{2}+2{q}_{x}{\phi }_{2}^{x}+q{\phi }_{2}^{{xx}})=0,\end{array}\right.\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{\phi }_{1}^{x} & = & {D}_{x}({\phi }_{1}-{\lambda }_{1}{p}_{x}-{\lambda }_{2}{p}_{y}-\tau {p}_{t})\\ & & +{\lambda }_{1}{p}_{{xx}}+{\lambda }_{2}{p}_{{xy}}+\tau {p}_{{xt}}\\ & = & {D}_{x}({\phi }_{1})-{p}_{x}{D}_{x}({\lambda }_{1})-{p}_{y}{D}_{x}({\lambda }_{2})-{p}_{t}{D}_{x}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{2}^{x} & = & {D}_{x}({\phi }_{2}-{\lambda }_{1}{q}_{x}-{\lambda }_{2}{q}_{y}-\tau {q}_{t})\\ & & +{\lambda }_{1}{q}_{{xx}}+{\lambda }_{2}{q}_{{xy}}+\tau {q}_{{xt}}\\ & = & {D}_{x}({\phi }_{2})-{q}_{x}{D}_{x}({\lambda }_{1})-{q}_{y}{D}_{x}({\lambda }_{2})-{q}_{t}{D}_{x}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{3}^{x} & = & {D}_{x}({\phi }_{3}-{\lambda }_{1}{v}_{x}-{\lambda }_{2}{v}_{y}-\tau {v}_{t})\\ & & +{\lambda }_{1}{v}_{{xx}}+{\lambda }_{2}{v}_{{xy}}+\tau {v}_{{xt}}\\ & = & {D}_{x}({\phi }_{3})-{v}_{x}{D}_{x}({\lambda }_{1})-{v}_{y}{D}_{x}({\lambda }_{2})-{v}_{t}{D}_{x}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{1}^{y} & = & {D}_{y}({\phi }_{1}-{\lambda }_{1}{p}_{x}-{\lambda }_{2}{p}_{y}-\tau {p}_{t})\\ & & +{\lambda }_{1}{p}_{{xy}}+{\lambda }_{2}{p}_{{yy}}+\tau {p}_{{yt}}\\ & = & {D}_{y}({\phi }_{1})-{p}_{x}{D}_{y}({\lambda }_{1})-{p}_{y}{D}_{y}({\lambda }_{2})-{p}_{t}{D}_{y}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{2}^{y} & = & {D}_{y}({\phi }_{2}-{\lambda }_{1}{q}_{x}-{\lambda }_{2}{q}_{y}-\tau {q}_{t})+{\lambda }_{1}{q}_{{xy}}+{\lambda }_{2}{q}_{{yy}}+\tau {q}_{{yt}}\\ & = & {D}_{y}({\phi }_{2})-{q}_{x}{D}_{y}({\lambda }_{1})-{q}_{y}{D}_{y}({\lambda }_{2})-{q}_{t}{D}_{y}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{3}^{y} & = & {D}_{y}({\phi }_{3}-{\lambda }_{1}{v}_{x}-{\lambda }_{2}{v}_{y}-\tau {v}_{t})\\ & & +{\lambda }_{1}{v}_{{xy}}+{\lambda }_{2}{v}_{{yy}}+\tau {v}_{{yt}}\\ & = & {D}_{y}({\phi }_{3})-{p}_{x}{D}_{y}({\lambda }_{1})-{p}_{y}{D}_{y}({\lambda }_{2})-{p}_{t}{D}_{y}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{1}^{{xx}} & = & {D}_{x}^{2}({\phi }_{1}-{\lambda }_{1}{p}_{x}-{\lambda }_{2}{p}_{y}-\tau {p}_{t})+{\lambda }_{1}{p}_{{xxx}}\\ & & +{\lambda }_{2}{p}_{{xxy}}+\tau {p}_{{xxt}}\\ & = & {D}_{x}({\phi }_{1}^{x})-{p}_{{xx}}{D}_{x}({\lambda }_{1})-{p}_{{xy}}{D}_{x}({\lambda }_{2})-{p}_{{xt}}{D}_{x}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{2}^{{xx}} & = & {D}_{x}^{2}({\phi }_{2}-{\lambda }_{1}{q}_{x}-{\lambda }_{2}{q}_{y}-\tau {q}_{t})+{\lambda }_{1}{q}_{{xxx}}\\ & & +{\lambda }_{2}{q}_{{xxy}}+\tau {q}_{{xxt}}\\ & = & {D}_{x}({\phi }_{2}^{x})-{q}_{{xx}}{D}_{x}({\lambda }_{1})-{q}_{{xy}}{D}_{x}({\lambda }_{2})-{q}_{{xt}}{D}_{x}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{3}^{{xx}} & = & {D}_{x}^{2}({\phi }_{3}-{\lambda }_{1}{v}_{x}-{\lambda }_{2}{v}_{y}-\tau {v}_{t})+{\lambda }_{1}{v}_{{xxx}}\\ & & +{\lambda }_{2}{v}_{{xxy}}+\tau {v}_{{xxt}}\\ & = & {D}_{x}({\phi }_{3}^{x})-{v}_{{xx}}{D}_{x}({\lambda }_{1})-{v}_{{xy}}{D}_{x}({\lambda }_{2})-{v}_{{xt}}{D}_{x}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{1}^{{yy}} & = & {D}_{y}^{2}({\phi }_{1}-{\lambda }_{1}{p}_{x}-{\lambda }_{2}{p}_{y}-\tau {p}_{t})+{\lambda }_{1}{p}_{{xyy}}\\ & & +{\lambda }_{2}{p}_{{yyy}}+\tau {p}_{{yyt}}\\ & = & {D}_{y}({\phi }_{1}^{y})-{p}_{{xy}}{D}_{y}({\lambda }_{1})-{p}_{{yy}}{D}_{y}({\lambda }_{2})-{p}_{{yt}}{D}_{y}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{2}^{{yy}} & = & {D}_{y}^{2}({\phi }_{2}-{\lambda }_{1}{q}_{x}-{\lambda }_{2}{q}_{y}-\tau {q}_{t})+{\lambda }_{1}{q}_{{xyy}}\\ & & +{\lambda }_{2}{q}_{{yyy}}+\tau {q}_{{yyt}}\\ & = & {D}_{y}({\phi }_{2}^{y})-{q}_{{xy}}{D}_{y}({\lambda }_{1})-{q}_{{yy}}{D}_{y}({\lambda }_{2})-{q}_{{yt}}{D}_{y}(\tau )\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{3}^{{yy}} & = & {D}_{y}^{2}({\phi }_{3}-{\lambda }_{1}{v}_{x}-{\lambda }_{2}{v}_{y}-\tau {v}_{t})+{\lambda }_{1}{v}_{{xyy}}\\ & & +{\lambda }_{2}{v}_{{yyy}}+\tau {v}_{{yyt}}\\ & = & {D}_{y}({\phi }_{3}^{y})-{v}_{{xy}}{D}_{y}({\lambda }_{1})-{v}_{{yy}}{D}_{y}({\lambda }_{2})-{v}_{{yt}}{D}_{y}(\tau )\end{array}\end{eqnarray}$
and Dx, Dy are denoted as the total derivatives with respect to x, y, respectively. The infinitesimal functions ${\phi }_{1}^{\gamma }$ and ${\phi }_{2}^{\gamma }$ can be represented as $\begin{eqnarray}\begin{array}{rcl}{\phi }_{1}^{\gamma } & = & {D}_{t}^{\gamma }({\phi }_{1})+{\lambda }_{1}{D}_{t}^{\gamma }({p}_{x})-{D}_{t}^{\gamma }({\lambda }_{1}{p}_{x})\\ & & +{\lambda }_{2}{D}_{t}^{\gamma }({p}_{y})-{D}_{t}^{\gamma }({\lambda }_{2}{p}_{y})\\ & & +{D}_{t}^{\gamma }({D}_{t}(\tau )p)-{D}_{t}^{\gamma +1}(\tau p)+\tau {D}_{t}^{\gamma +1}(p),\\ {\phi }_{2}^{\gamma } & = & {D}_{t}^{\gamma }({\phi }_{2})+{\lambda }_{1}{D}_{t}^{\gamma }({q}_{x})-{D}_{t}^{\gamma }({\lambda }_{1}{q}_{x})\\ & & +{\lambda }_{2}{D}_{t}^{\gamma }({q}_{y})-{D}_{t}^{\gamma }({\lambda }_{2}{q}_{y})\\ & & +{D}_{t}^{\gamma }({D}_{t}(\tau )q)-{D}_{t}^{\gamma +1}(\tau q)+\tau {D}_{t}^{\gamma +1}(q).\end{array}\end{eqnarray}$
By utilizing the Leibnitz rule of Riemann-Liouville derivative, equation ( $\begin{eqnarray}\begin{array}{rcl}{\phi }_{1}^{\gamma } & = & {D}_{t}^{\gamma }({\phi }_{1})-\gamma {D}_{t}(\tau )\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{1}){D}_{t}^{\gamma -n}({p}_{x})\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{2}){D}_{t}^{\gamma -n}({p}_{y})\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n+1}\right){D}_{t}^{n+1}(\tau ){D}_{t}^{\gamma -n}(p),\\ {\phi }_{2}^{\gamma } & = & {D}_{t}^{\gamma }({\phi }_{2})-\gamma {D}_{t}(\tau )\displaystyle \frac{{\partial }^{\gamma }q}{\partial {t}^{\gamma }}\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{1}){D}_{t}^{\gamma -n}({q}_{x})\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{2}){D}_{t}^{\gamma -n}({q}_{y})\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n+1}\right){D}_{t}^{n+1}(\tau ){D}_{t}^{\gamma -n}(q).\end{array}\end{eqnarray}$
According to the derivation rule for composite function, for functions f = f(x) and f = g(x), one obtains $\begin{eqnarray*}\displaystyle \frac{{{\rm{d}}}^{n}f[g(x)]}{{\rm{d}}{t}^{n}}=\displaystyle \displaystyle \sum _{i=0}^{n}\displaystyle \displaystyle \sum _{j=0}^{i}\left(\displaystyle \genfrac{}{}{0em}{}{i}{j}\right)\displaystyle \frac{1}{i!}{\left[-g(x)\right]}^{j}\displaystyle \frac{{{\rm{d}}}^{n}}{{\rm{d}}{t}^{n}}[{\left(g(x)\right)}^{i-j}]\displaystyle \frac{{{\rm{d}}}^{i}f(g)}{{\rm{d}}{g}^{i}}.\end{eqnarray*}$
By utilizing the preceding results, one acquires $\begin{eqnarray}\begin{array}{l}{D}_{t}^{\gamma }({\phi }_{1})=\displaystyle \frac{{\partial }^{\gamma }{\phi }_{1}}{\partial {t}^{\gamma }}+{\phi }_{1,p}\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}-p\displaystyle \frac{{\partial }^{\gamma }{\phi }_{1,p}}{\partial {t}^{\gamma }}+{\phi }_{1,q}\displaystyle \frac{{\partial }^{\gamma }q}{\partial {t}^{\gamma }}\\ \quad -q\displaystyle \frac{{\partial }^{\gamma }{\phi }_{1,q}}{\partial {t}^{\gamma }}+{\phi }_{1,v}\displaystyle \frac{{\partial }^{\gamma }v}{\partial {t}^{\gamma }}-v\displaystyle \frac{{\partial }^{\gamma }{\phi }_{1,v}}{\partial {t}^{\gamma }}\\ \quad +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{1,p}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(p)+\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{1,q}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(q)\\ \quad +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{1,v}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(v)\\ \quad +{\sigma }_{1}+{\sigma }_{2}+{\sigma }_{3},\\ {D}_{t}^{\gamma }({\phi }_{2})=\displaystyle \frac{{\partial }^{\gamma }{\phi }_{2}}{\partial {t}^{\gamma }}+{\phi }_{2,p}\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}-p\displaystyle \frac{{\partial }^{\gamma }{\phi }_{2,p}}{\partial {t}^{\gamma }}+{\phi }_{2,q}\displaystyle \frac{{\partial }^{\gamma }q}{\partial {t}^{\gamma }}\\ \quad -q\displaystyle \frac{{\partial }^{\gamma }{\phi }_{2,q}}{\partial {t}^{\gamma }}+{\phi }_{2,v}\displaystyle \frac{{\partial }^{\gamma }v}{\partial {t}^{\gamma }}-v\displaystyle \frac{{\partial }^{\gamma }{\phi }_{2,v}}{\partial {t}^{\gamma }}\\ \quad +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{2,p}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(p)+\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{2,q}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(q)\\ \quad +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{2,v}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(v)\\ \quad +{\sigma }_{4}+{\sigma }_{5}+{\sigma }_{6},\end{array}\end{eqnarray}$
Where $\begin{eqnarray*}\begin{array}{rcl}{\sigma }_{1} & = & \displaystyle \displaystyle \sum _{i=2}^{\infty }\displaystyle \displaystyle \sum _{j=2}^{i}\displaystyle \displaystyle \sum _{k=2}^{j}\displaystyle \displaystyle \sum _{m=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right)\left(\displaystyle \genfrac{}{}{0em}{}{i}{j}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{m}\right)\displaystyle \frac{1}{k!}\displaystyle \frac{{t}^{i-\gamma }}{{\rm{\Gamma }}(1+i-\gamma )}{\left(-p\right)}^{m}\\ & & \times \displaystyle \frac{{\partial }^{j}}{\partial {t}^{j}}({p}^{k-m})\displaystyle \frac{{\partial }^{i-j+k}{\phi }_{1}}{\partial {t}^{i-j}\partial {p}^{k}},\\ {\sigma }_{2} & = & \displaystyle \displaystyle \sum _{i=2}^{\infty }\displaystyle \displaystyle \sum _{j=2}^{i}\displaystyle \displaystyle \sum _{k=2}^{j}\displaystyle \displaystyle \sum _{m=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right)\left(\displaystyle \genfrac{}{}{0em}{}{i}{j}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{m}\right)\displaystyle \frac{1}{k!}\displaystyle \frac{{t}^{i-\gamma }}{{\rm{\Gamma }}(1+i-\gamma )}{\left(-q\right)}^{m}\\ & & \times \displaystyle \frac{{\partial }^{j}}{\partial {t}^{j}}({q}^{k-m})\displaystyle \frac{{\partial }^{i-j+k}{\phi }_{1}}{\partial {t}^{i-j}\partial {q}^{k}},\\ {\sigma }_{3} & = & \displaystyle \displaystyle \sum _{i=2}^{\infty }\displaystyle \displaystyle \sum _{j=2}^{i}\displaystyle \displaystyle \sum _{k=2}^{j}\displaystyle \displaystyle \sum _{m=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right)\left(\displaystyle \genfrac{}{}{0em}{}{i}{j}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{m}\right)\displaystyle \frac{1}{k!}\displaystyle \frac{{t}^{i-\gamma }}{{\rm{\Gamma }}(1+i-\gamma )}{\left(-v\right)}^{m}\\ & & \times \displaystyle \frac{{\partial }^{j}}{\partial {t}^{j}}({v}^{k-m})\displaystyle \frac{{\partial }^{i-j+k}{\phi }_{1}}{\partial {t}^{i-j}\partial {v}^{k}},\\ {\sigma }_{4} & = & \displaystyle \displaystyle \sum _{i=2}^{\infty }\displaystyle \displaystyle \sum _{j=2}^{i}\displaystyle \displaystyle \sum _{k=2}^{j}\displaystyle \displaystyle \sum _{m=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right)\left(\displaystyle \genfrac{}{}{0em}{}{i}{j}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{m}\right)\displaystyle \frac{1}{k!}\displaystyle \frac{{t}^{i-\gamma }}{{\rm{\Gamma }}(1+i-\gamma )}{\left(-p\right)}^{m}\\ & & \times \displaystyle \frac{{\partial }^{j}}{\partial {t}^{j}}({p}^{k-m})\displaystyle \frac{{\partial }^{i-j+k}{\phi }_{2}}{\partial {t}^{i-j}\partial {p}^{k}},\\ {\sigma }_{5} & = & \displaystyle \displaystyle \sum _{i=2}^{\infty }\displaystyle \displaystyle \sum _{j=2}^{i}\displaystyle \displaystyle \sum _{k=2}^{j}\displaystyle \displaystyle \sum _{m=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right)\left(\displaystyle \genfrac{}{}{0em}{}{i}{j}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{m}\right)\displaystyle \frac{1}{k!}\displaystyle \frac{{t}^{i-\gamma }}{{\rm{\Gamma }}(1+i-\gamma )}{\left(-q\right)}^{m}\\ & & \times \displaystyle \frac{{\partial }^{j}}{\partial {t}^{j}}({q}^{k-m})\displaystyle \frac{{\partial }^{i-j+k}{\phi }_{2}}{\partial {t}^{i-j}\partial {q}^{k}},\\ {\sigma }_{6} & = & \displaystyle \displaystyle \sum _{i=2}^{\infty }\displaystyle \displaystyle \sum _{j=2}^{i}\displaystyle \displaystyle \sum _{k=2}^{j}\displaystyle \displaystyle \sum _{m=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{i}\right)\left(\displaystyle \genfrac{}{}{0em}{}{i}{j}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{m}\right)\displaystyle \frac{1}{k!}\displaystyle \frac{{t}^{i-\gamma }}{{\rm{\Gamma }}(1+i-\gamma )}{\left(-v\right)}^{m}\\ & & \times \displaystyle \frac{{\partial }^{j}}{\partial {t}^{j}}({v}^{k-m})\displaystyle \frac{{\partial }^{i-j+k}{\phi }_{2}}{\partial {t}^{i-j}\partial {v}^{k}}.\end{array}\end{eqnarray*}$
Substituting equation ( $\begin{eqnarray}\begin{array}{rcl}{\phi }_{1}^{\gamma } & = & \displaystyle \frac{{\partial }^{\gamma }{\phi }_{1}}{\partial {t}^{\gamma }}+[{\phi }_{1,p}-\gamma {D}_{t}(\tau )]\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}-p\displaystyle \frac{{\partial }^{\gamma }{\phi }_{1,p}}{\partial {t}^{\gamma }}\\ & & +{\phi }_{1,q}\displaystyle \frac{{\partial }^{\gamma }q}{\partial {t}^{\gamma }}-q\displaystyle \frac{{\partial }^{\gamma }{\phi }_{1,q}}{\partial {t}^{\gamma }}+{\phi }_{1,v}\displaystyle \frac{{\partial }^{\gamma }v}{\partial {t}^{\gamma }}-v\displaystyle \frac{{\partial }^{\gamma }{\phi }_{1,v}}{\partial {t}^{\gamma }}\\ & & +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left[\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{1,p}}{\partial {t}^{n}}-\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n+1}\right){D}_{t}^{n+1}(\tau )\right]{D}_{t}^{\gamma -n}(p)\\ & & +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{1,q}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(q)+\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{1,v}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(v)\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{1}){D}_{t}^{\gamma -n}({p}_{x})\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{2}){D}_{t}^{\gamma -n}({p}_{y})\\ & & +{\sigma }_{1}+{\sigma }_{2}+{\sigma }_{3},\\ {\phi }_{2}^{\gamma } & = & \displaystyle \frac{{\partial }^{\gamma }{\phi }_{2}}{\partial {t}^{\gamma }}+{\phi }_{2,p}\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}-p\displaystyle \frac{{\partial }^{\gamma }{\phi }_{2,p}}{\partial {t}^{\gamma }}\\ & & +[{\phi }_{2,q}-\gamma {D}_{t}(\tau )]\displaystyle \frac{{\partial }^{\gamma }q}{\partial {t}^{\gamma }}-q\displaystyle \frac{{\partial }^{\gamma }{\phi }_{2,q}}{\partial {t}^{\gamma }}+{\phi }_{2,v}\displaystyle \frac{{\partial }^{\gamma }v}{\partial {t}^{\gamma }}\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl} & & -v\displaystyle \frac{{\partial }^{\gamma }{\phi }_{2,v}}{\partial {t}^{\gamma }}\\ & & +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{2,p}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(p)+\displaystyle \displaystyle \sum _{n=1}^{\infty }\left[\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{2,q}}{\partial {t}^{n}}\right.\\ & & \left.-\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n+1}\right){D}_{t}^{n+1}(\tau )\right]{D}_{t}^{\gamma -n}(q)\\ & & +\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right)\displaystyle \frac{{\partial }^{n}{\phi }_{2,v}}{\partial {t}^{n}}{D}_{t}^{\gamma -n}(v)-\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{1}){D}_{t}^{\gamma -n}({q}_{x})\\ & & -\displaystyle \displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\gamma }{n}\right){D}_{t}^{n}({\lambda }_{2}){D}_{t}^{\gamma -n}({q}_{y})\\ & & +{\sigma }_{4}+{\sigma }_{5}+{\sigma }_{6}.\end{array}\end{eqnarray}$
Substituting equations ( $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{1}(x,y,t,p,q,v)=\gamma {C}_{1}x+{C}_{2},\\ {\lambda }_{2}(x,y,t,p,q,v)=\gamma {C}_{1}y+{C}_{3},\\ \tau (x,y,t,p,q,v)=2{C}_{1}t,\\ {\phi }_{1}(x,y,t,p,q,v)=-\gamma {C}_{1}p+{C}_{4}q,\\ {\phi }_{2}(x,y,t,p,q,v)=-\gamma {C}_{1}q-{C}_{4}p,\\ {\phi }_{3}(x,y,t,p,q,v)=-2\gamma {C}_{1}v,\end{array}\right.\end{eqnarray}$
where C1, C2, C3 and C4 are arbitrary constants. Therefore, the vector field V can be presented as $\begin{eqnarray}\begin{array}{rcl}V & = & (\gamma {C}_{1}x+{C}_{2})\displaystyle \frac{\partial }{\partial x}+(\gamma {C}_{1}y+{C}_{3})\displaystyle \frac{\partial }{\partial y}+(2{C}_{1}t)\displaystyle \frac{\partial }{\partial t}\\ & & +(-\gamma {C}_{1}p+{C}_{4}q)\displaystyle \frac{\partial }{\partial p}+(-\gamma {C}_{1}q-{C}_{4}p)\displaystyle \frac{\partial }{\partial q}\\ & & +(-2\gamma {C}_{1}v)\displaystyle \frac{\partial }{\partial v}.\end{array}\end{eqnarray}$
Taking $\begin{eqnarray}\begin{array}{rcl}{V}_{1} & = & \gamma x\displaystyle \frac{\partial }{\partial x}+\gamma y\displaystyle \frac{\partial }{\partial y}+2t\displaystyle \frac{\partial }{\partial t}-\gamma p\displaystyle \frac{\partial }{\partial p}-\gamma q\displaystyle \frac{\partial }{\partial q}-2\gamma v\displaystyle \frac{\partial }{\partial v},\\ {V}_{2} & = & \displaystyle \frac{\partial }{\partial x},\\ {V}_{3} & = & \displaystyle \frac{\partial }{\partial x},\\ {V}_{4} & = & q\displaystyle \frac{\partial }{\partial p}-p\displaystyle \frac{\partial }{\partial q},\end{array}\end{eqnarray}$
then the vector field V can be written as $\begin{eqnarray}V={C}_{1}{V}_{1}+{C}_{2}{V}_{2}+{C}_{3}{V}_{3}+{C}_{4}{V}_{4},\end{eqnarray}$
and from that the following theorem is presented.The Lie algebra ${ \mathcal A }$, which is spanned by ${V}_{1},{V}_{2},{V}_{3}$ and V4, has the following commutation relations (table 1).
Table 1. Commutation relations of the Lie algebra ${ \mathcal A }$. |
| [Vi, Vj] | V1 | V2 | V3 | V4 |
|---|---|---|---|---|
| V1 | 0 | −γV2 | −γV3 | 0 |
| V2 | γV2 | 0 | 0 | 0 |
| V3 | γV3 | 0 | 0 | 0 |
| V4 | 0 | 0 | 0 | 0 |
Where [Vi, Vj] = ViVj − VjVi. It is obvious to see that the Lie algebra ${ \mathcal A }$ is closed. Consequently, the Lie point symmetries of equation (5 ) are generated by vector field V.
The symmetry groups can also be derived. The following theorem shows the result. 5 ). 22 ) results in
If $[p=p(x,y,t),q=q(x,y,t),v=v(x,y,t)]$ is a solution of equation (
$\begin{eqnarray*}\begin{array}{l}\left[{p}_{1},{q}_{1},{v}_{1}\right]\\ \quad =\left[{{\rm{e}}}^{-\gamma {\epsilon }}p({{\rm{e}}}^{-\gamma {\epsilon }}x,{{\rm{e}}}^{-\gamma {\epsilon }}y,{{\rm{e}}}^{-2{\epsilon }}t),{{\rm{e}}}^{-\gamma {\epsilon }}q\right.\\ \quad \times ({{\rm{e}}}^{-\gamma {\epsilon }}x,{{\rm{e}}}^{-\gamma {\epsilon }}y,{{\rm{e}}}^{-2{\epsilon }}t),\\ \quad \left.{{\rm{e}}}^{-2\gamma {\epsilon }}v({{\rm{e}}}^{-\gamma {\epsilon }}x,{{\rm{e}}}^{-\gamma {\epsilon }}y,{{\rm{e}}}^{-2{\epsilon }}t)\right],\\ \quad \left[{p}_{2},{q}_{2},{v}_{2}\right]\\ \quad =\left[p(x-{\epsilon },y,t),q(x-{\epsilon },y,t),v(x-{\epsilon },y,t)\right],\\ \quad \left[{p}_{3},{q}_{3},{v}_{3}\right]\\ \quad =\left[p(x,y-{\epsilon },t),q(x,y-{\epsilon },t),v(x,y-{\epsilon },t)\right]\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}\left[{p}_{4},{q}_{4},{v}_{4}\right]\\ \quad =\left[p(x,y,t)\cos {\epsilon }-q(x,y,t)\sin {\epsilon },p(x,y,t)\sin {\epsilon }\right.\\ \quad \left.+q(x,y,t)\cos {\epsilon },v(x,y,t)\right]\end{array}\end{eqnarray*}$
are also solutions of equation (Assuming symmetry group Gi satisfies:
$\begin{eqnarray}\begin{array}{l}{G}_{i}:(x,y,t,p,q,v)\to ({x}^{* },{y}^{* },{t}^{* },{p}^{* },{q}^{* },{v}^{* })\\ \quad \times (i=1,2,3,4).\end{array}\end{eqnarray}$
Consider initial value problem, which satisfies the conditions as follows: $\begin{eqnarray}\left\{\begin{array}{l}\displaystyle \frac{d}{d{\epsilon }}({x}^{* },{y}^{* },{t}^{* },{p}^{* },{q}^{* },{v}^{* })=V({x}^{* },{y}^{* },{t}^{* },{p}^{* },{q}^{* },{v}^{* }),\\ ({x}^{* },{y}^{* },{t}^{* },{p}^{* },{q}^{* },{v}^{* }){| }_{{\epsilon }=0}=(x,y,t,p,q,v),\end{array}\right.\end{eqnarray}$
where ε is a parameter. Soving equation ( $\begin{eqnarray}\begin{array}{l}{G}_{1}:(x,y,t,p,q,v)\to ({{\rm{e}}}^{\gamma {\epsilon }}x,{{\rm{e}}}^{\gamma {\epsilon }}y,{{\rm{e}}}^{2{\epsilon }}t,\\ \quad {{\rm{e}}}^{-\gamma {\epsilon }}p,{{\rm{e}}}^{-\gamma {\epsilon }}q,{{\rm{e}}}^{-2\gamma {\epsilon }}v),\\ {G}_{2}:(x,y,t,p,q,v)\to (x+{\epsilon },y,t,p,q,v),\\ {G}_{3}:(x,y,t,p,q,v)\to (x,y+{\epsilon },t,p,q,v),\\ {G}_{4}:(x,y,t,p,q,v)\to (x,y,t,\\ \quad p\cos {\epsilon }+q\sin {\epsilon },p\sin {\epsilon }-q\cos {\epsilon },v).\end{array}\end{eqnarray}$
Therefore, the solutions $[{p}_{1},{q}_{1},{v}_{1}],[{p}_{2},{q}_{2},{v}_{2}],[{p}_{3},{q}_{3},{v}_{3}]$ and $[{p}_{4},{q}_{4},{v}_{4}]$ can be derived. 4. Similarity reductions of the time-fractional DS equation
In the preceding section, the Lie point symmetries are presented. With the help of these results, similarity reductions can be provided.
I: For vector field (symmetry) $V=k\tfrac{\partial }{\partial x}+l\tfrac{\partial }{\partial y}$, the characteristic equation is24 ) into equation (5 ), one has
$\begin{eqnarray*}\displaystyle \frac{{\rm{d}}x}{k}=\displaystyle \frac{{\rm{d}}y}{l}=\displaystyle \frac{{\rm{d}}t}{0}=\displaystyle \frac{{\rm{d}}p}{0}=\displaystyle \frac{{\rm{d}}q}{0}=\displaystyle \frac{{\rm{d}}v}{0}.\end{eqnarray*}$
Taking similarity variables ξ = − lx + ky, η = t, and the group-invariant solutions $\begin{eqnarray}\left\{\begin{array}{l}p=p(x,y,t)=f(\xi ,\eta ),\\ q=q(x,y,t)=g(\xi ,\eta ),\\ v=v(x,y,t)=h(\xi ,\eta ).\end{array}\right.\end{eqnarray}$
Substituting equation ( $\begin{eqnarray}\left\{\begin{array}{l}{D}_{\eta }^{\gamma }f+({k}^{2}+{l}^{2}){g}_{\xi \xi }-({f}^{2}g+{g}^{3})+{gh}=0,\\ -{D}_{\eta }^{\gamma }g+({k}^{2}+{l}^{2}){f}_{\xi \xi }-({f}^{3}+{{fg}}^{2})+{fh}=0,\\ \left({k}^{2}+{l}^{2}\right){h}_{\xi \xi }+2{l}^{2}\left[{\left({f}_{\xi }\right)}^{2}+{{ff}}_{\xi \xi }+{\left({g}_{\xi }\right)}^{2}+{{gg}}_{\xi \xi }\right]=0,\end{array}\right.\end{eqnarray}$
where the subscripts are denoted as the partial derivative with respect to ξ, i.e. ${\left(\cdot \right)}_{\xi }=\tfrac{\partial }{\partial \xi }(\cdot ),{\left(\cdot \right)}_{\xi \xi }=\tfrac{{\partial }^{2}}{\partial {\xi }^{2}}(\cdot )$.II: For vector field (symmetry) corresponding to scalar transformations, i.e. $V=\gamma x\tfrac{\partial }{\partial x}+\gamma y\tfrac{\partial }{\partial y}$ +$2t\tfrac{\partial }{\partial t}-\gamma p\tfrac{\partial }{\partial p}-\gamma q\tfrac{\partial }{\partial q}$ $-2\gamma v\tfrac{\partial }{\partial v}$, the characteristic equation is26 ) into equation (5 ), one has
$\begin{eqnarray*}\displaystyle \frac{{\rm{d}}x}{\gamma x}=\displaystyle \frac{{\rm{d}}y}{\gamma y}=\displaystyle \frac{{\rm{d}}t}{2t}=\displaystyle \frac{{\rm{d}}p}{-\gamma p}=\displaystyle \frac{{\rm{d}}q}{-\gamma q}=\displaystyle \frac{{\rm{d}}v}{-2\gamma v}.\end{eqnarray*}$
Selecting similarity variable $\eta =t{\left(x+y\right)}^{-\tfrac{2}{\gamma }}$ and the group-invariant solutions $\begin{eqnarray}\left\{\begin{array}{l}p=p(x,y,t)={\left(x+y\right)}^{-1}f(\eta ),\\ q=q(x,y,t)={\left(x+y\right)}^{-1}g(\eta ),\\ v=v(x,y,t)={\left(x+y\right)}^{-1}h(\eta ).\end{array}\right.\end{eqnarray}$
Substituting equation ( $\begin{eqnarray}\left\{\begin{array}{l}{D}_{\eta }^{\gamma }f+4g+(\displaystyle \frac{8}{{\gamma }^{2}}+\displaystyle \frac{12}{\gamma })\eta {g}^{{\prime} }+\displaystyle \frac{8}{{\gamma }^{2}}{\eta }^{2}{g}^{{\prime\prime} }-({f}^{2}g+{g}^{3})+{gh}=0,\\ -{D}_{\eta }^{\gamma }g+4f+(\displaystyle \frac{8}{{\gamma }^{2}}+\displaystyle \frac{12}{\gamma })\eta {f}^{{\prime} }+\displaystyle \frac{8}{{\gamma }^{2}}{\eta }^{2}{g}^{{\prime\prime} }-({f}^{3}+{{fg}}^{2})+{fh}=0,\\ 6h+3({f}^{2}+{g}^{2})+(\displaystyle \frac{4}{{\gamma }^{2}}+\displaystyle \frac{10}{\gamma })\eta ({{ff}}^{{\prime} }+{{gg}}^{{\prime} }+{h}^{{\prime} })\\ +\displaystyle \frac{4}{{\gamma }^{2}}{\eta }^{2}\left[{\left({f}^{{\prime} }\right)}^{2}+{{ff}}^{{\prime\prime} }+{\left({g}^{{\prime} }\right)}^{2}+{{gg}}^{{\prime\prime} }+{h}^{{\prime\prime} }\right]=0,\end{array}\right.\end{eqnarray}$
where ${\left(\cdot \right)}^{{\prime} },{\left(\cdot \right)}^{{\prime\prime} }$ are denoted as the derivatives with respect to η, i.e. ${\left(\cdot \right)}^{{\prime} }=\tfrac{{\rm{d}}}{{\rm{d}}\eta }(\cdot ),{\left(\cdot \right)}^{{\prime\prime} }=\tfrac{{{\rm{d}}}^{2}}{{\rm{d}}{\eta }^{2}}(\cdot )$.III: The Erdélyi-Kober fractional integro-differential operators are powerful techniques for reducing fractional partial differential equations. In this part, the Erdélyi-Kober fractional integro-differential operators are utilized to reduce equation (5 ).
For vector field (symmetry) $V=\gamma x\tfrac{\partial }{\partial x}+\gamma y\tfrac{\partial }{\partial y}\,+2t\tfrac{\partial }{\partial t}-\gamma p\tfrac{\partial }{\partial p}-\gamma q\tfrac{\partial }{\partial q}-2\gamma v\tfrac{\partial }{\partial v}$, the characteristic equation3 ) and (4 ), one has29 ) leads to31 ) and (32 ) into equation (5 ), one has
$\begin{eqnarray*}\displaystyle \frac{{\rm{d}}x}{\gamma x}=\displaystyle \frac{{\rm{d}}y}{\gamma y}=\displaystyle \frac{{\rm{d}}t}{2t}=\displaystyle \frac{{\rm{d}}p}{-\gamma p}=\displaystyle \frac{{\rm{d}}q}{-\gamma q}=\displaystyle \frac{{\rm{d}}v}{-2\gamma v}.\end{eqnarray*}$
Choosing similarity variable $\mu =(x+y){t}^{-\tfrac{\gamma }{2}}$ and the group-invariant solution $\begin{eqnarray}\left\{\begin{array}{l}p=p(x,y,t)={t}^{-\tfrac{\gamma }{2}}f(\mu ),\\ q=q(x,y,t)={t}^{-\tfrac{\gamma }{2}}g(\mu ),\\ v=v(x,y,t)={t}^{-\gamma }h(\mu ).\end{array}\right.\end{eqnarray}$
According to equations ( $\begin{eqnarray}\begin{array}{l}{D}_{t}^{\gamma }\left[p(x,y,t)\right]=\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}=\displaystyle \frac{{\partial }^{n}}{\partial {t}^{n}}\\ \quad \times \left\{\displaystyle \frac{1}{{\rm{\Gamma }}(n-\gamma )}{\int }_{0}^{t}\displaystyle \frac{1}{{\left(t-\tau \right)}^{1-n+\gamma }}{\tau }^{-\tfrac{\gamma }{2}}f[(x+y){\tau }^{-\tfrac{\gamma }{2}}]{\rm{d}}\tau \right\}.\end{array}\end{eqnarray}$
Choosing variable transformation $\begin{eqnarray*}\varsigma =\displaystyle \frac{t}{\tau },\end{eqnarray*}$
then $\begin{eqnarray*}{\rm{d}}\tau =-\displaystyle \frac{t}{{\varsigma }^{2}}{\rm{d}}\varsigma .\end{eqnarray*}$
Therefore, equation ( $\begin{eqnarray}\begin{array}{l}{D}_{t}^{\gamma }\left[p(x,\,y,\,t)\right]=\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}=\displaystyle \frac{{\partial }^{n}}{\partial {t}^{n}}\left\{{t}^{n-\tfrac{3}{2}\gamma }\displaystyle \frac{1}{{\rm{\Gamma }}(n-\gamma )}\right.\\ \quad \left.\times {\int }_{1}^{\infty }\displaystyle \frac{1}{{\left(\varsigma -1\right)}^{1-n+\gamma }}{\varsigma }^{-(n-\tfrac{3}{2}\gamma +1)}f({\varsigma }^{\tfrac{\gamma }{2}}\mu ){\rm{d}}\varsigma \right\}\\ =\ \displaystyle \frac{{\partial }^{n}}{\partial {t}^{n}}\left[{t}^{n-\tfrac{3}{2}\gamma }\left({{ \mathcal K }}_{\tfrac{2}{\gamma }}^{1-\tfrac{\gamma }{2},\,n-\gamma }f\right)(\mu )\right]\\ =\ \displaystyle \frac{{\partial }^{n-1}}{\partial {t}^{n-1}}\left\{\displaystyle \frac{\partial }{\partial t}\left[{t}^{n-\tfrac{3}{2}\gamma }\left({{ \mathcal K }}_{\tfrac{2}{\gamma }}^{1-\tfrac{\gamma }{2},\,n-\gamma }f\right)(\mu )\right]\right\}\\ =\ \displaystyle \frac{{\partial }^{n-1}}{\partial {t}^{n-1}}\left[{t}^{n-\tfrac{3}{2}\gamma -1}\left(n-\displaystyle \frac{3}{2}\gamma -\displaystyle \frac{\gamma }{2}\mu \displaystyle \frac{{\rm{d}}}{{\rm{d}}\mu }\right)\left({{ \mathcal K }}_{\tfrac{2}{\gamma }}^{1-\tfrac{\gamma }{2},n-\gamma }f\right)(\mu )\right].\end{array}\end{eqnarray}$
Repeating the preceding process n − 1 times, one has $\begin{eqnarray}\begin{array}{l}{D}_{t}^{\gamma }\left[p(x,y,t)\right]=\displaystyle \frac{{\partial }^{\gamma }p}{\partial {t}^{\gamma }}=\displaystyle \frac{{\partial }^{n-1}}{\partial {t}^{n-1}}\\ \quad \times \left[{t}^{n-\tfrac{3}{2}\gamma -1}\left(n-\displaystyle \frac{3}{2}\gamma -\displaystyle \frac{\gamma }{2}\mu \displaystyle \frac{{\rm{d}}}{{\rm{d}}\mu }\right)\left({{ \mathcal K }}_{\tfrac{2}{\gamma }}^{1-\tfrac{\gamma }{2},n-\gamma }f\right)(\mu )\right]\\ \quad ={t}^{-\tfrac{3}{2}\gamma }\left[\displaystyle \displaystyle \prod _{i=0}^{n-1}\left(1-\displaystyle \frac{3}{2}\gamma +i-\displaystyle \frac{\gamma }{2}\mu \displaystyle \frac{{\rm{d}}}{{\rm{d}}\mu }\right)\left({{ \mathcal K }}_{\tfrac{2}{\gamma }}^{1-\tfrac{\gamma }{2},n-\gamma }f\right)(\mu )\right]\\ \quad ={t}^{-\tfrac{3}{2}\gamma }\left({{ \mathcal P }}_{\tfrac{2}{\gamma }}^{1-\tfrac{3}{2}\gamma ,\gamma }f\right)(\mu ),\end{array}\end{eqnarray}$
where the Erdélyi-Kober fractional integral operator is $\begin{eqnarray*}\begin{array}{l}\left({{ \mathcal K }}_{\delta }^{\xi ,\eta }f\right)(\mu )\\ \quad =\left\{\begin{array}{ll}\displaystyle \frac{1}{{\rm{\Gamma }}(\eta )}{\int }_{1}^{\infty }{\left(\zeta -1\right)}^{\eta -1}{\zeta }^{-(\xi +\eta )}f({\zeta }^{\tfrac{1}{\delta }}\mu ){\rm{d}}\zeta , & \eta \gt 0,\\ f(\mu ), & \eta =0,\end{array}\right.\end{array}\end{eqnarray*}$
and the Erdélyi-Kober fractional differential operator is $\begin{eqnarray*}\left({{ \mathcal P }}_{\delta }^{\xi ,\eta }f\right)(\mu )=\displaystyle \prod _{i=0}^{n-1}\left(\xi +i-\displaystyle \frac{1}{\delta }\mu \displaystyle \frac{{\rm{d}}}{{\rm{d}}\mu }\right)\left({{ \mathcal K }}_{\delta }^{\xi +\eta ,n-\eta }f\right)(\mu ),\end{eqnarray*}$
and $\begin{eqnarray*}n=\left\{\begin{array}{ll}[\eta ]+1, & \eta \notin {\mathbb{N}},\\ \eta , & \eta \in {\mathbb{N}}.\end{array}\right.\end{eqnarray*}$
Similarly, $\begin{eqnarray}{D}_{t}^{\gamma }\left[q(x,y,t)\right]={t}^{-\tfrac{3}{2}\gamma }\left({{ \mathcal P }}_{\tfrac{2}{\gamma }}^{1-\tfrac{3}{2}\gamma ,\gamma }g\right)(\mu ).\end{eqnarray}$
Substituting equations ( $\begin{eqnarray}\left\{\begin{array}{l}\left({{ \mathcal P }}_{\tfrac{2}{\gamma }}^{1-\tfrac{3}{2}\gamma ,\gamma }f\right)(\mu )+2{g}^{{\prime\prime} }-({f}^{2}g+{g}^{3})+{gh}=0,\\ -\left({{ \mathcal P }}_{\tfrac{2}{\gamma }}^{1-\tfrac{3}{2}\gamma ,\gamma }g\right)(\mu )+2{f}^{{\prime\prime} }-({f}^{3}+{{fg}}^{2})+{fh}=0,\\ {\left({f}^{{\prime} }\right)}^{2}+{{ff}}^{{\prime\prime} }+{\left({g}^{{\prime} }\right)}^{2}+{{gg}}^{{\prime\prime} }+{h}^{{\prime\prime} }=0,\end{array}\right.\end{eqnarray}$
where ${\left(\cdot \right)}^{{\prime} },{\left(\cdot \right)}^{{\prime\prime} }$ are denoted as the derivatives with respect to μ, i.e. ${\left(\cdot \right)}^{{\prime} }=\tfrac{{\rm{d}}}{{\rm{d}}\mu }(\cdot ),{\left(\cdot \right)}^{{\prime\prime} }=\tfrac{{{\rm{d}}}^{2}}{{\rm{d}}{\mu }^{2}}(\cdot )$.5. Nonlinear self-adjointness and conservation laws of the time-fractional DS equation
The conservation law is closely related to the integrability of NPDEs; therefore, it is an important aspect in the study of NPDEs. As is known to all, if an NPDE has N-soliton (N ≥ 3) solutions, it is integrable. This equation has infinitely many conservation laws. In addition, conservation laws can be used to reveal a number of physical laws. The three famous physical laws in nature, namely conservation of mass, conservation of momentum and conservation of energy, are all important representations of conservation laws. From this perspective, conservation laws link mathematical representations to physical laws skillfully. In recent decades, conservation laws have been studied widely by many researchers, and a large number of new natural laws and phenomena are explained. Many methods of constructing conservation laws have been presented. Among these achievements, Noether's theorem and Ibragimov's new conservation theorem are extensively used [64, 65]. In addition, making use of pairs of symmetries and adjoint symmetries, a general one-to-one correspondence between conservation laws and non-Lagrangian equations is established and has been applied to the computation of conservation densities of the nonlinear evolution equations, including heat equations, Burgers equation and Korteweg-de Vries equation [66].
5.1. Nonlinear self-adjointness of the time-fractional DS equation
Introducing new dependent variables α = α(x, y, t), β = β(x, y, t) and ζ = ζ(x, y, t) and taking34 ) is5 ), the adjoint equation is36 ) is nonlinear self-adjoint if Ξ1, Ξ2, Ξ3 are not all zeros, and there exists cij(1 ≤ i, j ≤ 3) so that the following equation holds:36 ), (37 ) and (38 ) leads to39 ) results in5 ) is nonlinear self-adjoint.
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal F } & = & \alpha {{\rm{\Delta }}}_{1}+\beta {{\rm{\Delta }}}_{2}+\zeta {{\rm{\Delta }}}_{3}\\ & = & \alpha \left[{D}_{t}^{\gamma }p+{q}_{{xx}}+{q}_{{yy}}-({p}^{2}q+{q}^{3})+{qv}\right]\\ & & +\beta \left[-{D}_{t}^{\gamma }q+{p}_{{xx}}+{p}_{{yy}}-({p}^{3}+{{pq}}^{2})+{pv}\right]\\ & & +\zeta \left[{v}_{{xx}}+{v}_{{yy}}+2({p}_{x}^{2}+{{pp}}_{{xx}}+{q}_{x}^{2}+{{qq}}_{{xx}})\right],\end{array}\end{eqnarray}$
where ${ \mathcal F }$ is called formal Lagrangian. The action integral of equation ( $\begin{eqnarray}\begin{array}{l}{\int }_{0}^{T}{\int }_{{{\rm{\Omega }}}_{2}}{\int }_{{{\rm{\Omega }}}_{1}}{ \mathcal F }(x,y,t,p,q,v,\alpha ,\beta ,\zeta ,\\ \quad {D}_{t}^{\gamma }p,{p}_{{xx}},{p}_{{yy}},{D}_{t}^{\gamma }q,{q}_{{xx}},{q}_{{yy}},{v}_{{xx}},{v}_{{yy}}){\rm{d}}x{\rm{d}}y{\rm{d}}t,\end{array}\end{eqnarray}$
where (x, y, t) ∈ Ω1 × Ω2 × T. For equation ( $\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Delta }}}_{1}^{* }=\displaystyle \frac{\delta { \mathcal F }}{\delta p}={\left({D}_{t}^{\gamma }\right)}^{* }\alpha -2\alpha {pq}+2{\zeta }_{{xx}}p+{\beta }_{{xx}}+{\beta }_{{yy}}-3\beta {p}^{2}-\beta {q}^{2}+\beta v=0,\\ {{\rm{\Delta }}}_{2}^{* }=\displaystyle \frac{\delta { \mathcal F }}{\delta q}=-{\left({D}_{t}^{\gamma }\right)}^{* }\beta -2\beta {pq}+2{\zeta }_{{xx}}q+{\alpha }_{{xx}}+{\alpha }_{{yy}}-\alpha {p}^{2}-3\alpha {q}^{2}+\alpha v=0,\\ {{\rm{\Delta }}}_{3}^{* }=\displaystyle \frac{\delta { \mathcal F }}{\delta v}=\alpha q+\beta p+{\zeta }_{{xx}}+{\zeta }_{{yy}}=0,\end{array}\right.\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{\delta }{\delta p} & = & \displaystyle \frac{\partial }{\partial p}+{\left({D}_{t}^{\gamma }\right)}^{* }\displaystyle \frac{\partial }{\partial ({D}_{t}^{\gamma }p)}+\displaystyle \displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^{n}{D}_{{i}_{1}{i}_{2}\cdot \cdot \cdot {i}_{n}}\displaystyle \frac{\partial }{\partial {p}_{{i}_{1}{i}_{2}\cdot \cdot \cdot {i}_{n}}}\\ & = & \displaystyle \frac{\partial }{\partial p}+{\left({D}_{t}^{\gamma }\right)}^{* }\displaystyle \frac{\partial }{\partial ({D}_{t}^{\gamma }p)}-{D}_{x}\displaystyle \frac{\partial }{\partial {p}_{x}}+{D}_{{xx}}\displaystyle \frac{\partial }{\partial {p}_{{xx}}}\\ & & -{D}_{y}\displaystyle \frac{\partial }{\partial {p}_{y}}+{D}_{{yy}}\displaystyle \frac{\partial }{\partial {p}_{{yy}}},\\ \displaystyle \frac{\delta }{\delta q} & = & \displaystyle \frac{\partial }{\partial q}+{\left({D}_{t}^{\gamma }\right)}^{* }\displaystyle \frac{\partial }{\partial ({D}_{t}^{\gamma }q)}+\displaystyle \displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^{n}{D}_{{i}_{1}{i}_{2}\cdot \cdot \cdot {i}_{n}}\displaystyle \frac{\partial }{\partial {q}_{{i}_{1}{i}_{2}\cdot \cdot \cdot {i}_{n}}}\\ & = & \displaystyle \frac{\partial }{\partial q}+{\left({D}_{t}^{\gamma }\right)}^{* }\displaystyle \frac{\partial }{\partial ({D}_{t}^{\gamma }q)}-{D}_{x}\displaystyle \frac{\partial }{\partial {q}_{x}}+{D}_{{xx}}\displaystyle \frac{\partial }{\partial {q}_{{xx}}}\\ & & -{D}_{y}\displaystyle \frac{\partial }{\partial {q}_{y}}+{D}_{{yy}}\displaystyle \frac{\partial }{\partial {q}_{{yy}}},\\ \displaystyle \frac{\delta }{\delta v} & = & \displaystyle \frac{\partial }{\partial v}+\displaystyle \displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^{n}{D}_{{i}_{1}{i}_{2}\cdot \cdot \cdot {i}_{n}}\displaystyle \frac{\partial }{\partial {v}_{{i}_{1}{i}_{2}\cdot \cdot \cdot {i}_{n}}}\\ & = & \displaystyle \frac{\partial }{\partial v}-{D}_{x}\displaystyle \frac{\partial }{\partial {v}_{x}}+{D}_{{xx}}\displaystyle \frac{\partial }{\partial {v}_{{xx}}}-{D}_{y}\displaystyle \frac{\partial }{\partial {v}_{y}}+{D}_{{yy}}\displaystyle \frac{\partial }{\partial {v}_{{yy}}},\end{array}\end{eqnarray*}$
and Dx, Dy are denoted as the total derivatives with respect to x, y, respectively. ${\left({D}_{t}^{\gamma }\right)}^{* }={I}_{T}^{n-\gamma }{D}_{t}^{n}$ is the adjoint operator of ${D}_{t}^{\gamma }$, and ${I}_{T}^{n-\gamma }$ is the (n − γ)th-order right-hand-sided Riemann-Liouville integral operator, which is presented as $\begin{eqnarray*}{I}_{T}^{n-\gamma }f(x,y,t)=\displaystyle \frac{{\left(-1\right)}^{n}}{{\rm{\Gamma }}(n-\gamma )}{\int }_{t}^{T}\displaystyle \frac{f(x,y,\xi )}{{\left(\xi -t\right)}^{1+\gamma -n}}{\rm{d}}\xi ,\end{eqnarray*}$
where Γ(γ) is the gamma function. Introducing dependent variables Ξ1, Ξ2, Ξ3 such that $\begin{eqnarray}\begin{array}{rcl}\alpha & = & \alpha (x,y,t)={{\rm{\Xi }}}_{1}(x,y,t,p,q,v),\\ \beta & = & \beta (x,y,t)={{\rm{\Xi }}}_{2}(x,y,t,p,q,v),\\ \zeta & = & \zeta (x,y,t)={{\rm{\Xi }}}_{3}(x,y,t,p,q,v).\end{array}\end{eqnarray}$
Equation ( $\begin{eqnarray}\left\{\begin{array}{l}\displaystyle \frac{\delta { \mathcal F }}{\delta p}{| }_{\alpha ={{\rm{\Xi }}}_{1},\beta ={{\rm{\Xi }}}_{2},\zeta ={{\rm{\Xi }}}_{3}}={c}_{11}{{\rm{\Delta }}}_{1}+{c}_{12}{{\rm{\Delta }}}_{2}+{c}_{13}{{\rm{\Delta }}}_{3},\\ \displaystyle \frac{\delta { \mathcal F }}{\delta q}{| }_{\alpha ={{\rm{\Xi }}}_{1},\beta ={{\rm{\Xi }}}_{2},\zeta ={{\rm{\Xi }}}_{3}}={c}_{21}{{\rm{\Delta }}}_{1}+{c}_{22}{{\rm{\Delta }}}_{2}+{c}_{23}{{\rm{\Delta }}}_{3},\\ \displaystyle \frac{\delta { \mathcal F }}{\delta v}{| }_{\alpha ={{\rm{\Xi }}}_{1},\beta ={{\rm{\Xi }}}_{2},\zeta ={{\rm{\Xi }}}_{3}}={c}_{31}{{\rm{\Delta }}}_{1}+{c}_{32}{{\rm{\Delta }}}_{2}+{c}_{33}{{\rm{\Delta }}}_{3}.\end{array}\right.\end{eqnarray}$
Combining equations ( $\begin{eqnarray}\left\{\begin{array}{l}{\left({D}_{t}^{\gamma }\right)}^{* }{{\rm{\Xi }}}_{1}-2{{\rm{\Xi }}}_{1}{pq}+2\left[{{\rm{\Xi }}}_{3,{xx}}+2({{\rm{\Xi }}}_{3,{xp}}{p}_{x}+{{\rm{\Xi }}}_{3,{xq}}{q}_{x}+{{\rm{\Xi }}}_{3,{xv}}{v}_{x})\right.\\ +({{\rm{\Xi }}}_{3,{pp}}{p}_{x}^{2}+{{\rm{\Xi }}}_{3,{pq}}{q}_{x}^{2}+{{\rm{\Xi }}}_{3,{vv}}{v}_{x}^{2})+2({{\rm{\Xi }}}_{3,{pq}}{p}_{x}{q}_{x}+{{\rm{\Xi }}}_{3,{pv}}{p}_{x}{v}_{x}+{{\rm{\Xi }}}_{3,{qv}}{q}_{x}{v}_{x})\\ \left.+{{\rm{\Xi }}}_{3,p}{p}_{{xx}}+{{\rm{\Xi }}}_{3,q}{q}_{{xx}}+{{\rm{\Xi }}}_{3,v}{v}_{{xx}}\right]p+\left[{{\rm{\Xi }}}_{2,{xx}}+2({{\rm{\Xi }}}_{2,{xp}}{p}_{x}+{{\rm{\Xi }}}_{2,{xq}}{q}_{x}+{{\rm{\Xi }}}_{2,{xv}}{v}_{x})\right.\\ +({{\rm{\Xi }}}_{2,{pp}}{p}_{x}^{2}+{{\rm{\Xi }}}_{2,{pq}}{q}_{x}^{2}+{{\rm{\Xi }}}_{2,{vv}}{v}_{x}^{2})+2({{\rm{\Xi }}}_{2,{pq}}{p}_{x}{q}_{x}+{{\rm{\Xi }}}_{2,{pv}}{p}_{x}{v}_{x}+{{\rm{\Xi }}}_{2,{qv}}{q}_{x}{v}_{x})\\ \left.+{{\rm{\Xi }}}_{2,p}{p}_{{xx}}+{{\rm{\Xi }}}_{2,q}{q}_{{xx}}+{{\rm{\Xi }}}_{2,v}{v}_{{xx}}\right]+\left[{{\rm{\Xi }}}_{2,{yy}}+2({{\rm{\Xi }}}_{2,{yp}}{p}_{y}+{{\rm{\Xi }}}_{2,{yq}}{q}_{y}+{{\rm{\Xi }}}_{2,{yv}}{v}_{y})\right.\\ +({{\rm{\Xi }}}_{2,{pp}}{p}_{y}^{2}+{{\rm{\Xi }}}_{2,{pq}}{q}_{y}^{2}+{{\rm{\Xi }}}_{2,{vv}}{v}_{y}^{2})+2({{\rm{\Xi }}}_{2,{pq}}{p}_{y}{q}_{y}+{{\rm{\Xi }}}_{2,{pv}}{p}_{y}{v}_{y}+{{\rm{\Xi }}}_{2,{qv}}{q}_{y}{v}_{y})\\ \left.+{{\rm{\Xi }}}_{2,p}{p}_{{yy}}+{{\rm{\Xi }}}_{2,q}{q}_{{yy}}+{{\rm{\Xi }}}_{2,v}{v}_{{yy}}\right]-3{{\rm{\Xi }}}_{2}{p}^{2}-{{\rm{\Xi }}}_{2}{q}^{2}+{{\rm{\Xi }}}_{2}v={c}_{11}{{\rm{\Delta }}}_{1}+{c}_{12}{{\rm{\Delta }}}_{2}+{c}_{13}{{\rm{\Delta }}}_{3},\\ \\ -{\left({D}_{t}^{\gamma }\right)}^{* }{{\rm{\Xi }}}_{2}-2{{\rm{\Xi }}}_{2}{pq}+2\left[{{\rm{\Xi }}}_{3,{xx}}+2({{\rm{\Xi }}}_{3,{xp}}{p}_{x}+{{\rm{\Xi }}}_{3,{xq}}{q}_{x}+{{\rm{\Xi }}}_{3,{xv}}{v}_{x})\right.\\ +({{\rm{\Xi }}}_{3,{pp}}{p}_{x}^{2}+{{\rm{\Xi }}}_{3,{pq}}{q}_{x}^{2}+{{\rm{\Xi }}}_{3,{vv}}{v}_{x}^{2})+2({{\rm{\Xi }}}_{3,{pq}}{p}_{x}{q}_{x}+{{\rm{\Xi }}}_{3,{pv}}{p}_{x}{v}_{x}+{{\rm{\Xi }}}_{3,{qv}}{q}_{x}{v}_{x})\\ \left.+{{\rm{\Xi }}}_{3,p}{p}_{{xx}}+{{\rm{\Xi }}}_{3,q}{q}_{{xx}}+{{\rm{\Xi }}}_{3,v}{v}_{{xx}}\right]q+\left[{{\rm{\Xi }}}_{1,{xx}}+2({{\rm{\Xi }}}_{1,{xp}}{p}_{x}+{{\rm{\Xi }}}_{1,{xq}}{q}_{x}+{{\rm{\Xi }}}_{1,{xv}}{v}_{x})\right.\\ +({{\rm{\Xi }}}_{1,{pp}}{p}_{x}^{2}+{{\rm{\Xi }}}_{1,{pq}}{q}_{x}^{2}+{{\rm{\Xi }}}_{1,{vv}}{v}_{x}^{2})+2({{\rm{\Xi }}}_{1,{pq}}{p}_{x}{q}_{x}+{{\rm{\Xi }}}_{1,{pv}}{p}_{x}{v}_{x}+{{\rm{\Xi }}}_{1,{qv}}{q}_{x}{v}_{x})\\ \left.+{{\rm{\Xi }}}_{1,p}{p}_{{xx}}+{{\rm{\Xi }}}_{1,q}{q}_{{xx}}+{{\rm{\Xi }}}_{1,v}{v}_{{xx}}\right]+\left[{{\rm{\Xi }}}_{1,{yy}}+2({{\rm{\Xi }}}_{1,{yp}}{p}_{y}+{{\rm{\Xi }}}_{1,{yq}}{q}_{y}+{{\rm{\Xi }}}_{1,{yv}}{v}_{y})\right.\\ +({{\rm{\Xi }}}_{1,{pp}}{p}_{y}^{2}+{{\rm{\Xi }}}_{1,{pq}}{q}_{y}^{2}+{{\rm{\Xi }}}_{1,{vv}}{v}_{y}^{2})+2({{\rm{\Xi }}}_{1,{pq}}{p}_{y}{q}_{y}+{{\rm{\Xi }}}_{1,{pv}}{p}_{y}{v}_{y}+{{\rm{\Xi }}}_{1,{qv}}{q}_{y}{v}_{y})\\ \left.+{{\rm{\Xi }}}_{1,p}{p}_{{yy}}+{{\rm{\Xi }}}_{1,q}{q}_{{yy}}+{{\rm{\Xi }}}_{1,v}{v}_{{yy}}\right]-{{\rm{\Xi }}}_{1}{p}^{2}-3{{\rm{\Xi }}}_{1}{q}^{2}+{{\rm{\Xi }}}_{1}v={c}_{21}{{\rm{\Delta }}}_{1}+{c}_{22}{{\rm{\Delta }}}_{2}+{c}_{23}{{\rm{\Delta }}}_{3},\\ \\ {{\rm{\Xi }}}_{1}q+{{\rm{\Xi }}}_{2}p+\left[{{\rm{\Xi }}}_{3,{xx}}+2({{\rm{\Xi }}}_{3,{xp}}{p}_{x}+{{\rm{\Xi }}}_{3,{xq}}{q}_{x}+{{\rm{\Xi }}}_{3,{xv}}{v}_{x})\right.\\ +({{\rm{\Xi }}}_{3,{pp}}{p}_{x}^{2}+{{\rm{\Xi }}}_{3,{pq}}{q}_{x}^{2}+{{\rm{\Xi }}}_{3,{vv}}{v}_{x}^{2})+2({{\rm{\Xi }}}_{3,{pq}}{p}_{x}{q}_{x}+{{\rm{\Xi }}}_{3,{pv}}{p}_{x}{v}_{x}+{{\rm{\Xi }}}_{3,{qv}}{q}_{x}{v}_{x})\\ \left.+{{\rm{\Xi }}}_{3,p}{p}_{{xx}}+{{\rm{\Xi }}}_{3,q}{q}_{{xx}}+{{\rm{\Xi }}}_{3,v}{v}_{{xx}}\right]+\left[{{\rm{\Xi }}}_{3,{yy}}+2({{\rm{\Xi }}}_{3,{yp}}{p}_{y}+{{\rm{\Xi }}}_{3,{yq}}{q}_{y}+{{\rm{\Xi }}}_{3,{yv}}{v}_{y})\right.\\ +({{\rm{\Xi }}}_{3,{pp}}{p}_{y}^{2}+{{\rm{\Xi }}}_{3,{pq}}{q}_{y}^{2}+{{\rm{\Xi }}}_{3,{vv}}{v}_{y}^{2})+2({{\rm{\Xi }}}_{3,{pq}}{p}_{y}{q}_{y}+{{\rm{\Xi }}}_{3,{pv}}{p}_{y}{v}_{y}+{{\rm{\Xi }}}_{3,{qv}}{q}_{y}{v}_{y})\\ \left.+{{\rm{\Xi }}}_{3,p}{p}_{{yy}}+{{\rm{\Xi }}}_{3,q}{q}_{{yy}}+{{\rm{\Xi }}}_{3,v}{v}_{{yy}}\right]={c}_{31}{{\rm{\Delta }}}_{1}+{c}_{32}{{\rm{\Delta }}}_{2}+{c}_{33}{{\rm{\Delta }}}_{3}.\end{array}\right.\end{eqnarray}$
Solving equation ( $\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Xi }}}_{1}(x,y,t,p,q,v)=0,\\ {{\rm{\Xi }}}_{2}(x,y,t,p,q,v)=0,\\ {{\rm{\Xi }}}_{3}(x,y,t,p,q,v)=F(t),\\ {c}_{{ij}}=0\quad {\rm{for}}\quad 1\leqslant i,j\leqslant 3.\end{array}\right.\end{eqnarray}$
where F(t) is an arbitrary function of t. Consequently, equation (5.2. Conservation laws of the time-fractional DS equation
Next, on the basis of the presented results, the conservation vectors and the conservation laws of equation (5 ) can be constructed. then V△ is called the conservation vector of equation (5 ), and equation (41 ) is called the conservation law equation, where Dx, Dy and Dt are denoted as the total derivatives with respect to x, y and t, respectively. By using Noether's theorem and Ibragimov's new conservation theorem, the conserved vector can be obtained as
Taking ${V}^{\bigtriangleup }=({V}^{x},{V}^{y},{V}^{t})$, if ${V}^{\bigtriangleup }$ satisfies the following equation:
$\begin{eqnarray}{D}_{t}({V}^{t})+{D}_{x}({V}^{x})+{D}_{y}({V}^{y}){| }_{(5)}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{V}^{t} & = & \tau { \mathcal F }+\displaystyle \displaystyle \sum _{i=0}^{n-1}{\left(-1\right)}^{i}{D}_{t}^{\gamma -1-i}({{ \mathcal N }}^{p}){D}_{t}^{i}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial ({D}_{t}^{\gamma }p)}\right]\\ & & -{\left(-1\right)}^{n}{ \mathcal J }\left[{{ \mathcal N }}^{p},{D}_{t}^{n}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial ({D}_{t}^{\gamma }p)}\right)\right]\\ & & +\displaystyle \displaystyle \sum _{i=0}^{n-1}{\left(-1\right)}^{i}{D}_{t}^{\gamma -1-i}({{ \mathcal N }}^{q}){D}_{t}^{i}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial ({D}_{t}^{\gamma }q)}\right]-{\left(-1\right)}^{n}{ \mathcal J }\\ & & \times \left[{{ \mathcal N }}^{q},{D}_{t}^{n}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial ({D}_{t}^{\gamma }q)}\right)\right]\\ & & +\displaystyle \displaystyle \sum _{i=0}^{n-1}{\left(-1\right)}^{i}{D}_{t}^{\gamma -1-i}({{ \mathcal N }}^{v}){D}_{t}^{i}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial ({D}_{t}^{\gamma }v)}\right]-{\left(-1\right)}^{n}{ \mathcal J }\\ & & \times \left[{{ \mathcal N }}^{v},{D}_{t}^{n}(\displaystyle \frac{\partial { \mathcal F }}{\partial ({D}_{t}^{\gamma }v)})\right],\\ {V}^{x} & = & {\lambda }_{1}{ \mathcal F }+{{ \mathcal N }}^{p}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial {p}_{x}}-{D}_{x}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {p}_{{xx}}}\right)\right]+{D}_{x}({{ \mathcal N }}^{p})\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {p}_{{xx}}}\right)\\ & & +{{ \mathcal N }}^{q}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial {q}_{x}}-{D}_{x}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {q}_{{xx}}}\right)\right]+{D}_{x}({{ \mathcal N }}^{q})\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {q}_{{xx}}}\right)\\ & & +{{ \mathcal N }}^{v}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial {v}_{x}}-{D}_{x}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {v}_{{xx}}}\right)\right]+{D}_{x}({{ \mathcal N }}^{v})\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {v}_{{xx}}}\right),\\ {V}^{y} & = & {\lambda }_{2}{ \mathcal F }+{{ \mathcal N }}^{p}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial {p}_{y}}-{D}_{y}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {p}_{{yy}}}\right)\right]+{D}_{y}({{ \mathcal N }}^{p})\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {p}_{{yy}}}\right)\\ & & +{{ \mathcal N }}^{q}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial {q}_{y}}-{D}_{y}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {q}_{{yy}}}\right)\right]+{D}_{y}({{ \mathcal N }}^{q})\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {q}_{{yy}}}\right)\\ & & +{{ \mathcal N }}^{v}\left[\displaystyle \frac{\partial { \mathcal F }}{\partial {v}_{y}}-{D}_{y}\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {v}_{{yy}}}\right)\right]+{D}_{y}({{ \mathcal N }}^{v})\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {v}_{{yy}}}\right),\end{array}\end{eqnarray}$
where n = [γ] + 1; ${{ \mathcal N }}^{p}={\phi }_{1}-{\lambda }_{1}{p}_{x}-{\lambda }_{2}{p}_{y}-\tau {p}_{t}$; ${{ \mathcal N }}^{q}={\phi }_{2}-{\lambda }_{1}{q}_{x}-{\lambda }_{2}{q}_{y}-\tau {q}_{t}$; ${{ \mathcal N }}^{v}={\phi }_{3}-{\lambda }_{1}{v}_{x}-{\lambda }_{2}{v}_{y}-\tau {v}_{t}$; and λ1, λ2, $\tau$, φ1, φ2 and φ3 satisfy vector field (symmetry) $V={\lambda }_{1}\tfrac{\partial }{\partial x}+{\lambda }_{2}\tfrac{\partial }{\partial y}+\tau \tfrac{\partial }{\partial t}+{\phi }_{1}\tfrac{\partial }{\partial p}+{\phi }_{2}\tfrac{\partial }{\partial q}+{\phi }_{3}\tfrac{\partial }{\partial v}$ and $\begin{eqnarray*}{ \mathcal J }({f}_{1},{f}_{2})=\displaystyle \frac{1}{{\rm{\Gamma }}(n-\gamma )}{\int }_{0}^{t}{\int }_{t}^{T}\displaystyle \frac{{f}_{1}(x,y,\xi ){f}_{2}(x,y,\eta )}{{\left(\eta -\xi \right)}^{1+\gamma -n}}{\rm{d}}\eta {\rm{d}}\xi .\end{eqnarray*}$
I: For vector field (symmetry) $V=\gamma x\tfrac{\partial }{\partial x}+\gamma y\tfrac{\partial }{\partial y}+2t\tfrac{\partial }{\partial t}-\gamma p\tfrac{\partial }{\partial p}-\gamma q\tfrac{\partial }{\partial q}-2\gamma v\tfrac{\partial }{\partial v}$,
$\begin{eqnarray*}\left\{\begin{array}{l}{\lambda }_{1}=\gamma x,{\lambda }_{2}=\gamma y,\tau =2t,{\phi }_{1}=-\gamma p,{\phi }_{2}=-\gamma q,{\phi }_{3}=-2\gamma v,\\ {{ \mathcal N }}^{p}=-\gamma p-\gamma {{xp}}_{x}-\gamma {{yp}}_{y}-2{{tp}}_{t},\\ {{ \mathcal N }}^{q}=-\gamma q-\gamma {{xq}}_{x}-\gamma {{yq}}_{y}-2{{tq}}_{t},\\ {{ \mathcal N }}^{v}=-2\gamma v-\gamma {{xv}}_{x}-\gamma {{yv}}_{y}-2{{tv}}_{t}.\end{array}\right.\end{eqnarray*}$
Therefore, $\begin{eqnarray}\begin{array}{rcl}{V}^{t} & = & 0,\\ {V}^{x} & = & -2{p}_{x}F(\gamma p+\gamma {{xp}}_{x}+\gamma {{yp}}_{y}+2{{tp}}_{t})\\ & & -2{pF}(2\gamma {p}_{x}+\gamma {{xp}}_{{xx}}+\gamma {{yp}}_{{xy}}+2{{tp}}_{{xt}})\\ & & -2{q}_{x}F(\gamma q+\gamma {{xq}}_{x}+\gamma {{yq}}_{y}+2{{tq}}_{t})\\ & & -2{qF}(2\gamma {q}_{x}+\gamma {{xq}}_{{xx}}+\gamma {{yq}}_{{xy}}+2{{tq}}_{{xt}})\\ & & -F(3\gamma {v}_{x}+\gamma {{xv}}_{{xx}}+\gamma {{yv}}_{{xy}}+2{{tv}}_{{xt}}),\\ {V}^{y} & = & -F(3\gamma {v}_{y}+\gamma {{xv}}_{{xy}}+\gamma {{yv}}_{{yy}}+2{{tv}}_{{yt}}).\end{array}\end{eqnarray}$
II: For vector field (symmetry) $V=\tfrac{\partial }{\partial x}$,
$\begin{eqnarray*}\left\{\begin{array}{l}{\lambda }_{1}=1,{\lambda }_{2}=\tau ={\phi }_{1}={\phi }_{2}={\phi }_{3}=0,\\ {{ \mathcal N }}^{p}=-{p}_{x},\\ {{ \mathcal N }}^{q}=-{q}_{x},\\ {{ \mathcal N }}^{v}=-{v}_{x}.\end{array}\right.\end{eqnarray*}$
Therefore, $\begin{eqnarray}\begin{array}{rcl}{V}^{t} & = & 0,\\ {V}^{x} & = & -F[2({p}_{x}^{2}+{{pp}}_{{xx}}+{q}_{x}^{2}+{{qq}}_{{xx}})+{v}_{{xx}}]={{Fv}}_{{yy}},\\ {V}^{y} & = & -{{Fv}}_{{xy}}.\end{array}\end{eqnarray}$
III: For vector field (symmetry) $V=\tfrac{\partial }{\partial y}$,
$\begin{eqnarray*}\left\{\begin{array}{l}{\lambda }_{1}=\tau ={\phi }_{1}={\phi }_{2}={\phi }_{3}=0,{\lambda }_{2}=1,\\ {{ \mathcal N }}^{p}=-{p}_{y},\\ {{ \mathcal N }}^{q}=-{q}_{y},\\ {{ \mathcal N }}^{v}=-{v}_{y}.\end{array}\right.\end{eqnarray*}$
Therefore, $\begin{eqnarray}\begin{array}{rcl}{V}^{t} & = & 0,\\ {V}^{x} & = & -F[2({p}_{x}{p}_{y}+{{pp}}_{{xy}}+{q}_{x}{q}_{y}+{{qq}}_{{xy}})+{v}_{{xy}}],\\ {V}^{y} & = & -{{Fv}}_{{yy}}.\end{array}\end{eqnarray}$
IV: For vector field (symmetry) $V=q\tfrac{\partial }{\partial p}-p\tfrac{\partial }{\partial q}$,43 )-(46 ) all satisfy equation (41 ).
$\begin{eqnarray*}\left\{\begin{array}{l}{\lambda }_{1}={\lambda }_{2}=\tau ={\phi }_{3}=0,{\phi }_{1}=q,{\phi }_{2}=-p,\\ {{ \mathcal N }}^{p}=q,\\ {{ \mathcal N }}^{q}=-p,\\ {{ \mathcal N }}^{v}=0.\end{array}\right.\end{eqnarray*}$
Therefore, $\begin{eqnarray}\begin{array}{rcl}{V}^{t} & = & 0,\\ {V}^{x} & = & 0,\\ {V}^{y} & = & 0.\end{array}\end{eqnarray}$
It is verified that equations (6. Traveling wave solutions of the time-fractional DS equation
In this section, the traveling wave solutions of equation (5 ) can be provided.
For variable transformation49 ) leads to50 ) into equation (5 ), one obtains52 ), one acquires53 ) into equation (52 ) comes55 ) with respect to ξ, one obtains56 ) derives57 ), one achieves
$\begin{eqnarray}T=n\displaystyle \frac{{t}^{\gamma }}{{\rm{\Gamma }}(1+\gamma )},\end{eqnarray}$
one has $\begin{eqnarray}{D}_{t}^{\gamma }=n\displaystyle \frac{\partial }{{\partial }_{T}},\end{eqnarray}$
where n is a nonzero constant. Taking $\begin{eqnarray}\begin{array}{rcl}p(x,y,t) & = & {p}^{* }(x,y,T)=F(\xi )\cos \mu ,\\ q(x,y,t) & = & {q}^{* }(x,y,T)=F(\xi )\sin \mu ,\\ v(x,y,t) & = & {v}^{* }(x,y,T)=G(\xi ).\end{array}\end{eqnarray}$
where F and G are two real-valued functions, and $\begin{eqnarray*}\begin{array}{rcl}\xi & = & {k}_{1}x+{k}_{2}y+{k}_{3}T,\\ \mu & = & {k}_{4}x+{k}_{5}y+{k}_{6}T,\end{array}\end{eqnarray*}$
where k1, k2, k3, k4, k5 and k6 are nonzero real-valued constants to be determined. Equation ( $\begin{eqnarray}\begin{array}{rcl}{D}_{t}^{\gamma }p & = & {{np}}_{T}=n({k}_{3}{F}^{{\prime} }\cos \mu -{k}_{6}F\sin \mu ),\\ {p}_{x} & = & {k}_{1}{F}^{{\prime} }\cos \mu -{k}_{4}F\sin \mu ,\\ {p}_{{xx}} & = & ({k}_{1}^{2}{F}^{{\prime\prime} }-{k}_{4}^{2}F)\cos \mu -2{k}_{1}{k}_{4}{F}^{{\prime} }\sin \mu ,\\ {p}_{y} & = & {k}_{2}{F}^{{\prime} }\cos \mu -{k}_{5}F\sin \mu ,\\ {p}_{{yy}} & = & ({k}_{2}^{2}{F}^{{\prime\prime} }-{k}_{5}^{2}F)\cos \mu -2{k}_{2}{k}_{5}{F}^{{\prime} }\sin \mu ,\\ {D}_{t}^{\gamma }q & = & {{nq}}_{T}=n({k}_{3}{F}^{{\prime} }\sin \mu +{k}_{6}F\cos \mu ),\\ {q}_{x} & = & {k}_{1}{F}^{{\prime} }\sin \mu +{k}_{4}F\cos \mu ,\\ {q}_{{xx}} & = & ({k}_{1}^{2}{F}^{{\prime\prime} }-{k}_{4}^{2}F)\sin \mu +2{k}_{1}{k}_{4}{F}^{{\prime} }\cos \mu ,\\ {q}_{y} & = & {k}_{2}{F}^{{\prime} }\sin \mu +{k}_{5}F\cos \mu ,\\ {q}_{{yy}} & = & ({k}_{2}^{2}{F}^{{\prime\prime} }-{k}_{5}^{2}F)\sin \mu +2{k}_{2}{k}_{5}{F}^{{\prime} }\cos \mu ,\\ {v}_{x} & = & {k}_{1}{G}^{{\prime} },\quad \quad \quad {v}_{{xx}}={k}_{1}^{2}{G}^{{\prime\prime} },\\ {v}_{y} & = & {k}_{2}{G}^{{\prime} },\quad \quad \quad {v}_{{yy}}={k}_{2}^{2}{G}^{{\prime\prime} },\end{array}\end{eqnarray}$
where the subscripts are denoted as the derivatives with respect to ξ. Substituting equation ( $\begin{eqnarray}\left\{\begin{array}{l}n({k}_{3}{F}^{{\prime} }\cos \mu -{k}_{6}F\sin \mu )+({k}_{1}^{2}{F}^{{\prime\prime} }-{k}_{4}^{2}F)\sin \mu +2{k}_{1}{k}_{4}{F}^{{\prime} }\cos \mu \\ +({k}_{2}^{2}{F}^{{\prime\prime} }-{k}_{5}^{2}F)\sin \mu +2{k}_{2}{k}_{5}{F}^{{\prime} }\cos \mu -{F}^{3}\sin \mu +{FG}\sin \mu =0,\\ -n({k}_{3}{F}^{{\prime} }\sin \mu +{k}_{6}F\cos \mu )+({k}_{1}^{2}{F}^{{\prime\prime} }-{k}_{4}^{2}F)\cos \mu -2{k}_{1}{k}_{4}{F}^{{\prime} }\sin \mu \\ +({k}_{2}^{2}{F}^{{\prime\prime} }-{k}_{5}^{2}F)\cos \mu -2{k}_{2}{k}_{5}{F}^{{\prime} }\sin \mu -{F}^{3}\cos \mu +{FG}\cos \mu =0,\\ ({k}_{1}^{2}+{k}_{2}^{2}){G}^{{\prime\prime} }+{k}_{1}^{2}{\left({F}^{2}\right)}^{{\prime\prime} }=0.\end{array}\right.\end{eqnarray}$
Accordingly, $\begin{eqnarray}\left\{\begin{array}{l}{{nk}}_{3}{F}^{{\prime} }+2{k}_{1}{k}_{4}{F}^{{\prime} }+2{k}_{2}{k}_{5}{F}^{{\prime} }=0,\\ -{{nk}}_{6}F+({k}_{1}^{2}+{k}_{2}^{2}){F}^{{\prime\prime} }-({k}_{4}^{2}+{k}_{5}^{2})F-{F}^{3}+{FG}=0,\\ ({k}_{1}^{2}+{k}_{2}^{2}){G}^{{\prime\prime} }+{k}_{1}^{2}{\left({F}^{2}\right)}^{{\prime\prime} }=0.\end{array}\right.\end{eqnarray}$
From the third equality of equation ( $\begin{eqnarray}G=-\displaystyle \frac{{k}_{1}^{2}}{{k}_{1}^{2}+{k}_{2}^{2}}{F}^{2}.\end{eqnarray}$
Substituting equation ( $\begin{eqnarray}\begin{array}{l}({k}_{1}^{2}+{k}_{2}^{2}){F}^{{\prime\prime} }-({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})F\\ \quad -\left(1+\displaystyle \frac{{k}_{1}^{2}}{{k}_{1}^{2}+{k}_{2}^{2}}\right){F}^{3}=0.\end{array}\end{eqnarray}$
Accordingly, $\begin{eqnarray}\begin{array}{l}({k}_{1}^{2}+{k}_{2}^{2}){F}^{{\prime\prime} }{F}^{{\prime} }-({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2}){{FF}}^{{\prime} }\\ \quad -\left(1+\displaystyle \frac{{k}_{1}^{2}}{{k}_{1}^{2}+{k}_{2}^{2}}\right){F}^{3}{F}^{{\prime} }=0.\end{array}\end{eqnarray}$
Integrating equation ( $\begin{eqnarray}({k}_{1}^{2}+{k}_{2}^{2}){\left({F}^{{\prime} }\right)}^{2}=({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2}){F}^{2}+\displaystyle \frac{2{k}_{1}^{2}+{k}_{2}^{2}}{2({k}_{1}^{2}+{k}_{2}^{2})}{F}^{4}.\end{eqnarray}$
Equation ( $\begin{eqnarray}{F}^{{\prime} }=\displaystyle \frac{F\sqrt{2({k}_{1}^{2}+{k}_{2}^{2})({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})+(2{k}_{1}^{2}+{k}_{2}^{2}){F}^{2}}}{\sqrt{2}({k}_{1}^{2}+{k}_{2}^{2})}.\end{eqnarray}$
Solving equation ( $\begin{eqnarray}\begin{array}{l}\xi =\sqrt{2}({k}_{1}^{2}+{k}_{2}^{2})\\ \quad \times \int \displaystyle \frac{{\rm{d}}F}{F\sqrt{2({k}_{1}^{2}+{k}_{2}^{2})({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})+(2{k}_{1}^{2}+{k}_{2}^{2}){F}^{2}}},\end{array}\end{eqnarray}$
i.e. $\begin{eqnarray}\begin{array}{l}{k}_{1}x+{k}_{2}y+{k}_{3}T=\sqrt{2}({k}_{1}^{2}+{k}_{2}^{2})\\ \quad \times \int \displaystyle \frac{{\rm{d}}F}{F\sqrt{2({k}_{1}^{2}+{k}_{2}^{2})({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})+(2{k}_{1}^{2}+{k}_{2}^{2}){F}^{2}}}.\end{array}\end{eqnarray}$
Therefore, $\begin{eqnarray}F=\sqrt{\displaystyle \frac{2({k}_{1}^{2}+{k}_{2}^{2})({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})}{2{k}_{1}^{2}+{k}_{2}^{2}}}{\rm{sech}} \displaystyle \frac{{k}_{1}x+{k}_{2}y+{k}_{3}T}{\sqrt{{k}_{1}^{2}+{k}_{2}^{2}}}.\end{eqnarray}$
Finally,
$\begin{eqnarray}\begin{array}{l}p(x,y,t)=F(\xi )\cos \mu \\ \quad =\sqrt{\displaystyle \frac{2({k}_{1}^{2}+{k}_{2}^{2})({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})}{2{k}_{1}^{2}+{k}_{2}^{2}}}\\ \quad {\rm{sech}} \displaystyle \frac{{k}_{1}x+{k}_{2}y+{k}_{3}n\tfrac{{t}^{\gamma }}{{\rm{\Gamma }}(1+\gamma )}}{\sqrt{{k}_{1}^{2}+{k}_{2}^{2}}}\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\quad \cos \left({k}_{4}x+{k}_{5}y+{k}_{6}n\displaystyle \frac{{t}^{\gamma }}{{\rm{\Gamma }}(1+\gamma )}\right),\\ \quad q(x,y,t)=F(\xi )\sin \mu \\ \quad =\sqrt{\displaystyle \frac{2({k}_{1}^{2}+{k}_{2}^{2})({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})}{2{k}_{1}^{2}+{k}_{2}^{2}}}\\ \quad {\rm{sech}} \displaystyle \frac{{k}_{1}x+{k}_{2}y+{k}_{3}n\tfrac{{t}^{\gamma }}{{\rm{\Gamma }}(1+\gamma )}}{\sqrt{{k}_{1}^{2}+{k}_{2}^{2}}}\\ \quad \sin \left({k}_{4}x+{k}_{5}y+{k}_{6}n\displaystyle \frac{{t}^{\gamma }}{{\rm{\Gamma }}(1+\gamma )}\right),\\ \quad v(x,y,t)=G(\xi )=-\displaystyle \frac{2{k}_{1}^{2}({{nk}}_{6}+{k}_{4}^{2}+{k}_{5}^{2})}{2{k}_{1}^{2}+{k}_{2}^{2}}\\ {{\rm{sech}} }^{2}\displaystyle \frac{{k}_{1}x+{k}_{2}y+{k}_{3}n\tfrac{{t}^{\gamma }}{{\rm{\Gamma }}(1+\gamma )}}{\sqrt{{k}_{1}^{2}+{k}_{2}^{2}}},\end{array}\end{eqnarray}$
where k1, k2, k3, k4, k5 and k6 are nonzero real-valued constants.Figure 1 shows the traveling wave solutions of p, q and v when the parameter values are taken as $t=2,{k}_{1}={k}_{2}\,={k}_{3}={k}_{4}={k}_{5}={k}_{6}=1,n=1\,\mathrm{and}\,\gamma =\tfrac{1}{2}$. In figure 2, t = 200, and the values of the other parameters are the same as those in figure 1. It can be observed that when the waves of p, q and v propagate, their shapes and amplitudes remain the same. From sectional drawings (b), (d) and (f) in figures 1 and 2, it can be seen that the wave shapes of p, q and v have symmetry properties. It can also be seen that when x → ∞ or y → ∞ , the solutions p → 0, q → 0 and v → 0.
Figure 1. (a) Three-dimensional drawing of the traveling wave solution p, (b) sectional drawing of p, (c) three-dimensional drawing of the traveling wave solution q, (d) sectional drawing of q, (e) three-dimensional drawing of the traveling wave solution v and (f) sectional drawing of v. The parameter values are taken as $t=2,{k}_{1}={k}_{2}={k}_{3}={k}_{4}={k}_{5}={k}_{6}=1,n=1\,\mathrm{and}\,\gamma =\tfrac{1}{2}$. |
Figure 2. (a) Three-dimensional drawing of the traveling wave solution p, (b) sectional drawing of p, (c) three-dimensional drawing of the traveling wave solution q, (d) sectional drawing of q, (e) three-dimensional drawing of the traveling wave solution v and (f) sectional drawing of v. The parameter values are taken as $t=200,{k}_{1}={k}_{2}={k}_{3}={k}_{4}={k}_{5}={k}_{6}=1,n=1\,\mathrm{and}\,\gamma =\tfrac{1}{2}$. |
7. Conclusions
In this paper, on the basis of the Riemann-Liouville fractional derivative, the time-fractional DS equation was investigated. By making use of the Lie symmetry analysis approach, the Lie point symmetries, symmetry groups, similarity reductions and traveling wave solutions were presented. For the first symmetry, the time-fractional DS equation was converted to a (1+1)-dimensional system of fractional partial differential equations. For the second symmetry, the equation was transformed into a system of fractional ordinary differential equations. Meanwhile, the equation can also be converted to a system of fractional ordinary differential equations in the sense of the Erdélyi-Kober fractional integro-differential operators. By application of Noether's theorem and Ibragimov's new conservation theorem, the conserved vectors and conservation laws were derived. Finally, the traveling wave solutions were deduced. The results profoundly revealed that the Lie symmetry analysis approach can be effectively applied to the theoretical research for NFPDEs. It should also be interesting to see if traveling wave solutions could be similarly determined for nonlocal integrable equations, for example, nonlocal NLS models [67].