Now, we construct conservation laws for system (2) by using the Lie point symmetry (7).
The vectors \(C_{i}=(C_{i}^{t},C_{i}^{x})\), \((i=1,2,3)\) are called conserved vectors for system (2), if they satisfy the conservation equations,
$$\begin{aligned} &D_{t}\bigl(C_{1}^{t}\bigr)+D_{x} \bigl(C_{1}^{x}\bigr)|_{D_{t}^{\alpha }\mathbf{u}+ \mathbf{u}_{xxx}+3u\mathbf{u}_{x}+3\mathbf{w}\mathbf{w}_{x}=0}=0, \\ &D_{t}\bigl(C_{2}^{t}\bigr)+D_{x} \bigl(C_{2}^{x}\bigr)|_{D_{t}^{\alpha }\mathbf{v}+ \mathbf{v}_{xxx}+3v\mathbf{v}_{x}+3\mathbf{w}\mathbf{w}_{x}=0}=0, \\ &D_{t}\bigl(C_{3}^{t}\bigr)+D_{x} \bigl(C_{3}^{x}\bigr)|_{ D_{t}^{\alpha }\mathbf{w}+ \mathbf{w}_{xxx}+\frac{3}{2}(\mathbf{uw})_{x}+\frac{3}{2}(\mathbf{vw})_{x}=0}=0. \end{aligned}$$
For system (2), a formal Lagrangian can be introduced as
$$\begin{aligned} L&=\Lambda ^{1}(x,t)\bigl[D_{t}^{\alpha }u+ \mathbf{u}_{xxx}+3u\mathbf{u}_{x}+3w \mathbf{w}_{x} \bigr]+\Lambda ^{2}(x,t)\bigl[D_{t}^{\alpha }v+ \mathbf{v}_{xxx}+3v \mathbf{v}_{x}+3\mathbf{w} \mathbf{w}_{x}\bigr] \\ &\quad {}+\Lambda ^{3}(x,t)\biggl[D_{t}^{\alpha }w+ \mathbf{w}_{xxx}+\frac{3}{2}( \mathbf{uw})_{x}+ \frac{3}{2}(\mathbf{vw})_{x}\biggr]=0, \end{aligned}$$
(11)
where \(\Lambda ^{i}(x,t), i=1,2,3\), are new dependent variables.
The Euler–Lagrange operators are defined by
$$\begin{aligned} &\frac{\delta }{\delta u}=\frac{\partial }{\partial \mathbf{u}}+\bigl(D_{t}^{ \alpha } \bigr)^{\star }\frac{\partial }{\partial D_{t}^{\alpha }\mathbf{u}}-D_{x} \frac{\partial }{\partial \mathbf{u}_{x}}+D_{x}^{2} \frac{\partial }{\partial \mathbf{u}_{xx}}-D_{x}^{3} \frac{\partial }{\partial \mathbf{u}_{xxx}}, \\ &\frac{\delta }{\delta v}=\frac{\partial }{\partial \mathbf{v}}+\bigl(D_{t}^{ \alpha } \bigr)^{\star }\frac{\partial }{\partial D_{t}^{\alpha }\mathbf{v}}-D_{x} \frac{\partial }{\partial \mathbf{v}_{x}}+D_{x}^{2} \frac{\partial }{\partial \mathbf{v}_{xx}}-D_{x}^{3} \frac{\partial }{\partial \mathbf{v}_{xxx}}, \\ &\frac{\delta }{\delta w}=\frac{\partial }{\partial \mathbf{w}}+\bigl(D_{t}^{ \alpha } \bigr)^{\star }\frac{\partial }{\partial D_{t}^{\alpha }\mathbf{w}}-D_{x} \frac{\partial }{\partial \mathbf{w}_{x}}+D_{x}^{2} \frac{\partial }{\partial \mathbf{w}_{xx}}-D_{x}^{3} \frac{\partial }{\partial \mathbf{w}_{xxx}}, \end{aligned}$$
here \((D_{t}^{\alpha })^{\star }\) is the adjoint operator of \(D_{t}^{\alpha }\).
For the RL-fractional operators
$$ \bigl(D_{t}^{\alpha }\bigr)^{\star }=(-1)^{n}I_{T}^{n-\alpha } \bigl(D_{t}^{n}\bigr)=_{t}^{C}D_{T}^{ \alpha }, $$
where
$$ I_{T}^{n-\alpha }f(t,x)=\frac{1}{\Gamma (n-\alpha )} \int _{t}^{\tau } \frac{f(\tau ,x)}{(\tau -t)^{1+\alpha -n}} \,\mathrm{d}\tau , \quad n=[ \alpha ]+1. $$
The adjoint equations to the system (2) are written as
$$ F_{1}^{\star }=\frac{\delta L}{\delta u}=0,\qquad F_{2}^{\star }= \frac{\delta L}{\delta v}=0,\qquad F_{3}^{\star }= \frac{\delta L}{\delta w}=0. $$
(12)
Replacing the formal Lagrangian (11) into (12), we have
$$\begin{aligned} &F_{1}^{\star }=\bigl(D_{t}^{\alpha } \bigr)^{\star }\Lambda ^{1}-3u\Lambda ^{1}_{x}- \Lambda ^{1}_{xxx}+\frac{3}{2}\mathbf{w}_{x} \Lambda ^{3}-\frac{3}{2}w \Lambda ^{3}_{x}=0, \\ & F_{2}^{\star }=\bigl(D_{t}^{\alpha } \bigr)^{\star }\Lambda ^{2}-3v\Lambda ^{2}_{x}- \Lambda ^{2}_{xxx}+\frac{3}{2}\mathbf{w}_{x} \Lambda ^{3}-\frac{3}{2}w \Lambda ^{3}_{x}=0, \\ & F_{3}^{\star }=\bigl(D_{t}^{\alpha } \bigr)^{\star }\Lambda ^{3}-3w\Lambda ^{1}_{x}-3w \Lambda ^{2}_{x}+\frac{3}{2}(\mathbf{u}_{x}+ \mathbf{v}_{x})\Lambda ^{3}- \frac{3}{2}(u+v)\Lambda ^{3}_{x}-\Lambda ^{3}_{xxx}=0. \end{aligned}$$
(13)
Since in the system(2), there are no fractional derivatives involved w.r.t. x, we have
$$\begin{aligned} &X^{(\alpha )}+D_{1t}(\tau )L+D_{1x}(\xi )L= \mathbf{w}_{i} \frac{\partial }{\partial \mathbf{u}}+D_{1t}N^{t}_{1}+D_{1x}N^{x}_{1}, \\ &X^{(\alpha )}+D_{2t}(\tau )L+D_{2x}(\xi )L= \mathbf{w}_{i} \frac{\partial }{\partial \mathbf{v}}+D_{2t}N^{t}_{2}+D_{2x}N^{x}_{2}, \\ &X^{(\alpha )}+D_{3t}(\tau )L+D_{3x}(\xi )L= \mathbf{w}_{i} \frac{\partial }{\partial \mathbf{w}}+D_{3t}N^{t}_{3}+D_{3x}N^{x}_{3}, \end{aligned}$$
where
$$ W_{i}=\bigl(\eta ^{\mathbf{u}}+\eta ^{\mathbf{v}}+\eta ^{\mathbf{w}}\bigr)-\xi _{i}( \mathbf{u}_{x}+ \mathbf{v}_{x}+\mathbf{w}_{x})-\tau _{i}( \mathbf{u}_{t}+ \mathbf{v}_{t}+\mathbf{w}_{t}), \quad i=1,2, $$
are the Lie characteristic functions corresponding to the Lie symmetries \(X_{1}\) and \(X_{2}\).
If we have the RL-time-fractional derivative in the system (2) then the operators \(N^{t}\) are given by
$$\begin{aligned} &N_{1}^{t}=\sum_{k=0}^{n-1}(-1)^{k}D_{1t}^{\alpha -1-k}( \mathbf{w}_{i})D_{1t}^{k} \frac{\partial }{(\partial D_{t}^{\alpha }\mathbf{u})}-(-1)^{n}J \biggl(W_{i},D_{1t}^{n} \frac{\partial }{(\partial D_{t}^{\alpha }u)} \biggr), \\ &N_{2}^{t}=\sum_{k=0}^{n-1}(-1)^{k}D_{2t}^{\alpha -1-k}(W_{i})D_{2t}^{k} \frac{\partial }{(\partial D_{t}^{\alpha }\mathbf{v})}-(-1)^{n}J \biggl(W_{i},D_{2t}^{n} \frac{\partial }{(\partial D_{t}^{\alpha }v)} \biggr), \\ &N_{3}^{t}=\sum_{k=0}^{n-1}(-1)^{k}D_{1t}^{\alpha -1-k}(W_{i})D_{3t}^{k} \frac{\partial }{(\partial D_{t}^{\alpha }\mathbf{w})}-(-1)^{n}J \biggl(W_{i},D_{3t}^{n} \frac{\partial }{(\partial D_{t}^{\alpha }w)} \biggr), \end{aligned}$$
where J is the integral
$$ J(f,g)=\frac{1}{\Gamma (n-\alpha )} \int _{0}^{t} \int _{t}^{\tau } \frac{f(\tau ,x)g(\mu ,x)}{(\mu -\tau )^{\alpha +1-n}} \,\mathrm{d} \mu \,\mathrm{d}\tau ; $$
the operators \(N^{x}\) are defined by
$$\begin{aligned} &N_{1}^{x}=W_{i}\frac{\partial L}{\partial \mathbf{u}_{x}}+D_{1x}(W_{i}) \frac{\partial L}{\partial \mathbf{u}_{xx}}+D_{1x}^{2}(W_{i}) \frac{\partial L}{\partial \mathbf{u}_{xxx}}, \\ &N_{2}^{x}=W_{i}\frac{\partial L}{\partial \mathbf{v}_{x}}+D_{2x}(W_{i}) \frac{\partial L}{\partial \mathbf{v}_{xx}}+D_{2x}^{2}(W_{i}) \frac{\partial L}{\partial \mathbf{v}_{xxx}}, \\ &N_{3}^{x}=W_{i}\frac{\partial L}{\partial \mathbf{w}_{x}}+D_{3x}(W_{i}) \frac{\partial L}{\partial \mathbf{w}_{xx}}+D_{3x}^{2}(W_{i}) \frac{\partial L}{\partial \mathbf{w}_{xxx}}. \end{aligned}$$
For any generator X of system (2), we have
$$\begin{aligned} &\bigl(X^{(\alpha )}L+D_{1t}(\tau )L+D_{1x}(\xi )L \bigr)|_{D_{t}^{\alpha }u+ \mathbf{u}_{xxx}+3u\mathbf{u}_{x}+3w\mathbf{w}_{x}=0}=0, \\ &\bigl(X^{(\alpha )}L+D_{2t}(\tau )L+D_{2x}(\xi )L \bigr)|_{D_{t}^{\alpha }v+ \mathbf{v}_{xxx}+3v\mathbf{v}_{x}+3w\mathbf{w}_{x}=0}=0, \\ &\bigl(X^{(\alpha )}L+D_{3t}(\tau )L+D_{3x}(\xi )L \bigr)|_{ D_{t}^{\alpha }w+ \mathbf{w}_{xxx}+\frac{3}{2}(uw)_{x}+\frac{3}{2}(vw)_{x}=0}=0. \end{aligned}$$
These equalities yield the conservation laws
$$\begin{aligned} &D_{1t}\bigl(N_{1}^{t}L\bigr)+D_{1x} \bigl(N_{1}^{x}L\bigr)=0, \\ &D_{2t}\bigl(N_{2}^{t}L\bigr)+D_{2x} \bigl(N_{2}^{x}L\bigr)=0, \\ &D_{3t}\bigl(N_{3}^{t}L\bigr)+D_{3x} \bigl(N_{3}^{x}L\bigr)=0. \end{aligned}$$
For the case, when \(\alpha \in (0,1)\), using \(N^{t}_{i}\) and \(N^{x}_{i}\) (\(i=1,2,3\)), one can get the components of the conserved vectors
$$\begin{aligned} &C_{1}^{t}=(-1)^{0}D_{1t}^{\alpha -1}(W_{i})D_{1t}^{0} \frac{\partial L}{\partial D_{t}^{\alpha }u}-(-1)^{1}J \biggl(W_{i},D_{1t}^{1} \frac{\partial L}{\partial D_{t}^{\alpha }u} \biggr)=\Lambda ^{1}D_{1t}^{ \alpha -1}(W_{i})+J \bigl(W_{i},\Lambda _{t}^{1}\bigr), \\ &C_{2}^{t}=(-1)^{0}D_{2t}^{\alpha -1}(W_{i})D_{2t}^{0} \frac{\partial L}{\partial D_{t}^{\alpha }v}-(-1)^{1}J \biggl(W_{i},D_{2t}^{1} \frac{\partial L}{\partial D_{t}^{\alpha }v} \biggr)=\Lambda ^{2}D_{2t}^{ \alpha -1}(W_{i})+J \bigl(W_{i},\Lambda _{t}^{2}\bigr), \\ &C_{3}^{t}=(-1)^{0}D_{3t}^{\alpha -1}(W_{i})D_{3t}^{0} \frac{\partial L}{\partial D_{t}^{\alpha }w}-(-1)^{1}J \biggl(W_{i},D_{3t}^{1} \frac{\partial L}{\partial D_{t}^{\alpha }w} \biggr)=\Lambda ^{3}D_{1t}^{ \alpha -1}(W_{i})+J \bigl(W_{i},\Lambda _{t}^{3}\bigr), \end{aligned}$$
and
$$\begin{aligned}& \begin{aligned}[b] C_{1}^{x}&=W_{i}\biggl(\frac{\partial L}{\partial \mathbf{u}_{x}}-D_{1x} \frac{\partial L}{\partial \mathbf{u}_{xx}}+D_{1x}^{2} \frac{\partial L}{\partial \mathbf{u}_{xxx}} \biggr)+D_{1x}(W_{i}) \biggl( \frac{\partial L}{\partial \mathbf{u}_{xx}}-D_{1x} \frac{\partial L}{\partial \mathbf{u}_{xxx}}\biggr)\\ &\quad {}+D_{1x}^{2}(W_{i}) \frac{\partial }{\partial \mathbf{u}_{xxx}} \\ & =W_{i}\biggl(3u\Lambda ^{1}+\frac{3}{2}w \Lambda ^{3}+\Lambda ^{1}_{xx} \biggr)-D_{1x}(W_{i}) \Lambda ^{1}_{x}+D_{1x}^{2}(W_{i}) \Lambda ^{1} , \end{aligned} \end{aligned}$$
(14)
$$\begin{aligned}& \begin{aligned}[b] C_{2}^{x}&=W_{i}\biggl(\frac{\partial L}{\partial \mathbf{v}_{x}}-D_{2x} \frac{\partial L}{\partial \mathbf{v}_{xx}}+D_{2x}^{2} \frac{\partial L}{\partial \mathbf{v}_{xxx}} \biggr)+D_{2x}(W_{i}) \biggl( \frac{\partial L}{\partial \mathbf{v}_{xx}}-D_{2x} \frac{\partial L}{\partial \mathbf{v}_{xxx}}\biggr)\\ &\quad {}+D_{2x}^{2}(W_{i}) \frac{\partial }{\partial \mathbf{v}_{xxx}} \\ & =\mathbf{w}_{i}\biggl(3v\Lambda ^{2}+ \frac{3}{2}w\Lambda ^{3}+ \Lambda ^{2}_{xx} \biggr)-D_{2x}(W_{i})\Lambda ^{2}_{x}+D_{2x}^{2}(W_{i}) \Lambda ^{2} , \end{aligned} \end{aligned}$$
(15)
$$\begin{aligned}& \begin{aligned}[b] C_{3}^{x}&=W_{i}\biggl(\frac{\partial L}{\partial \mathbf{w}_{x}}-D_{3x} \frac{\partial L}{\partial \mathbf{w}_{xx}}+D_{3x}^{2} \frac{\partial L}{\partial \mathbf{w}_{xxx}}\biggr)+ D_{3x}(W_{i}) \biggl( \frac{\partial L}{\partial \mathbf{w}_{xx}}-D_{3x} \frac{\partial L}{\partial \mathbf{w}_{xxx}}\biggr)\\ &\quad {}+D_{3x}^{2}(W_{i}) \frac{\partial }{\partial \mathbf{w}_{xxx}} \\ & =W_{i}\biggl(3w\Lambda ^{1}+3w\Lambda ^{2}+\frac{3}{2}u\Lambda ^{3}+ \frac{3}{2}v \Lambda ^{3}+\Lambda ^{3}_{xx} \biggr)-D_{3x}(W_{i})\Lambda ^{3}_{x}+D_{3x}^{2}(W_{i}) \Lambda ^{3}, \end{aligned} \end{aligned}$$
(16)
where \(i=1,2\) and the functions \(W_{i}\) are
$$\begin{aligned} &W_{1}=-(\mathbf{u}_{x}+ \mathbf{v}_{x}+\mathbf{w}_{x}), \\ &W_{2}=-2\alpha u-2\alpha v-2\alpha w-\alpha x(\mathbf{u}_{x}+ \mathbf{v}_{x}+\mathbf{w}_{x})-3t(\mathbf{u}_{t}+ \mathbf{v}_{t}+ \mathbf{w}_{t}). \end{aligned}$$
(17)
Also, when \(\alpha \in (1,2)\), we get the components of the conserved vectors
$$\begin{aligned} &C_{1}^{t}=\Lambda ^{1}D_{1t}^{\alpha -1}( \mathbf{w}_{i})+J\bigl(W_{i}, \Lambda ^{1}_{t} \bigr)-\Lambda ^{1}_{t}D_{1t}^{\alpha -2}(W_{i})-J \bigl(W_{i}, \Lambda ^{1}_{tt}\bigr), \\ &C_{2}^{t}=\Lambda ^{2}D_{2t}^{\alpha -1}(W_{i})+J \bigl(W_{i},\Lambda ^{2}_{t}\bigr)- \Lambda ^{2}_{t}D_{2t}^{\alpha -2}(W_{i})-J \bigl(W_{i},\Lambda ^{2}_{tt}\bigr), \\ &C_{3}^{t}=\Lambda ^{3}D_{3t}^{\alpha -1}(W_{i})+J \bigl(W_{i},\Lambda ^{3}_{t}\bigr)- \Lambda ^{3}_{t}D_{3t}^{\alpha -2}(W_{i})-J \bigl(W_{i},\Lambda ^{3}_{tt}\bigr), \end{aligned}$$
where \(i=1,2\) and the functions \(W_{i}\) in the form (17); also the conserved vectors \(C_{1}^{x}\), \(C_{2}^{x}\), \(C_{3}^{x}\) coincide with (14), (15) and (16).