Some new exact solutions of $(3+1)$ -dimensional Burgers system via Lie symmetry analysis

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 2 The commutator table of $L(G)$

[⋅,⋅]	$X_{1}$	$X_{2}$	$X_{3}$	$X_{4}$	$X_{5}$	$A_{\tilde{f}}$	$B_{\tilde{g}}$	$C_{\tilde{h}}$
$X_{1}$	0	$-X_{3}$	$-X_{1}$	0	$-\frac{1}{2}X_{4}$	$A_{t(t\tilde{f}'-\tilde{f})/2}$	$B_{(t\tilde{g}'-\tilde{g})/2}$	$C_{t^{2}\tilde{h}'/2}$
$X_{2}$	$X_{3}$	0	0	$X_{5}$	0	$A_{\tilde{f}'}$	$B_{\tilde{g}'}$	$C_{\tilde{h}'}$
$X_{3}$	$X_{1}$	0	0	$\frac{1}{2}X_{4}$	$-\frac{1}{2}X_{5}$	$A_{t\tilde{f}'-\tilde{f}/2}$	$B_{t\tilde{g}'-\tilde{g}/2}$	$C_{t\tilde{h}'}$
$X_{4}$	0	$-X_{5}$	$-\frac{1}{2}X_{4}$	0	0	0	0	0
$X_{5}$	$\frac{1}{2}X_{4}$	0	$\frac{1}{2}X_{5}$	0	0	0	0	0
$A_{f}$	$-A_{(tf'-f)/2}$	$-A_{f'}$	$-A_{tf'-f/2}$	0	0	0	0	$B_{\tilde{h}f}$
$B_{g}$	$-B_{(tg'-g)/2}$	$-B_{g'}$	$-B_{tg'-g/2}$	0	0	0	0	$-A_{\tilde{h}g}$
$C_{h}$	$-C_{t^{2}h'/2}$	$-C_{h'}$	$-C_{th'}$	0	0	$-B_{h\tilde{f}}$	$A_{h\tilde{g}}$	0

[⋅,⋅]	\(X_{1}\)	\(X_{2}\)	\(X_{3}\)	\(X_{4}\)	\(X_{5}\)	\(A_{\tilde{f}}\)	\(B_{\tilde{g}}\)	\(C_{\tilde{h}}\)
\(X_{1}\)	0	\(-X_{3}\)	\(-X_{1}\)	0	\(-\frac{1}{2}X_{4}\)	\(A_{t(t\tilde{f}'-\tilde{f})/2}\)	\(B_{(t\tilde{g}'-\tilde{g})/2}\)	\(C_{t^{2}\tilde{h}'/2}\)
\(X_{2}\)	\(X_{3}\)	0	0	\(X_{5}\)	0	\(A_{\tilde{f}'}\)	\(B_{\tilde{g}'}\)	\(C_{\tilde{h}'}\)
\(X_{3}\)	\(X_{1}\)	0	0	\(\frac{1}{2}X_{4}\)	\(-\frac{1}{2}X_{5}\)	\(A_{t\tilde{f}'-\tilde{f}/2}\)	\(B_{t\tilde{g}'-\tilde{g}/2}\)	\(C_{t\tilde{h}'}\)
\(X_{4}\)	0	\(-X_{5}\)	\(-\frac{1}{2}X_{4}\)	0	0	0	0	0
\(X_{5}\)	\(\frac{1}{2}X_{4}\)	0	\(\frac{1}{2}X_{5}\)	0	0	0	0	0
\(A_{f}\)	\(-A_{(tf'-f)/2}\)	\(-A_{f'}\)	\(-A_{tf'-f/2}\)	0	0	0	0	\(B_{\tilde{h}f}\)
\(B_{g}\)	\(-B_{(tg'-g)/2}\)	\(-B_{g'}\)	\(-B_{tg'-g/2}\)	0	0	0	0	\(-A_{\tilde{h}g}\)
\(C_{h}\)	\(-C_{t^{2}h'/2}\)	\(-C_{h'}\)	\(-C_{th'}\)	0	0	\(-B_{h\tilde{f}}\)	\(A_{h\tilde{g}}\)	0