Symmetry reductions of the $(3+1)$ -dimensional modified Zakharov–Kuznetsov equation

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 2 Adjoint representation of the Lie algebra of Eq. (2)

	$\mathcal{V}_{1}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$\mathcal{V}_{5}$	$\mathcal{V}_{6}$
$\mathcal{V}_{1}$	$\mathcal{V}_{1}$	$\Delta _{1} $	$e^{\frac{\varepsilon }{3}}\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$e^{\frac{\varepsilon }{3}}\mathcal{V}_{5}$	$e^{\frac{\varepsilon }{3}}\mathcal{V}_{6}$
$\mathcal{V}_{2}$	$\mathcal{V}_{1}-\varepsilon \mathcal{V}_{2}+\frac{2k}{3}\varepsilon \mathcal{V}_{3}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$\mathcal{V}_{5}$	$\mathcal{V}_{6}$
$\mathcal{V}_{3}$	$\mathcal{V}_{1}-\frac{\varepsilon }{3}\mathcal{V}_{3}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$\mathcal{V}_{5}$	$\mathcal{V}_{6}$
$\mathcal{V}_{4}$	$\mathcal{V}_{1}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$\Delta _{2} $	$\Delta _{3} $
$\mathcal{V}_{5}$	$\mathcal{V}_{1}-\frac{1}{3}\varepsilon \mathcal{V}_{5}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}+\varepsilon \mathcal{V}_{6}$	$\mathcal{V}_{5}$	$\mathcal{V}_{6}$
$\mathcal{V}_{6}$	$\mathcal{V}_{1}-\frac{1}{3}\varepsilon \mathcal{V}_{6}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}-\varepsilon \mathcal{V}_{5}$	$\mathcal{V}_{5}$	$\mathcal{V}_{6}$

	\(\mathcal{V}_{1}\)	\(\mathcal{V}_{2}\)	\(\mathcal{V}_{3}\)	\(\mathcal{V}_{4}\)	\(\mathcal{V}_{5}\)	\(\mathcal{V}_{6}\)
\(\mathcal{V}_{1}\)	\(\mathcal{V}_{1}\)	\(\Delta _{1} \)	\(e^{\frac{\varepsilon }{3}}\mathcal{V}_{3}\)	\(\mathcal{V}_{4}\)	\(e^{\frac{\varepsilon }{3}}\mathcal{V}_{5}\)	\(e^{\frac{\varepsilon }{3}}\mathcal{V}_{6}\)
\(\mathcal{V}_{2}\)	\(\mathcal{V}_{1}-\varepsilon \mathcal{V}_{2}+\frac{2k}{3}\varepsilon \mathcal{V}_{3}\)	\(\mathcal{V}_{2}\)	\(\mathcal{V}_{3}\)	\(\mathcal{V}_{4}\)	\(\mathcal{V}_{5}\)	\(\mathcal{V}_{6}\)
\(\mathcal{V}_{3}\)	\(\mathcal{V}_{1}-\frac{\varepsilon }{3}\mathcal{V}_{3}\)	\(\mathcal{V}_{2}\)	\(\mathcal{V}_{3}\)	\(\mathcal{V}_{4}\)	\(\mathcal{V}_{5}\)	\(\mathcal{V}_{6}\)
\(\mathcal{V}_{4}\)	\(\mathcal{V}_{1}\)	\(\mathcal{V}_{2}\)	\(\mathcal{V}_{3}\)	\(\mathcal{V}_{4}\)	\(\Delta _{2} \)	\(\Delta _{3} \)
\(\mathcal{V}_{5}\)	\(\mathcal{V}_{1}-\frac{1}{3}\varepsilon \mathcal{V}_{5}\)	\(\mathcal{V}_{2}\)	\(\mathcal{V}_{3}\)	\(\mathcal{V}_{4}+\varepsilon \mathcal{V}_{6}\)	\(\mathcal{V}_{5}\)	\(\mathcal{V}_{6}\)
\(\mathcal{V}_{6}\)	\(\mathcal{V}_{1}-\frac{1}{3}\varepsilon \mathcal{V}_{6}\)	\(\mathcal{V}_{2}\)	\(\mathcal{V}_{3}\)	\(\mathcal{V}_{4}-\varepsilon \mathcal{V}_{5}\)	\(\mathcal{V}_{5}\)	\(\mathcal{V}_{6}\)