The transformations \(u(x,y,t)=U(\zeta )\) and \(v(x,y,t)=V(\zeta )\), where \(\zeta =\alpha x+\beta y-\mu t\) in Eq. (1), lead to the NLODE system presented below.
$$\begin{aligned}{}&\beta U^{\prime }(\zeta )=\alpha V^{\prime }(\zeta ) \\&\quad \frac{3}{2} a^2 \alpha U(\zeta )^2 U^{\prime }(\zeta )+3 a \alpha V(\zeta ) U^{\prime }(\zeta )-6 \alpha b U(\zeta ) U^{\prime }(\zeta )-\mu U^{\prime }(\zeta )-\alpha ^3 U^{(3)}(\zeta )-3 \beta V^{\prime }(\zeta )=0. \end{aligned}$$
The first equation is integrated, and the result is
$$\begin{aligned} \beta U(\zeta )=\alpha V(\zeta )….
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