Non-classical symmetry and new exact solutions of the Kudryashov-Sinelshchikov and modified KdV-ZK equations

In this paper, by applying the non-classical symmetry method, nonclassical symmetries of the Kodryashov-Sinleschikov (K-S) and modified Korteweg-de Vries-Zaharov-Kuznetsov (mKdV-ZK) equations are obtained. Apart from classical symmetries, this theory can be effective in finding a few other solutions for a system of PDEs and ODEs. Also, non-classical symmetries of a system of PDEs can be applied to reduce the number of independent variables. By adding the invariance surface condition to the assumed equations and applying the classical symmetry method for them, non-classical symmetries are calculated. Finally, some of the group invariant solutions and the similarity reduced equations associated to non-classical symmetry are obtained.