Codimension two Lie invariant solutions of the modified Khokhlov–Zabolotskaya–Kuznetsov equation

Abstract

Invariant solutions for the modified Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation are obtained by using classical Lie symmetries. The complete set of local point symmetries is established for modified KZK equation governing the propagation of finite amplitude. An optimal set of two‐dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. The optimality among the subalgebras of Lie algebra is proved in a constructive manner by using rank of coefficient matrix of general two‐dimensional element and successive application of adjoint actions. On the basis of these subalgebras, we carry out group invariant reductions and compute exact solutions for different classes of subalgebras in an optimal system. Mathematical and physical behaviors of different invariant solutions are shown graphically demonstrating that classical Lie symmetries are capable of solving the modified KZK equation.