Analysis of Lie symmetries and traveling wave solutions for the (2+1)-dimensional Boussinesq equation with general nonlinearity

Abstract

In this study, we investigate Lie symmetries of the (2+1)-dimensional Boussinesq equation, which has been proposed to model the propagation of gravity waves on the water surface, with particular emphasis on the head-on collision of oblique waves. We consider this equation in a more general form involving an arbitrary function f(u) and establish a complete Lie symmetry classification with respect to the admissible forms of the nonlinearity. For the canonical equations arising from the classification, we construct reductions to ordinary differential equations by using an optimal system of two-dimensional subalgebras. Furthermore, we examine the exact solutions of the equation and analyze the stability of the traveling wave solutions.