Lie symmetry analysis of (2+1)-dimensional KdV equations with variable coefficients

Abstract

In this paper, symmetry groups are used to obtain symmetry reductions of (2+1)-dimensional KdV equations with variable coefficients. Despite the fact that these equations emerge in a nonlocal form, by using suitable transformations, they can be written as systems of partial differential equations, and in potential form, as fourth-order partial differential equations. We show that the point symmetries of the potential equation involve a large number of arbitrary functions. Moreover, these symmetries are used to transform the fourth-order partial differential equations into (1+1)-dimensional fourth-order differential equations. Furthermore, we have determined all two-dimensional solvable symmetry subalgebras, under certain restrictions, which the potential equation admits. Finally, by way of example, taking into account a two-dimensional abelian subalgebra, we obtain a direct reduction of the potential equation to an ordinary differential equation.