Symmetries of variable coefficient Korteweg–de Vries equations

Abstract

The Lie point symmetries of the equation ut+f(x, t)uux+g(x,t)uxxx=0 are studied. The symmetry group is shown to be, at most, four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the KdV equation with f=g=1. For nine different classes of functions f and g, the symmetry group turns out to be three dimensional. Two-dimensional and one-dimensional symmetry groups occur for 11 and 15 classes of equations, respectively.