On invariant solutions of the generalised Korteweg-de Vries equation in a variable medium—I

Abstract

The invariance under continuous group of transformations of the generalised Korteweg-de Vries equation u t + α(t)f(u)u x + β(t)u xxx = 0 , where α, β and f are arbitrary functions of their respective arguments, has been studied via a generalised procedure. Possible invariant solutions of this equation for physically realisable forms of α(t), β(t) and f(u) are also obtained. This new procedure beside yielding new invariants has provided soliton-like solutions for variable coefficients KdV, modified KdV and the KdV with higher order non-linearity and other new exact solutions which have hitherto been unexplored.