On the integrability properties of variable coefficient Korteweg – de Vries equations

Abstract

All variable coefficient Korteweg – de Vries (KdV) equations with three-dimensional Lie point symmetry groups are investigated. For such an equation to have the Painlevé property, its coefficients must satisfy seven independent partial differential equations. All of them are satisfied only for equations equivalent to the KdV equation itself. However, most of them are satisfied in all cases. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time. Symmetry reduction is used to obtain particular solutions. The reduced ordinary differential equations are classified.