Factorization and Painleve analysis of a class of nonlinear third-order partial differential equations

Abstract

A class of fully nonlinear third-order partial differential equations (PDES) is considered. This class contains several examples which have recently appeared in the literature and for which rather unusual travelling-wave solutions have been given. These solutions consist essentially of sums of exponentials; we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling-wave reduction. In addition, we consider the Painleve analysis of this set of equations, both for the original PDE and also for reductions to ordinary differential equations (ODES). No equation in the class considered survives the combination of PDE and ODE tests. Also, an equation in the class considered which is known to be integrable is shown to possess only the weak Painleve property. Our analysis, therefore, confirms the limitations of the Painleve test as a test for complete integrability when applied to fully nonlinear PDES.