Painlevé Analysis and the Complete Integrability of a Generalized Variable-Coefficient Kadomtsev-Petviashvili Equation

Abstract

In this paper, we discuss the complete integrability of a generalized Kadomtsev-Petviashvili (GKP) equation with space- and time-dependent coefficients, which is known to have a physical application in the propagation of surface waves in straits or channels with varying depth and width. It is shown that the constraints which the variable-coefficient functions must satisfy for the GKP equation to pass the Painlevé tests for complete integrability are precisely the same as those in order that the equation may be transformed into either the Kadomtsev-Petviashvili equation or the Korteweg-de Vries equation, both of which are known to be completely integrable. Therefore we conclude that these are necessary and sufficient conditions for the GKP equation to be completely integrable, and when satisfied we obtain a class of completely integrable equations involving two arbitrary functions of one of the spatial variables (y) and time together with three arbitrary functions of time.