Auto-Bäcklund Transformation, Lax Pairs and Painlevé Property of $u_t+p(t)uu_x+q(t)u_{xxx}+r(t)u=0$ (Journal of Informatics and Mathematical Sciences, 2012)
Using the Painlevé property (PP) of partial differential equations, the auto-Bäcklund transformation (ABT) and Lax pairs for Korteweg-de Vries (KdV) equation with time-dependent coefficients are obtained. The Lax pair criterion also makes it possible for some new models of the variable coefficient KdV equation to be found that can represent nonsoliton dynamical systems. This can explain the wave breaking phenomenon in variable depth shallow water.
Keywords: Korteweg-de Vries (KdV) equation; Painlevé property; Resonances; Exact solutions; Auto-Bäcklund transformation and Lax pairs