Auto-Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg–de Vries equation. I.

Using the Painlevé property of partial differential equations, the auto-Bäcklund transformation and Lax pairs for a 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. [ABSTRACT FROM AUTHOR]