Integrability properties of a generalized reduction of the KdV6 equation

摘要

We consider the integrability properties of a generalized version of a similarity reduction of the so-called KdV6 equation, an equation that has recently generated much interest. We give a linear problem for this generalized reduction and show that it satisfies the requirements of the Ablowitz–Ramani–Segur algorithm. In addition we give a Bäcklund transformation to a related equation, giving also an auto-Bäcklund transformation for this last. Our results mirror those for the Korteweg–de Vries equation itself, which has a similarity reduction to an ordinary differential equation which is related by a Bäcklund transformation to the second Painlevé equation, this last having an auto-Bäcklund transformation.