The Bogoyavlenskii-Schiff hierarchy and integrable equations in (2 + 1) dimensions

In this paper we discuss two problems. Firstly, integrable equations have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. We consider whether the integrable equation in (2 + 1) dimensions — the Bogoyavlenskii-Schiff (BS) equation — has also the analogous hierarchy. We derive the Lax pair of the BS hierarchy. We also investigate the integrability of the 5th order BS equation and find that this equation admits the Painlevé property. Secondly, the study of higher dimensional integrable systems is one of the central themes in integrability. A typical way of constructing higher dimensional integrable systems is to modify Lax operators of basic equation, in this paper — the potential KdV(p-KdV) equation. We modified the L and T operators of the p-KdV equation for the search of (3 + 1) dimensional integrable equation. However, the Lax equation is eventually reduced to (2 + 1)-dimensional one. In addition, we also propose the modified equation and the Lax pair.