Can we always distinguish between positive and negative hierarchies?

Abstract

It is a common belief that nonlinearizable PDEs in (1 + 1) dimensions cannot possess two mutually inverse positive-order recursion operators and that the negative hierarchies for such PDEs, unlike the positive ones, contain at most a finite number of local symmetries. We show that the equation , a generalization of the Hunter–Saxton equation considered by Manna and Neveu, provides a counterexample for both of these assertions. Namely, we find two positive-order integro-differential recursion operators for this equation and show that the corresponding positive and negative hierarchies consist solely of local symmetries. The recursion operators in question turn out to be mutually inverse on symmetries of the equation under study.