On $(t_2,t_3)-$Zakharov-Shabat equations of generalized Kadomtsev-Petviashvili hierarchies

Abstract

We review the integration of the KP hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation; an algebra of functions equipped with a derivation that generalizes the gradient operator; an algebra of quaternion-valued functions; a differential Lie algebra; an algebra of polynomials equipped with the Pincherle differential; a Moyal algebra. In all these cases we can formulate and solve the Cauchy problem of the KP hierarchy. Also, in each of these cases we derive different Zakharov-Shabat $(t_2,t_3)$-equations -- that is, different Kadomtsev-Petviashvili equations -- and we prove existence of solutions arising from solutions to the corresponding KP hierarchy.