On (t2, t3)-Zakharov–Shabat equations of generalized Kadomtsev–Petviashvili hierarchies

Abstract

We review the integration of the Kadomtsev–Petviashvili (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, and a Moyal algebra. In all these cases, we can formulate and solve the Cauchy problem of the KP hierarchy. In addition, in each of these cases, we derive different Zakharov–Shabat (t2, t3)-equations—that is, different Kadomtsev–Petviashvili equations—and we prove the existence of solutions arising from solutions to the corresponding KP hierarchy.