On (t(2), t(3))-Zakharov-Shabat equations of generalized Kadomtsev-Petviashvili hierarchies

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.