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

Abstract

We review the integration of the KP hierarchy in several nonstandard 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.

Lie Bialgebroid of Pseudo-differential Operators

“…Hence, we obtain non-commutative KP equations in a more general sense and not by using non-commutativity of the coordinates i.e., [x k , x l ] = i kl . Up to now, in special cases, non-commutative KP equations have been solved or it has been shown the existence and uniqueness of solutions of them using Moyal product, see [18] and references there in. The algebraic procedure described during this paper provides the possibility of solving non-commutative KP equations very generally.…”

“…Solving KP equations, in particular showing existence and uniqueness of solutions of KP equations has been an interesting subject of study. For instance, in a recent work in [18] (also see references there in), the Cauchy problem of the KP hierarchy has been solved and formulated in several non-standard cases such as non-commutative case described by Moyal product. Now, with the algebraic construction in this paper, one can expect that the non-commutative KP equations can be solved more generally.…”