ON (t 2 , t 3 )−ZAKHAROV-SHABAT EQUATIONS OF GENERALIZED KADOMTSEV-PETVIASHVILI HIERARCHIES
Résumé
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.
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