Solvability of the super KP equation and a generalization of the Birkhoff decomposition

Summary The unique solvability of the initial value problem for the total hierarchy of the super Kadomtsev-Petviashvili system is established. To prove the existence we use a generalization of the Birkhoff decomposition which is obtained by replacing the loop variable and loop groups in the original setting by a super derivation operator and groups of infinite order super micro- (i.e. pseudo-) differential operators. To show the uniqueness we generalize the fact that every flat connection admits horizontal sections to the case of an infinite dimensional super algebra bundle defined over an infinite dimensional super space. The usual KP system with non-commutative coefficients is also studied. The KP system is obtained from the super KP system by reduction modulo odd variables. On the other hand, the first modified KP equation can be obtained from the super KP system by elimination of odd variables. Thus the super KP system is a natural unification of the KP system and the modified KP systems.Research supported in part by NSF Grant No. DMS 86-03175