ON THE KADOMTSEV-PETVIASHVILI HIERARCHY IN AN EXTENDED CLASS OF FORMAL PSEUDO-DIFFERENTIAL OPERATORS

摘要

We study the existence and uniqueness of the Kadomtsev-Petviashvili (KP) hierarchy solutions in the algebra FCl(S-1, K-n) of formal classical pseudodifferential operators. The classical algebra psi DO(S-1, K-n), where the KP hierarchy is well known, appears as a subalgebra of FCl(S-1, K-n). We investigate algebraic properties of FCl(S-1, K-n) such as splittings, r-matrices, extension of the Gelfand-Dickey bracket, and almost complex structures. We then prove the existence and uniqueness of the KP hierarchy solutions in FCl(S-1, K-n) with respect to extended classes of initial values. Finally, we extend this KP hierarchy to complex-order formal pseudodifferential operators and describe their Hamiltonian structures similarly to the previously known formal case.