On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators
Résumé
We study the existence and uniqueness of the Kadomtsev-Petviashvili (KP) hierarchy solutions in the algebra of F Cl(S 1 , K n) of formal classical pseudo-differential operators. The classical algebra ΨDO(S 1 , K n) where the KP hierarchy is well-known appears as a subalgebra of F Cl(S 1 , K n). We investigate algebraic properties of F Cl(S 1 , K n) such as splittings, r-matrices, extension of the Gelfand-Dickii bracket, almost complex structures. Then, we prove the existence and uniqueness of the KP hierarchy solutions in F Cl(S 1 , K n) with respect to extended classes of initial values. Finally, we extend this KP hierarchy to complex order formal pseudo-differential operators and we describe their Hamiltonian structures similarly to previously known formal case. .
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