On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators

We study the existence and uniqueness of the Kadomtsev–Petviashvili (KP) hierarchy solutions in the algebra $\mathcal FCl(S^1,\mathbb K^n)$ of formal classical pseudodifferential operators. The classical algebra $\Psi DO(S^1,\mathbb K^n)$, where the KP hierarchy is well known, appears as a subalgebra of $\mathcal FCl(S^1,\mathbb K^n)$. We investigate algebraic properties of $\mathcal FCl(S^1,\mathbb K^n)$ 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 $\mathcal FCl(S^1,\mathbb K^n)$ 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.