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

Abstract

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.

arXiv commentAccepted for publication in Theoretical and Mathematical Physics

References in corpus (3)

• Poisson-Lie group of pseudodifferential symbols
• A One-Parameter Family of Hamiltonian Structures for the KP Hierarchy and a Continuous Deformation of the Nonlinear $\W_{\rm KP}$ Algebra
• Non-Local Matrix Generalizations of W-Algebras

Cited by in corpus (1)

• On $(t_2,t_3)-$Zakharov-Shabat equations of generalized Kadomtsev-Petviashvili hierarchies