2021/01/09 by Jean-Pierre Magnot, Vladimir Roubtsov, Vladimir Rubtsov
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic number #Algebraic structures and combinatorial models #Class (philosophy) #Combinatorics #Differential operator #Discrete mathematics #Hamiltonian (control theory) #Hierarchy #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Subalgebra #Uniqueness #math-ph #math.MP #math.OA #msc:37K10 #msc:37K20 #msc:37K30 #nlin.SI
paper · pdf · doi:10.1134/s004057792106009x
Accepted for publication in Theoretical and Mathematical Physics
arxiv created 2021/01/09 · openalex publication_date 2021/06/01 · arxiv updated 2021/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the existence and uniqueness of the Kadomtsev–Petviashvili (KP ) hierarchy solutions in the algebra \mathcal FCl(S1,\mathbb Kn) of formal classical pseudodifferential operators. The classical algebra Ψ DO(S1,\mathbb Kn) , where the KP hierarchy is well known, appears as a subalgebra of \mathcal FCl(S1,\mathbb Kn) . We investigate algebraic properties of \mathcal FCl(S1,\mathbb Kn) 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(S1,\mathbb Kn) 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.