Partial bihamiltonian structures (Q6592015)

\textit{F. Magri} [J. Math. Phys. 19, 1156--1162 (1978; Zbl 0383.35065)] introduced the notion of a bihamiltonian structure for integrable systems. An extension of Poisson structures within the realm of Banach spaces was addressed in \textit{A. Odzijewicz} and \textit{T. S. Ratiu}, World Sci. Monogr. Ser. Math. 8, 113--127 (2005; Zbl 1102.53057)], where the authors considered a Poisson bracket on the algebra \(C^{\infty}\left( M\right) \)\ for a smooth manifold \(M\). The notion of partial Poisson structures within the realm of Banach manifolds was considered in [\textit{A. B. Tumpach}, Commun. Math. Phys. 373, No. 3, 795--858 (2020; Zbl 1437.37087)], while its extension to the locally convex framework was investigated in [\textit{K. H. Neeb} et al., Springer Proc. Math. Stat. 111, 105--135 (2014; Zbl 1319.58006)]. The notion of partial Poisson structures within the realm of adaptive spaces in the sense of Frölicher, Kriegl and Michor was studied in [\textit{F. Pelletier} and \textit{P. Cabau}, J. Geom. Phys. 136, 173--194 (2019; Zbl 1414.53074)].\N\NThis paper introduces three notions of partial bihamiltonian structures on an adaptive manifold \(M\)\ equipped with\N\begin{itemize}\N\item[(1)] two compatible partial Poisson structures \(P\)\ and \(Q\)\ defined on the same subfiber \(T^{\flat}M\);\N\item[(2)] a partial Poisson structure \(P\)\ defined over \(T^{\flat}M\)\ and a Nijenhuis tensor compatible with \(P\);\N\item[(3)] a partial Poisson structure \(P\)\ defined over \(T^{\flat}M\)\ and a weak symplectic structure \(\Omega\)\ with values in \(T^{\flat}M\), which abide by the condition\N\[\Nd\left( \Omega P\Omega\right) =0.\N\]\N\end{itemize}