Adler-Kostant-Symes construction, bi-Hamiltonian manifolds, and KdV equations

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This paper focuses a relation between Adler-Kostant-Symes (AKS) theory applied to Fordy-Kulish scheme, and bi-Hamiltonian manifolds. The spirit of this paper is closely related to Casati-Magri-Pedroni work on Hamiltonian formulation of the KP equation. Here the KdV equation is deduced via the superposition of the Fordy-Kulish scheme and AKS construction on the underlying current algebra C(S1,g⊗C[[λ]]). This method is different from the Drinfeld-Sokolov reduction method. It is known that AKS construction is endowed with bi-Hamiltonian structure. In this paper we show that if one applies the Fordy-Kulish construction in the Adler-Kostant-Symes scheme to construct an integrable equation associated with symmetric spaces then this superposition method becomes closer to Casati-Magri-Pedroni's bi-Hamiltonian method of the KP equation. We also add a self-contained Appendix, where we establish a direct relation between AKS scheme and bi-Hamiltonian methods.

Original languageBritish English
Pages (from-to)5167-5182
Number of pages16
JournalJournal of Mathematical Physics
Issue number10
StatePublished - Oct 1997


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