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

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.