Liouville integrable N-Hamiltonian structures, involutive solutions and separation of variables associated with Kaup–Newell hierarchy

Abstract

In this paper, firstly, N-Hamiltonian structures for Kaup–Newell (KN) hierarchy is presented from an isospectral problem via the N pairs of Hamiltonian operators (Lenard operators) and it is shown that the Hamiltonian KN system is Liouville integrable. Secondly, a Lax representation in terms of 2×2 matrices for the completely integrable finite-dimensional Hamiltonian system (CIFHS) ( H) is produced through the non-linearization procedure for the KN hierarchy. By making use of the known r-matrix and matrix trace equality, a system of finite-dimensional involutive functions F m ( m=0,1,2,…, F 0= H) which guarantees the integrability of Hamiltonian system ( H), and the Lax representations in terms of 2×2 matrices for the whole Hamiltonian hierarchies ( F m ) ( m=0,1,2,…) are obtained. Moreover, the involutive solutions of the KN hierarchy are given. Finally, it is found that the Hamilton–Jacobi equation for the Hamiltonian system ( H) can be separated under a group of new coordinates introduced by the 2×2 Lax matrix. In addition, the separation equations are given.