Stationary coupled KdV hierarchies and related Poisson structures

Abstract

In this paper we continue our analysis of the stationary flows of M component, coupled KdV (cKdV) hierarchies and their modifications. We describe the general structure of the t1 and t2 flows, using the case M=3 as our main example. One of our stationary reductions gives N degrees of freedom, superintegrable systems. When N=1 (for t1) and N=2 (for t2), we have Poisson maps, which give multi-Hamiltonian representations of the flows. We discuss the general structure of these Poisson tensors and give explicit forms for the case M=3. In this case there are 3 modified hierarchies, each with 4 Poisson brackets.The stationary t2 flow (for N=2) is separable in parabolic coordinates. Each Poisson bracket has rank 4, with M+1 Casimirs. The 4×4 “core” of the Poisson tensors are nonsingular and related by a “recursion operator”. The remaining part of each tensor is built out of the two commuting Hamiltonian vector fields, depending upon the specific Casimirs. The Poisson brackets are generalised to include the entire class of potential, separable in parabolic coordinates. The Jacobi identity imposes specific dependence on some parameters, representing the Casimirs of the extended canonical bracket. This general case is no longer a stationary cKdV flow, with Lax representation. We give a recursive procedure for constructing the Lax representation of the stationary flow for all values of M, without having to go through the stationary reduction.