Stationary Coupled KdV Hierarchies and Related Poisson Structures

TL;DR

This paper analyzes stationary flows of coupled KdV hierarchies, describing their Poisson structures, superintegrable reductions, and providing recursive methods for Lax representations across different hierarchy levels.

Contribution

It introduces explicit Poisson structures for stationary coupled KdV hierarchies, explores superintegrable reductions, and develops recursive procedures for Lax representations without stationary reduction.

Findings

Poisson brackets with rank 4 and Casimirs for stationary flows

Explicit forms of Poisson tensors for M=3 case

Recursive construction of Lax representations for all M

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 $t_1$ and $t_2$ 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 $t_1$) and $N=2$ (for $t_2$), 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 $t_2$ flow (for $N=2$) is separable in parabolic coordinates. Each Poisson bracket has rank 4, with $M+1$ Casimirs. The $4\times 4$ ``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 $M$, {\em without} having to go through the stationary reduction.