The Pfaff lattice on symplectic matrices

TL;DR

This paper explores the Pfaff lattice on symplectic matrices, revealing that even hierarchy members relate to the indefinite Toda lattice and analyzing fixed points and associated skew-orthogonal polynomials.

Contribution

It introduces the Pfaff lattice for symplectic lower Hessenberg matrices, showing even hierarchy members are equivalent to the indefinite Toda lattice and analyzing their fixed points.

Findings

Odd Pfaff lattice hierarchy members are trivial.

Even hierarchy members are equivalent to the indefinite Toda lattice.

Fixed points are block diagonal matrices with zero diagonals.

Abstract

The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our recent paper [{\it Intern. Math. Res. Notices}, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of a symplectic lower Hessenberg Lax matrix, this forces the Lax matrix to take a tridiagonal shape. We then show that the odd members of the Pfaff lattice hierarchy are trivial, while the even members are equivalent to the indefinite Toda lattice hierarchy defined in [Y. Kodama and J. Ye, {\it Physica D}, {\bf 91} (1996) 321-339]. This is analogous to the case of the Toda lattice hierarchy in the relation to the Kac-van Moerbeke system. In the case with initial matrix having only real or imaginary eigenvalues, the fixed points of the even flows are given by $2\times 2$ block diagonal matrices with zero diagonals. We also consider a family of skew-orthogonal polynomials with symplectic recursion relation related to the Pfaff lattice, and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite Toda lattice.