The Pfaff lattice on symplectic matrices

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 earlier paper (Kodama and Pierce 2007 Int. Math. Res. Not. (arXiv:0705.0510)), 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 2 × 2 block 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 Kodama and Ye (1996 Physica D 91 321–39). This is analogous to the case of the Toda lattice hierarchy in relation to the Kac–van Moerbeke system. In the case with the initial matrix having only real or imaginary eigenvalues, the fixed points of the even flows are given by 2 × 2 block diagonal matrices with zero diagonals. We also consider a family of skew-orthogonal polynomials with a 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.