The Toda lattice, old and new

TL;DR

This paper reviews the Toda lattice's historical development, mathematical structure, and recent advances, highlighting its integrability, asymptotic behavior, and applications in numerical linear algebra.

Contribution

It provides a comprehensive overview of the Toda lattice, including new coordinate systems and their implications for asymptotic analysis and numerical algorithms.

Findings

Bidiagonal coordinates simplify asymptotic analysis.

The Toda lattice's integrability is linked to coadjoint orbits.

Numerical algorithms for the flow are explicitly described.

Abstract

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also non-Jacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra. The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.