On the Integrability of the Toda Lattice

Abstract

This paper presents a variety of computer g.enerated evidence indicating that the Toda lattice behaves remarkably like· an integrable, nonlinear system, where here integrability means that the system Hamiltonian can be brought to an obviously integrable form. In particular, we investigate Toda lattices having three and six particles, using periodic boundary conditions. While computer calculations cannot rigorously prove integrability,. the evidence presented here is sufficiently strong to provide incentive for seeking the general, closed form, analytic so­ lution or, at least, some approximation to it. § I. Introduction The. T oda lattice 1J is of considerable interest to physical scientists because it serves as an example of a nonlinear lattice system which may rigorously be shown to propagafe certain wave forms without change of shape. Moreover the propagation of such unchanging wave forms has been observed experimentally 2> in various physical systems. Additionally in the continuum limit, the T thereby connecting the theory1> o{ the Toda lattice with the KdV soliton theory8> developed by Kruskal, Zabusky, and others. For the KdV equation the unchanging wave form, called a soliton, has been shown8> both theoretically and empirically (on a computer) to be remarkably stable. · Indeed, a recent paper by Zakharov and Faddeev4> definitively establishes the' validity and the source of this stability. In particular these authors' prove that· the KdV equation is a completely integrable Hamiltonian system, where integrability 6> here means that the system Hamiltonian can be reduced to an obvi­ ously integrable (or solvable) form6> by a canonical transformation analytic in the position and momentum variables. Because of the intimate connection7> between the Toda lattice and the KdV equation, the Zakharov-Faddeev result suggests that the Toda lattice may also be integrable and its associated solutions also highly stable; however, extreme caution is needed here. For the class of nonlinear Hamiltonian oscillator systems to which the Toda lattice belongs, Siegel 8) has shown that non-integrability is overwhelmingly the general case. Moreover, Saito9> and coworkers10> have presented computer evi