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The Toda lattice. I: Existence of integrals. (English) Zbl 0942.37504
Summary: Following recent computer studies, which suggested that the equations of motion of Toda’s exponential lattice should be completely integrable, M. Hénon [ibid. 9, 1921-1923 (1974)] discovered analytical expressions for the constants of the motion. In the present paper, the existence of integrals is proved by a different method. Our approach shows the Toda lattice to be a finite-dimensional analog of the Korteweg-de Vries partial differential equation. Certain integrals of the Toda equations are the counterparts of the conserved quantities of the Korteweg-de Vries equation, and the theory initiated here has been used elsewhere to obtain solutions of the infinite lattice by inverse-scattering methods.

MSC:
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K60 Lattice dynamics; integrable lattice equations
82C99 Time-dependent statistical mechanics (dynamic and nonequilibrium)
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References:
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