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Toda Lattice

In 1967, Morikazu Toda introduced a simple model for a one-dimensional crystal, known as the Toda lattice, consisting of a chain of particles with exponential nearest-neighbor interactions.

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From the Hamiltonian
,
we can derive the equations of motion
,
,
where represents the displacement of the particle from its equilibrium position and is its momentum. The Toda lattice shows solitonic behavior that makes it a significant example in the theory of integrable systems.
Reference
[1] G. Teschl, "Almost Everything You Always Wanted to Know about the Toda Equation," Jahresbericht der Deutschen Mathematiker-Vereinigung, 103(4), 2001 pp. 149–162. www.mat.univie.ac.at/~gerald/ftp/articles/Toda.html
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