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.

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