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.
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.
 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