The Toda lattice, introduced by Morikazu Toda (1967), is a simple model for a one-dimensional crystal in solid state physics.
It is famous because it is one of the earliest examples of a non-linear completely integrable system.
It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian and the equations of motion where
Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way.
The general N-soliton solution of the equation is where with where
The Toda lattice is a prototypical example of a completely integrable system.
To see this one uses Flaschka's variables such that the Toda lattice reads To show that the system is completely integrable, it suffices to find a Lax pair, that is, two operators L(t) and P(t) in the Hilbert space of square summable sequences
such that the Lax equation (where [L, P] = LP - PL is the Lie commutator of the two operators) is equivalent to the time derivative of Flaschka's variables.
has the property that its eigenvalues are invariant in time.
In particular, the Toda lattice can be solved by virtue of the inverse scattering transform for the Jacobi operator L. The main result implies that arbitrary (sufficiently fast) decaying initial conditions asymptotically for large t split into a sum of solitons and a decaying dispersive part.