Toda lattice

The Toda lattice, named after Morikazu Toda, is a simple model of a one-dimensional crystal in solid state physics. It models a series of particles, in which only the closest neighbors interact with the associated equation of motion:

The deflection of the - th particle from its rest position and its momentum (the mass ).

The Toda lattice is an example of a completely integrable system with Solitonenlösungen. To see the used one Flaschka variables

In which the Toda lattice by

Is given. Then one can verify easily that the Toda lattice is equivalent to the Lax equation

Is. Herein, [P, L] = PL - LP the commutator of two operators. The operators L and P, the Lax pair of linear operators in Hilbert space of quadratsummierbaren consequences that are by

Are given. In particular, the Toda lattice can be solved using the inverse scattering transform ( IST) for the Jacobi operator L. The main result states that any sufficiently strong decrease in initial conditions is asymptotically for large times t is given by a sum of solitons and a decaying dispersive component.