The Toda Lattice , named after Morikazu Toda , is a simple model of a one-dimensional crystal in solid state physics . It models a linear chain of particles in which only nearest neighbors interact with each other, using the associated equation of motion :
It is
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the momentum of the -th particle (the mass is normalized to )
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the deflection of the particle from its rest position .
The toda lattice is an example of a fully integrable system with soliton solutions. To see this, one uses Flaschka variables
in which the Toda grid is given by:
Then you can check that the Toda grid is equivalent to the Lax equation :
Here denotes the commutator of two operators and . These, the Lax pair , are linear operators in the Hilbert space of the square summable sequences , which are given by:
In particular, the Toda lattice can be solved for the Jacobi operator with the aid of the inverse scattering transformation (IST) . The central result says that arbitrary, sufficiently strongly decreasing initial conditions are asymptotically divided into a sum of solitons and a decaying dispersive part for large times .
literature
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G. Teschl , Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs 72, Amer. Math. Soc., Providence, 2000. ISBN 0-8218-1940-2 ( free online version )
- M. Toda, Theory of Nonlinear Lattices, 2nd edition, Springer, Berlin, 1989. ISBN 978-0387102245
Web links