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 :
![{\ displaystyle {\ begin {aligned} {\ frac {\ mathrm {d}} {\ mathrm {d} t}} p (n, t) & = e ^ {- \ left (q (n, t) - q (n-1, t) \ right)} - e ^ {- \ left (q (n + 1, t) -q (n, t) \ right)} \\ {\ frac {\ mathrm {d} } {\ mathrm {d} t}} q (n, t) & = p (n, t). \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56e284110b402ea5997c8c22e1458c67a23b6aa8)
It is
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the momentum of the -th particle (the mass is normalized to )![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![m = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173)
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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
![{\ displaystyle {\ begin {aligned} a (n, t) & = {\ frac {1} {2}} \ cdot {\ rm {e}} ^ {- \ left (q (n + 1, t) -q (n, t) \ right) / 2} \\ b (n, t) & = - {\ frac {1} {2}} \ cdot p (n, t) \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71896168b74a91d392e3a4c079087c2a5eca90e8)
in which the Toda grid is given by:
![{\ displaystyle {\ begin {aligned} {\ frac {\ mathrm {d}} {\ mathrm {d} t}} a (n, t) & = a (n, t) \ cdot {\ Big (} b (n + 1, t) -b (n, t) {\ Big)} \\ {\ frac {\ mathrm {d}} {\ mathrm {d} t}} b (n, t) & = 2 \ cdot {\ Big (} a (n, t) ^ {2} -a (n-1, t) ^ {2} {\ Big)} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c25a848cd8a42f3f6049ae8d24a00f44164942)
Then you can check that the Toda grid is equivalent to the Lax equation :
![{\ displaystyle \ Leftrightarrow {\ frac {\ mathrm {d}} {\ mathrm {d} t}} L (t) = [P (t), L (t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f83369ee2248d41178784e93414ffcc3e39516a6)
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:
![L.](https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8)
![{\ displaystyle \ ell ^ {2} (\ mathbb {Z})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c6c1a10e78acb22d6137ef1fad2dec4c15ac01)
![{\ displaystyle {\ begin {aligned} L (t) f (n) & = a (n, t) f (n + 1) + a (n, t) f (n-1) + b (n, t ) f (n) \\ P (t) f (n) & = a (n, t) f (n + 1) -a (n, t) f (n-1) \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22525436e9f310661fb80b6a5274c845d9fe74af)
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 .
![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
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