Toda grid

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 :

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}p(n,t)&=e^{-\left(q(n,t)-q(n-1,t)\right)}-e^{-\left(q(n+1,t)-q(n,t)\right)}\\{\frac {\mathrm {d} }{\mathrm {d} t}}q(n,t)&=p(n,t).\end{aligned}}}$

It is

• ${\displaystyle p(n,t)}$the momentum of the -th particle (the mass is normalized to ) ${\displaystyle n}$ ${\displaystyle m=1}$
• ${\displaystyle q(n,t)}$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

${\displaystyle {\begin{aligned}a(n,t)&={\frac {1}{2}}\cdot {\rm {e}}^{-\left(q(n+1,t)-q(n,t)\right)/2}\\b(n,t)&=-{\frac {1}{2}}\cdot p(n,t)\end{aligned}}}$

in which the Toda grid is given by:

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}a(n,t)&=a(n,t)\cdot {\Big (}b(n+1,t)-b(n,t){\Big )}\\{\frac {\mathrm {d} }{\mathrm {d} t}}b(n,t)&=2\cdot {\Big (}a(n,t)^{2}-a(n-1,t)^{2}{\Big )}\end{aligned}}}$

Then you can check that the Toda grid is equivalent to the Lax equation :

${\displaystyle \Leftrightarrow {\frac {\mathrm {d} }{\mathrm {d} t}}L(t)=[P(t),L(t)]}$

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: ${\displaystyle [P,L]=PL-LP}$ ${\displaystyle L}$ ${\displaystyle P}$ ${\displaystyle \ell ^{2}(\mathbb {Z} )}$

${\displaystyle {\begin{aligned}L(t)f(n)&=a(n,t)f(n+1)+a(n,t)f(n-1)+b(n,t)f(n)\\P(t)f(n)&=a(n,t)f(n+1)-a(n,t)f(n-1)\end{aligned}}}$

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 . ${\displaystyle L}$ ${\displaystyle t}$

literature

• 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

• EW Weisstein, Toda Lattice on ScienceWorld
• G. Teschl, The Toda Lattice