# Toda lattice

The Toda lattice, introduced by Morikazu Toda (1967), is a simple model for a one-dimensional crystal in solid state physics. It is given by a chain of particles with nearest neighbor interaction described by the equations of motion

{\displaystyle {\begin{aligned}{\frac {d}{dt}}p(n,t)&=e^{-(q(n,t)-q(n-1,t))}-e^{-(q(n+1,t)-q(n,t))},\\{\frac {d}{dt}}q(n,t)&=p(n,t),\end{aligned}}}

where ${\displaystyle q(n,t)}$ is the displacement of the ${\displaystyle n}$-th particle from its equilibrium position, and ${\displaystyle p(n,t)}$ is its momentum (mass ${\displaystyle m=1}$).

The Toda lattice is a prototypical example of a completely integrable system with soliton solutions. To see this one uses Flaschka's variables

${\displaystyle a(n,t)={\frac {1}{2}}{\rm {e}}^{-(q(n+1,t)-q(n,t))/2},\qquad b(n,t)=-{\frac {1}{2}}p(n,t)}$

such that the Toda lattice reads

{\displaystyle {\begin{aligned}{\dot {a}}(n,t)&=a(n,t){\Big (}b(n+1,t)-b(n,t){\Big )},\\{\dot {b}}(n,t)&=2{\Big (}a(n,t)^{2}-a(n-1,t)^{2}{\Big )}.\end{aligned}}}

Then one can verify that the Toda lattice is equivalent to the Lax equation

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

where [LP] = LP - PL is the commutator of two operators. The operators L and P, the Lax pair, are linear operators in the Hilbert space of square summable sequences ${\displaystyle \ell ^{2}(\mathbb {Z} )}$ given by

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

The matrix ${\displaystyle L(t)}$ has the property that its eigenvalues are invariant in time. These eigenvalues constitute independent integrals of motion, therefore the Toda lattice is completely integrable. 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.