The Toda lattice, introduced by Morikazu Toda (1967), is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system. It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian
H ( p , q ) = ∑ n ∈ Z ( p ( n , t ) 2 2 + V ( q ( n + 1 , t ) − q ( n , t ) ) ) {\displaystyle {\begin{aligned}H(p,q)&=\sum _{n\in \mathbb {Z} }\left({\frac {p(n,t)^{2}}{2}}+V(q(n+1,t)-q(n,t))\right)\end{aligned}}}
and the equations of motion
d d t p ( n , t ) = − ∂ H ( p , q ) ∂ q ( n , t ) = e − ( q ( n , t ) − q ( n − 1 , t ) ) − e − ( q ( n + 1 , t ) − q ( n , t ) ) , d d t q ( n , t ) = ∂ H ( p , q ) ∂ p ( n , t ) = p ( n , t ) , {\displaystyle {\begin{aligned}{\frac {d}{dt}}p(n,t)&=-{\frac {\partial H(p,q)}{\partial q(n,t)}}=e^{-(q(n,t)-q(n-1,t))}-e^{-(q(n+1,t)-q(n,t))},\\{\frac {d}{dt}}q(n,t)&={\frac {\partial H(p,q)}{\partial p(n,t)}}=p(n,t),\end{aligned}}}
where q ( n , t ) {\displaystyle q(n,t)} is the displacement of the n {\displaystyle n} -th particle from its equilibrium position, and p ( n , t ) {\displaystyle p(n,t)} is its momentum (mass m = 1 {\displaystyle m=1} ), and the Toda potential V ( r ) = e − r + r − 1 {\displaystyle V(r)=e^{-r}+r-1} .
Soliton solutions Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way. The general N-soliton solution of the equation is
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