On periodic motions of a two dimensional Toda type chain

In this paper we consider a chain of strings with fixed end points coupled by a nearest-neighbour interaction potential of exponential type, i.e. $$ \phi_{tt}^i-\phi_{xx}^i=\exp(\phi^{i+1}-\phi^i)-\exp(\phi^i-\phi^{i-1}),\quad i\in\Bbb Z,$$ $$0<x<\pi, t\in\Bbb R,$$ with periodicity conditions $ \phi^i(0,t)=\phi^i(\pi,t)=0$ for all $t$ and all $i$. We consider the case of `closed chains', i.e. $\phi^{i+N}=\phi^i$ for all $i\in\Bbb Z$ and some $N\in\Bbb N$, and look for solutions which are periodic in time. The existence of periodic solutions for the dual problem is proved in an Orlicz space setting