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On periodic motions of a two dimensional Toda type chain
Published online by Cambridge University Press: 15 December 2004
Abstract
In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{l} \varphi^{i}_{tt} - \varphi^{i}_{xx} = \exp(\varphi^{i+1} -\varphi^{i}) - \exp( \varphi^{i} - \varphi{i-1} ) \quad 0<x<\pi, \quad t \in \rm I\hskip-1.8pt R, i \in Z\!\!\!Z\quad (TC) \varphi^i (0,t) = \varphi^i (\pi,t) = 0 \quad \forall t, i. \end{array}\right.$$ We consider the case of “closed chains" i.e.$ \varphi^{i+N} = \varphi^i \forall i \in Z\!\!\!Z$ and some $ N \in {I\!\!N}$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.
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- Research Article
- Information
- ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 1 , January 2005 , pp. 72 - 87
- Copyright
- © EDP Sciences, SMAI, 2005