On periodic motions of a two dimensional Toda type chain

Author： Mancini Gianni   Srikanth P. N.  

Publisher： Edp Sciences

E-ISSN： 1262-3377|11|1|72-87

ISSN： 1292-8119

Source： ESAIM: Control, Optimisation and Calculus of Variations, Vol.11, Iss.1, 2010-03, pp. : 72-87

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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 0We 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 peirodicin time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

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