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On periodic motions of a two dimensional Toda type chain

Published online by Cambridge University Press:  15 December 2004

Gianni Mancini  and
P. N. Srikanth
Gianni Mancini
Affiliation:
Dipartimento di Mathematica, Universita di Roma Tre, Via S. Leonardo Murialdo 1, Roma, Italy; mancini@matrm3.mat.uniroma3.it
P. N. Srikanth
Affiliation:
TIFR Centre, IISc Campus, PB 1234, Bangalore – 560 012, India; srikanth@math.tifrbng.res.in
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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.$$\left\{\begin{array}{l} \varphi^{i}_{tt} - \varphi^{i}_{xx} = \exp(\varphi^{i+1} -\varphi^{i}) - \exp( \varphi^{i} - \varphi{i-1} ) \quad 0<x<\pi, \quad t \in \rm I\hskip-1.8pt R, i \in Z\!\!\!Z\quad (TC) \varphi^i (0,t) = \varphi^i (\pi,t) = 0 \quad \forall t, i. \end{array}\right.$$ 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.

Keywords

PeriodicToda type chain.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 1 , January 2005 , pp. 72 - 87
Copyright
© EDP Sciences, SMAI, 2005

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