On periodic motions of a two dimensional Toda type chain

Gianni Mancini; P. N. Srikanth

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 1, page 72-87
  • ISSN: 1292-8119

Abstract

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In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e. ϕ t t i - ϕ x x i = exp ( ϕ i + 1 - ϕ i ) - exp ( ϕ i - ϕ i - 1 ) 0 < x < π , t I R , i Z Z ( TC ) ϕ i ( 0 , t ) = ϕ i ( π , t ) = 0 t , i . We consider the case of “closed chains" i.e. ϕ i + N = ϕ i i Z Z and some N 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.

How to cite

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Mancini, Gianni, and Srikanth, P. N.. "On periodic motions of a two dimensional Toda type chain." ESAIM: Control, Optimisation and Calculus of Variations 11.1 (2010): 72-87. <http://eudml.org/doc/90757>.

@article{Mancini2010,
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. },
author = {Mancini, Gianni, Srikanth, P. N.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Periodic; Toda type chain.; Toda chain; periodic solutions; mountain pass},
language = {eng},
month = {3},
number = {1},
pages = {72-87},
publisher = {EDP Sciences},
title = {On periodic motions of a two dimensional Toda type chain},
url = {http://eudml.org/doc/90757},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Mancini, Gianni
AU - Srikanth, P. N.
TI - On periodic motions of a two dimensional Toda type chain
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 1
SP - 72
EP - 87
AB - 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.
LA - eng
KW - Periodic; Toda type chain.; Toda chain; periodic solutions; mountain pass
UR - http://eudml.org/doc/90757
ER -

References

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  9. J. Moser, On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, K12 (1962) 1.  Zbl0107.29301
  10. A.V. Mikhailov, Integrability of a Two-Dimensional Generalization of the Toda Chain. JETP Lett.30 (1979) 414–413.  
  11. L. Nirenberg, Variational Methods in nonlinear problems. M. Giaquinta Ed., Springer-Verlag, Lect. Notes Math.1365 (1987).  Zbl0679.58021
  12. P.H. Rabinowitz, Periodic solutions of Hamiltonian Systems. Comm. Pure Appl. Math.31 (1978) 157–184.  Zbl0358.70014
  13. B. Ruf and P.N. Srikanth, On periodic Motions of Lattices of Toda Type via Critical Point Theory. Arch. Ration. Mech. Anal.126 (1994) 369–385.  Zbl0809.34056
  14. M. Toda, Theory of Nonlinear Lattices. Springer-Verlag (1989).  Zbl0694.70001

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