On periodic motions of a two dimensional Toda type chain

Gianni Mancini; P. N. Srikanth

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • 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 ∈ ℝ , i ∈ ℤ ( T C ) 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 and some N ∈ 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 (2005): 72-87. <http://eudml.org/doc/244737>.

@article{Mancini2005,
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.\[ \{\left\lbrace \begin\{array\}\{ll\} \varphi ^\{i\}\_\{tt\} - \varphi ^\{i\}\_\{xx\} = \exp (\varphi ^\{i+1\} -\varphi ^\{i\}) - \exp ( \varphi ^\{i\} - \varphi \{i-1\} ) & 0&lt;x&lt;\pi , \quad t \in \mathbb \{R\}, i \in \mathbb \{Z\}\quad (TC)\\ \varphi ^i (0,t) = \varphi ^i (\pi ,t) = 0 &\forall t, i. \end\{array\}\right.\} \]We consider the case of “closed chains” i.e. $ \varphi ^\{i+N\} = \varphi ^i \forall i \in \mathbb \{Z\}$ and some $ N \in \mathbb \{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},
number = {1},
pages = {72-87},
publisher = {EDP-Sciences},
title = {On periodic motions of a two dimensional Toda type chain},
url = {http://eudml.org/doc/244737},
volume = {11},
year = {2005},
}

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
PY - 2005
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.\[ {\left\lbrace \begin{array}{ll} \varphi ^{i}_{tt} - \varphi ^{i}_{xx} = \exp (\varphi ^{i+1} -\varphi ^{i}) - \exp ( \varphi ^{i} - \varphi {i-1} ) & 0&lt;x&lt;\pi , \quad t \in \mathbb {R}, i \in \mathbb {Z}\quad (TC)\\ \varphi ^i (0,t) = \varphi ^i (\pi ,t) = 0 &\forall t, i. \end{array}\right.} \]We consider the case of “closed chains” i.e. $ \varphi ^{i+N} = \varphi ^i \forall i \in \mathbb {Z}$ and some $ N \in \mathbb {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/244737
ER -

References

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  9. [9] J. Moser, On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, K1 2 (1962) 1. Zbl0107.29301MR147741
  10. [10] A.V. Mikhailov, Integrability of a Two-Dimensional Generalization of the Toda Chain. JETP Lett. 30 (1979) 414–413. 
  11. [11] L. Nirenberg, Variational Methods in nonlinear problems. M. Giaquinta Ed., Springer-Verlag, Lect. Notes Math. 1365 (1987). Zbl0679.58021MR994020
  12. [12] P.H. Rabinowitz, Periodic solutions of Hamiltonian Systems. Comm. Pure Appl. Math. 31 (1978) 157–184. Zbl0358.70014
  13. [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. [14] M. Toda, Theory of Nonlinear Lattices. Springer-Verlag (1989). Zbl0694.70001MR971987

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