# On periodic motions of a two dimensional Toda type chain

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

top

top
In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$\left\{\begin{array}{cc}{\varphi }_{tt}^{i}-{\varphi }_{xx}^{i}=exp\left({\varphi }^{i+1}-{\varphi }^{i}\right)-exp\left({\varphi }^{i}-\varphi i-1\right)\hfill & 0<x<\pi ,\phantom{\rule{1.0em}{0ex}}t\in ℝ,i\in ℤ\phantom{\rule{1.0em}{0ex}}\left(TC\right)\hfill \\ {\varphi }^{i}\left(0,t\right)={\varphi }^{i}\left(\pi ,t\right)=0\hfill & \forall t,i.\hfill \end{array}\right$/extract_itex]We consider the case of “closed chains” i.e. ${\varphi }^{i+N}={\varphi }^{i}\forall i\in ℤ$ and some $N\in ℕ$ 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 top 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

top
1. [1] R.A. Adams, Sobolev Spaces. A.P (1975). Zbl0314.46030MR450957
2. [2] V.I. Arnold, Proof of a Theorem of A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18 (1963) 9–36. Zbl0129.16606
3. [3] H. Brezis and L. Nirenberg, Forced vibrations for a nonlinear wave equation. CPAM, XXXI(1) (1978) 1–30. Zbl0378.35040
4. [4] H. Brezis, J.M. Coron and L. Nirenberg, Free Vibrations for a Nonlinear Wave Equation and a Theorem of P. Rabinowitz. CPAM, XXXIII (1980) 667–684. Zbl0484.35057
5. [5] G. Friesecke and A.D. Wattis Jonathan, Existence Theorem for Solitary Waves on Lattices. Commun. Math. Phys. 161 (1994) 391–418. Zbl0807.35121
6. [6] G. Iooss, Travelling waves in the Fermi-Pasta-Ulam lattice. Nonlinearity 13 (2000) 849–866. Zbl0960.37038
7. [7] M.A. Krasnoselsky and Y.B. Rutitsky, Convex Functions and Orlicz Spaces. Internat. Monogr. Adv. Math. Phys. Hindustan Publishing Corpn., India (1962).
8. [8] H. Lovicarova’, Periodic solutions of a weakly nonlinear wave equation in one dimension. Czechmath. J. 19 (1969) 324–342. Zbl0181.10901
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

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.