# 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

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topMancini, 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

top- R.A. Adams, Sobolev Spaces. A.P (1975).
- 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.
- H. Brezis and L. Nirenberg, Forced vibrations for a nonlinear wave equation. CPAM, XXXI(1) (1978) 1–30.
- 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.
- G. Friesecke and A.D. Wattis Jonathan, Existence Theorem for Solitary Waves on Lattices. Commun. Math. Phys.161 (1994) 391–418.
- G. Iooss, Travelling waves in the Fermi-Pasta-Ulam lattice. Nonlinearity13 (2000) 849–866.
- M.A. Krasnoselsky and Y.B. Rutitsky, Convex Functions and Orlicz Spaces. Internat. Monogr. Adv. Math. Phys. Hindustan Publishing Corpn., India (1962).
- H. Lovicarova', Periodic solutions of a weakly nonlinear wave equation in one dimension. Czechmath. J.19 (1969) 324–342.
- J. Moser, On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, K12 (1962) 1.
- A.V. Mikhailov, Integrability of a Two-Dimensional Generalization of the Toda Chain. JETP Lett.30 (1979) 414–413.
- L. Nirenberg, Variational Methods in nonlinear problems. M. Giaquinta Ed., Springer-Verlag, Lect. Notes Math.1365 (1987).
- P.H. Rabinowitz, Periodic solutions of Hamiltonian Systems. Comm. Pure Appl. Math.31 (1978) 157–184.
- 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.
- M. Toda, Theory of Nonlinear Lattices. Springer-Verlag (1989).

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