# 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

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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 (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.\[ \{\left\lbrace \begin\{array\}\{ll\} \varphi ^\{i\}\_\{tt\} - \varphi ^\{i\}\_\{xx\} = \exp (\varphi ^\{i+1\} -\varphi ^\{i\}) - \exp ( \varphi ^\{i\} - \varphi \{i-1\} ) & 0<x<\pi , \quad t \in \mathbb \{R\}, i \in \mathbb \{Z\}\quad (TC)\\ \varphi ^i (0,t) = \varphi ^i (\pi ,t) = 0 &\forall t, i. \end\{array\}\right.\} \]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.\[ {\left\lbrace \begin{array}{ll} \varphi ^{i}_{tt} - \varphi ^{i}_{xx} = \exp (\varphi ^{i+1} -\varphi ^{i}) - \exp ( \varphi ^{i} - \varphi {i-1} ) & 0<x<\pi , \quad t \in \mathbb {R}, i \in \mathbb {Z}\quad (TC)\\ \varphi ^i (0,t) = \varphi ^i (\pi ,t) = 0 &\forall t, i. \end{array}\right.} \]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 -

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- [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). Zbl0679.58021MR994020
- [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
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