M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Method for Solving the Sine-Gordon Equation, Physical Review Letters, vol.11, issue.25, pp.1262-1264, 1973.
DOI : 10.1364/AO.11.002572

J. Argyris, M. Haase, and J. C. Heinrich, Finite element approximation to two-dimensional sine-Gordon solitons, Computer Methods in Applied Mechanics and Engineering, vol.86, issue.1, pp.1-26, 1991.
DOI : 10.1016/0045-7825(91)90136-T

V. I. Arnold, Mathematical Methods of Classical Mechanics, 1997.

P. N. Bibikov and L. V. Prokhorov, Mechanics not on a manifold, Journal of Physics A: Mathematical and Theoretical, vol.42, issue.4, pp.45302-45330, 2009.
DOI : 10.1088/1751-8113/42/4/045302

J. L. Bona and R. Cascaval, Nonlinear dispersive waves on trees, Canadian Applied Mathematics Quarterly, vol.16, issue.1 5, pp.1-18, 2008.

A. Bressan, S. Cani?, M. Garavello, M. Herty, and B. Piccoli, Flows on networks: recent results and perspectives, EMS Surveys in Mathematical Sciences, vol.1, issue.1, pp.47-111, 2014.
DOI : 10.4171/EMSS/2

J. Caputo and D. Dutykh, Nonlinear waves in networks: Model reduction for the sine-Gordon equation, Physical Review E, vol.16, issue.2, pp.22912-2014
DOI : 10.1063/1.2979714

J. Caputo, D. Dutykh, and B. Gleyse, Coupling conditions for the nonlinear shallow water equations in forks, pp.1-24, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01206504

R. Courant, K. Friedrichs, and H. Lewy, ???ber die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, vol.98, issue.6, pp.32-74, 1928.
DOI : 10.1002/zamm.19260060408

R. F. Dashen, B. Hasslacher, and A. Neveu, Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods, Physical Review D, vol.10, issue.12, pp.4114-4129, 1974.
DOI : 10.1103/PhysRevD.10.2428

M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Mathematics and Computers in Simulation, vol.79, issue.3, pp.700-715, 2008.
DOI : 10.1016/j.matcom.2008.04.018

D. Dutykh and E. Pelinovsky, Numerical simulation of a solitonic gas in KdV and KdV???BBM equations, Physics Letters A, vol.378, issue.42, pp.3102-3110, 1923.
DOI : 10.1016/j.physleta.2014.09.008
URL : https://hal.archives-ouvertes.fr/hal-00913960

L. D. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons, 1987.
DOI : 10.1007/978-3-540-69969-9

D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, Journal of Computational and Applied Mathematics, vol.134, issue.1-2, pp.37-57, 2001.
DOI : 10.1016/S0377-0427(00)00527-6
URL : https://doi.org/10.1016/s0377-0427(00)00527-6

R. Gould, Graph Theory, p.15, 2012.

A. Grunnet-jepsen, F. N. Fahrendorf, S. A. Hattel, N. Grønbech-jensen, and M. R. Samuelsen, Fluxons in three long coupled Josephson junctions, Physics Letters A, vol.175, issue.2, pp.116-120, 1993.
DOI : 10.1016/0375-9601(93)90132-J

D. Gulevich and F. Kusmartsev, Flux Cloning in Josephson Transmission Lines, Physical Review Letters, vol.97, issue.1, pp.17004-17025, 2006.
DOI : 10.1109/77.622126

D. Gulevich, F. Kusmartsev, S. Savel-'ev, V. , and F. Nori, Shape Waves in 2D Josephson Junctions: Exact Solutions and Time Dilation, Physical Review Letters, vol.25, issue.12, p.127002, 1921.
DOI : 10.1103/PhysRevB.69.064502
URL : http://arxiv.org/pdf/0808.1514

A. Haefliger, Feuilletages sur les vari??t??s ouvertes, Topology, vol.9, issue.2, pp.183-194, 1970.
DOI : 10.1016/0040-9383(70)90040-6
URL : https://doi.org/10.1016/0040-9383(70)90040-6

S. A. Hattel, A. Grunnet-jepsen, and M. R. Samuelsen, Dynamics of three coupled long Josephson junctions, Physics Letters A, vol.221, issue.1-2, pp.115-123, 1996.
DOI : 10.1016/0375-9601(96)00562-2

M. Ilati and M. Dehghan, The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations. Engineering Analysis with Boundary Elements, pp.99-109, 2015.

A. L. Islas and C. M. Schober, Multi-symplectic Spectral Methods for the Sine-Gordon Equation, Computational Science -ICCS 2003, pp.101-110, 2003.
DOI : 10.1007/3-540-44862-4_12

G. Khakimzyanov and D. Dutykh, On supraconvergence phenomenon for second order centered finite differences on non-uniform grids, Journal of Computational and Applied Mathematics, vol.326, pp.1-14, 2017.
DOI : 10.1016/j.cam.2017.05.006
URL : https://hal.archives-ouvertes.fr/hal-01223522

B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol.14, issue.13, p.14, 2005.
DOI : 10.1017/CBO9780511614118
URL : http://cds.cern.ch/record/835066/files/0521772907_TOC.pdf

A. Lew, J. Marsden, M. Ortiz, and M. West, An overview of variational integrators, Finite Element Methods: 1970s and beyond, pp.18-31, 2003.

J. E. Marsden, G. W. Patrick, and S. Shkoller, Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs, Communications in Mathematical Physics, vol.199, issue.2, pp.351-395, 1998.
DOI : 10.1007/s002200050505
URL : http://arxiv.org/pdf/math/9807080

F. A. Mehmeti and V. Régnier, Splitting of energy of dispersive waves in a star-shaped network, ZAMM, vol.83, issue.2, pp.105-118, 2003.
DOI : 10.1002/zamm.200310010

B. Moore and S. Reich, Backward error analysis for multi-symplectic integration methods, Numerische Mathematik, vol.95, issue.4, pp.625-652, 2003.
DOI : 10.1007/s00211-003-0458-9
URL : http://www.ma.ic.ac.uk/~sreich/01_4.pdf

K. Nakajima and Y. Onodera, Logic design of Josephson network. II, Journal of Applied Physics, vol.14, issue.5, p.2958, 1978.
DOI : 10.1063/1.1663738

K. Nakajima, Y. Onodera, and Y. Ogawa, Logic design of Josephson network, Journal of Applied Physics, vol.47, issue.4, pp.1620-1627, 1976.
DOI : 10.1109/PROC.1975.9825

J. Rashidinia and R. Mohammadi, Tension spline solution of nonlinear sine-Gordon equation, Numerical Algorithms, vol.203, issue.1, pp.129-142, 2011.
DOI : 10.1017/CBO9781139172059

L. H. Ryder, Quantum Field Theory, p.28, 1996.
DOI : 10.1017/CBO9780511813900

A. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structures, 2003.

A. Scott, Encyclopedia of Nonlinear Science, p.8, 2004.

H. Susanto and S. A. Van-gils, Existence and stability analysis of solitary waves in a tricrystal junction, Physics Letters A, vol.338, issue.3-5, pp.239-246, 2005.
DOI : 10.1016/j.physleta.2005.02.058

L. A. Takhtadzhyan and L. D. Faddeev, Essentially nonlinear one-dimensional model of classical field theory, Theoretical and Mathematical Physics, vol.262, issue.No. 4, pp.1046-1057, 1974.
DOI : 10.1007/BF01035551

Y. Vassilevskii, S. Simakov, V. Salamatova, Y. Ivanov, and T. Dobroserdova, Numerical issues of modelling blood flow in networks of vessels with pathologies, Russian Journal of Numerical Analysis and Mathematical Modelling, vol.26, issue.6, pp.605-622, 2011.
DOI : 10.1515/rjnamm.2011.036

J. A. Wattis, Variational approximations to breathers in the discrete sine - Gordon equation II: moving breathers and Peierls - Nabarro energies, Nonlinearity, vol.9, issue.6, pp.1583-1598, 1996.
DOI : 10.1088/0951-7715/9/6/011

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yankov, Soliton turbulence, JETP Lett, vol.48, issue.2, pp.79-82, 1988.

E. Zuazua, Control and Stabilization of Waves on 1-d Networks, Lecture Notes in Mathematics, vol.2062, issue.5, pp.463-493, 2013.
DOI : 10.1007/978-3-642-32160-3_9

D. Dutykh, Campus Scientifique , F-73376 Le Bourget-du-Lac Cedex

. Savoie-mont-blanc and L. Cnrs, France E-mail address: Denys.Dutykh@univ-savoie.fr URL

J. Caputo, Avenue de l'Université, Saint-Etienne du Rouvray, F-76801 France E-mail address: caputo@insa-rouen.fr URL: https