E. Afshari and A. Hajimiri, Nonlinear transmission lines for pulse shaping in silicon, IEEE Journal of Solid-State Circuits, vol.40, pp.744-752, 2005.

K. Ahnert and A. Pikovsky, Compactons and chaos in strongly nonlinear lattices, Phys. Rev. E, vol.79, p.26209, 2009.

F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, The limit as p ? ? in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles, Calc. Var, vol.35, pp.279-316, 2009.

D. Bambusi and A. Ponno, On metastability in FPU, Comm. Math. Phys, vol.264, pp.539-561, 2006.

I. Bialynicki-birula and J. Mycielski, Nonlinear wave mechanics, Annals of Physics, vol.100, p.62, 1976.

B. Bidégaray-fesquet, E. Dumas, and G. James, From Newton's cradle to the discrete p-Schrödinger equation, SIAM J. Math. Anal, vol.45, pp.3404-3430, 2013.

A. Björck, Numerical methods for least squares problems, 1996.

L. Bonanomi, G. Theocharis, and C. Daraio, Wave propagation in granular chains with local resonances, Phys. Rev. E, vol.91, p.33208, 2015.

J. P. Boyd and G. Chen, Weakly nonlinear wavepackets in the Korteweg-de Vries equation: the KdV/NLS connection, vol.55, pp.317-328, 2001.

, The Fermi-Pasta-Ulam problem : the first 50 years, Chaos, vol.15, 2005.

T. Cazenave and A. Haraux, Equations d'´ evolution avec non linéarité logarithmique, Annales Faculté des Sciences Toulouse, vol.2, pp.21-51, 1980.

A. Chatterjee, Asymptotic solution for solitary waves in a chain of elastic spheres, Phys. Rev. E, vol.59, pp.5912-5919, 1999.

Q. Dou, J. Cuevas, J. C. Eilbeck, and F. M. Russell, Breathers and kinks in a simulated breather experiment, Discrete Contin. Dyn. Syst. Ser. S, vol.4, pp.1107-1118, 2011.

O. A. Dubovsky and A. V. Orlov, Emission of supersonic soliton wave beams-generators of restructuring of nanocrystals under atom bombardment, and the self-organization of a dynamic superlattice of complexes of soliton atomic vibrations, Phys. of solid state, vol.52, pp.899-903, 2010.

J. M. English and R. L. Pego, On the solitary wave pulse in a chain of beads, Proc. Amer. Math. Soc, vol.133, issue.6, pp.1763-1768, 2005.

M. Fe?kan and V. M. Rothos, Travelling waves of discrete nonlinear Schrödinger equations with nonlocal interactions, Applicable Analysis, vol.89, pp.1387-1411, 2010.

M. Fe?kan and V. M. Rothos, Travelling waves of forced discrete nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. S, vol.4, pp.1129-1145, 2011.

S. Flach, Conditions of the existence of localized excitations in nonlinear discrete systems, Phys. Rev. E, vol.50, pp.3134-3142, 1994.

S. Flach, Existence of localized excitations in nonlinear Hamiltonian lattices, Phys. Rev. E, vol.51, pp.1503-1507, 1995.

S. Flach and A. Gorbach, Discrete breathers : advances in theory and applications, Physics Reports, vol.467, pp.1-116, 2008.

S. Flach and K. Kladko, Moving discrete breathers ?, Physica D, vol.127, pp.61-72, 1999.
DOI : 10.1016/s0167-2789(98)00274-7

URL : http://arxiv.org/pdf/cond-mat/9709049

S. Flach and C. R. Willis, Discrete Breathers, Physics Reports, vol.295, pp.181-264, 1998.

G. Friesecke and J. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys, vol.161, pp.391-418, 1994.
DOI : 10.1007/bf02099784

G. Gantzounis, M. Serra-garcia, K. Homma, J. M. Mendoza, and C. Daraio, Granular metamaterials for vibration mitigation, Journal of Applied Physics, vol.114, p.93514, 2013.
DOI : 10.1063/1.4820521

URL : https://authors.library.caltech.edu/42260/1/1.4820521.pdf

O. V. Gendelman and L. I. Manevitch, Discrete breathers in vibroimpact chains: Analytic solutions, Phys. Rev. E, vol.78, p.26609, 2008.
DOI : 10.1103/physreve.78.026609

J. Giannoulis and A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials, Discr. Cont. Dyn. Syst. B, vol.6, pp.493-523, 2006.

J. Giannoulis, M. Herrmann, and A. Mielke, Lagrangian and Hamiltonian two-scale reduction, Journal of Mathematical Physics, vol.49, pp.1-42, 2008.
DOI : 10.1063/1.2956487

URL : http://arxiv.org/pdf/0802.2820

M. Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. R. Soc. Edinb. Sect. A-Math, vol.140, pp.753-785, 2010.
DOI : 10.1017/s0308210509000146

URL : http://arxiv.org/pdf/0901.3736

G. Iooss and K. Kirchgässner, Travelling waves in a chain of coupled nonlinear oscillators, Commun. Math. Phys, vol.211, pp.439-464, 2000.
URL : https://hal.archives-ouvertes.fr/hal-01271086

G. Iooss and G. James, Localized waves in nonlinear oscillator chains, Chaos, vol.15, p.15113, 2005.
DOI : 10.1063/1.1836151

URL : https://hal.archives-ouvertes.fr/hal-00012887

G. James, Nonlinear waves in Newton's cradle and the discrete p-Schrödinger equation, Math. Models Meth. Appl. Sci, vol.21, pp.2335-2377, 2011.

G. James, P. G. Kevrekidis, and J. Cuevas, Breathers in oscillator chains with Hertzian interactions, Physica D, vol.251, pp.39-59, 2013.
DOI : 10.1016/j.physd.2013.01.017

URL : https://hal.archives-ouvertes.fr/hal-00638750

G. James and Y. Sire, Center manifold theory in the context of infinite one-dimensional lattices, The Fermi-Pasta-Ulam Problem. A Status Report, G. Gallavotti Ed, vol.728, pp.207-238, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00390032

G. James, Y. Starosvetsky-;-r.-carretero-gonzález, J. Cuevas-maraver, D. Frantzeskakis, N. Karachalios et al., Breather solutions of the discrete p-Schrödinger equation, in Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity, vol.7, pp.77-115, 2014.

G. James and D. Pelinovsky, Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials, Proc. R. Soc. A, vol.470, 2014.
DOI : 10.1098/rspa.2013.0462

URL : https://hal.archives-ouvertes.fr/hal-00844170

L. A. Kalyakin, Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems, Russian Math. Surveys, vol.44, pp.3-42, 1989.

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, 2009.

P. G. Kevrekidis, A. G. Stefanov, and H. Xu, Traveling Waves for the Mass in Mass Model of Granular Chains, Letters in Mathematical Physics, vol.106, pp.1067-1088, 2016.

P. G. Kevrekidis, A. Vainchtein, M. S. Garcia, and C. Daraio, Interaction of traveling waves with masswith-mass defects within a Hertzian chain, Phys. Rev. E, vol.87, p.42911, 2013.
DOI : 10.1103/physreve.87.042911

URL : https://link.aps.org/accepted/10.1103/PhysRevE.87.042911

D. Khatri, D. Ngo, and C. Daraio, Highly nonlinear solitary waves in chains of cylindrical particles, Granular Matter, vol.14, pp.63-69, 2012.
DOI : 10.1007/s10035-011-0297-9

E. Kim and J. Yang, Wave propagation in single column woodpile phononic crystal: Formation of tunable band gaps, Journal of the Mechanics and Physics of Solids, vol.71, pp.33-45, 2014.

E. Kim, F. Li, C. Chong, G. Theocharis, J. Yang et al., Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures, Phys. Rev. Lett, vol.114, p.118002, 2015.

. S. Yu and . Kivshar, Intrinsic localized modes as solitons with a compact support, Phys. Rev. E, vol.48, pp.43-45, 1993.

L. Liu, G. James, P. Kevrekidis, and A. Vainchtein, Nonlinear waves in a strongly resonant granular chain, Nonlinearity, vol.29, pp.3496-3527, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01161719

R. S. Mackay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, vol.7, pp.1623-1643, 1994.

R. S. Mackay, Solitary waves in a chain of beads under Hertz contact, Phys. Lett. A, vol.251, pp.191-192, 1999.

T. R. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, Radiationless traveling waves in saturable nonlinear Schrödinger lattices, Phys. Rev. Lett, vol.97, p.124101, 2006.

T. R. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: Existence, stability and dynamics, Physica D, vol.237, pp.551-567, 2008.

V. F. Nesterenko, Dynamics of heterogeneous materials, 2001.

D. Ngo, S. Griffiths, D. Khatri, and C. Daraio, Highly nonlinear solitary waves in chains of hollow spherical particles, Granular Matter, vol.15, pp.149-155, 2013.

N. Nguyen and B. Brogliato, Multiple impacts in dissipative granular chains, Lecture Notes in Applied and Computational Mechanics, vol.72, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00908612

O. F. Oxtoby and I. V. Barashenkov, Moving solitons in the discrete nonlinear Schrödinger equation, Phys. Rev. E, vol.76, p.36603, 2007.

M. Peyrard and G. James, Intrinsic localized modes in nonlinear models inspired by DNA, Nonlinear Theory and its Applications (NOLTA), vol.3, pp.27-51, 2012.
URL : https://hal.archives-ouvertes.fr/ensl-00934647

M. Porter, P. G. Kevrekidis, and C. Daraio, Granular crystals : Nonlinear dynamics meets materials engineering, Physics Today, vol.68, p.44, 2015.

P. Rosenau, Hamiltonian dynamics of dense chains and lattices: or how to correct the continuum, Phys. Lett. A, vol.311, pp.39-52, 2003.

P. Rosenau, Compactification of patterns by a singular convection or stress, Phys. Rev. Lett, vol.99, p.234102, 2007.

G. Schneider, Bounds for the nonlinear Schrödinger approximation of the Fermi-Pasta-Ulam system, Appl. Anal, vol.89, pp.1523-1539, 2010.

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, Differential Equations Appl, vol.5, issue.1, pp.69-82, 1998.

K. Sekimoto, Newton's cradle versus nonbinary collisions, Phys. Rev. Lett, vol.104, p.124302, 2010.

S. Sen, J. Hong, J. Bang, E. Avalos, and R. Doney, Solitary waves in the granular chain, Physics Reports, vol.462, pp.21-66, 2008.

A. J. Sievers and J. B. Page, Unusual anharmonic local mode systems, Dynamical Properties of Solids, vol.7, 1995.

Y. Sire, Travelling breathers in Klein-Gordon lattices as homoclinic orbits to p-tori, J. Dyn. Diff. Eqs, vol.17, pp.779-823, 2005.

Y. Starosvetsky, M. A. Hasan, A. F. Vakakis, and L. I. Manevitch, Strongly nonlinear beat phenomena and energy exchanges in weakly coupled granular chains on elastic foundations, SIAM J. Appl. Math, vol.72, p.337, 2012.

Y. Starosvetsky, K. R. Jayaprakash, M. A. Hasan, and A. F. Vakakis, Topics on the nonlinear dynamics and acoustics of ordered granular media, 2017.

A. Stefanov and P. G. Kevrekidis, On the existence of solitary traveling waves for generalized Hertzian chains, J. Nonlinear Sci, vol.22, pp.327-349, 2012.

D. Sun, C. Daraio, and S. Sen, Nonlinear repulsive force between two solids with axial symmetry, Phys. Rev. E, vol.83, p.66605, 2011.

D. Sun and S. Sen, Nonlinear grain-grain forces and the width of the solitary wave in granular chains : a numerical study, Granular Matter, vol.15, pp.157-161, 2013.

T. Tao, The high exponent limit p ? ? for the one-dimensional nonlinear wave equation, Analysis and PDE, vol.2, pp.235-259, 2009.

K. Vorotnikov, Y. Starosvetsky, G. Theocharis, and P. G. Kevrekidis, Wave propagation in a strongly nonlinear locally resonant granular crystal, Physica D, vol.365, pp.27-41, 2018.

G. B. Whitham, Linear and nonlinear waves, 1974.

H. Xu, P. G. Kevrekidis, and A. Stefanov, Traveling Waves and their Tails in Locally Resonant Granular Systems, J. Phys. A: Math. Theor, vol.48, 2015.