×

zbMATH — the first resource for mathematics

Breathers in a locally resonant granular chain with precompression. (English) Zbl 1364.74028
Summary: We study a locally resonant granular material in the form of a precompressed Hertzian chain with linear internal resonators. Using an asymptotic reduction, we derive an effective nonlinear Schrödinger (NLS) modulation equation. This, in turn, leads us to provide analytical evidence, subsequently corroborated numerically, for the existence of two distinct types of discrete breathers related to acoustic or optical modes: (a) traveling bright breathers with a strain profile exponentially vanishing at infinity and (b) stationary and traveling dark breathers, exponentially localized, time-periodic states mounted on top of a non-vanishing background. The stability and bifurcation structure of numerically computed exact stationary dark breathers is also examined. Stationary bright breathers cannot be identified using the NLS equation, which is defocusing at the upper edges of the phonon bands and becomes linear at the lower edge of the optical band.

MSC:
74E20 Granularity
35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
35B35 Stability in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Nesterenko, V. F., Dynamics of heterogeneous materials, (2001), Springer New York
[2] Sen, S.; Hong, J.; Bang, J.; Avalos, E.; Doney, R., Solitary waves in the granular chain, Phys. Rep., 462, 2, 21-66, (2008)
[3] Coste, C.; Falcon, E.; Fauve, S., Solitary waves in a chain of beads under Hertz contact, Phys. Rev. E, 56, 5, 6104-6117, (1997)
[4] Herbold, E. B.; Nesterenko, V. F., Shock wave structure in a strongly nonlinear lattice with viscous dissipation, Phys. Rev. E, 75, (2007)
[5] Molinari, A.; Daraio, C., Stationary shocks in periodic highly nonlinear granular chains, Phys. Rev. E, 80, (2009)
[6] Boechler, N.; Theocharis, G.; Job, S.; Kevrekidis, P. G.; Porter, M. A.; Daraio, C., Discrete breathers in one-dimensional diatomic granular crystals, Phys. Rev. Lett., 104, (2010)
[7] Chong, C.; Kevrekidis, P. G.; Theocharis, G.; Daraio, C., Dark breathers in granular crystals, Phys. Rev. E, 87, (2013)
[8] Chong, C.; Li, F.; Yang, J.; Williams, M. O.; Kevrekidis, I. G.; Kevrekidis, P. G.; Daraio, C., Damped-driven granular chains: an ideal playground for dark breathers and multibreathers, Phys. Rev. E, 89, (2014)
[9] James, G., Nonlinear waves in newton’s cradle and the discrete p-Schrödinger equation, Math. Models Methods Appl. Sci., 21, 11, 2335-2377, (2011) · Zbl 1331.70042
[10] James, G.; Kevrekidis, P. G.; Cuevas, J., Breathers in oscillator chains with Hertzian interactions, Physica D, 251, 39-59, (2013) · Zbl 1278.37053
[11] Job, S.; Santibanez, F.; Tapia, F.; Melo, F., Wave localization in strongly nonlinear Hertzian chains with mass defect, Phys. Rev. E, 80, (2009)
[12] Hoogeboom, G.; Kevrekidis, P. G., Breathers in periodic granular chain with multiple band gaps, Phys. Rev. E, 86, (2012)
[13] Theocharis, G.; Kavousanakis, M.; Kevrekidis, P. G.; Daraio, C.; Porter, M. A.; Kevrekidis, I. G., Localized breathing modes in granular crystals with defects, Phys. Rev. E, 80, (2009)
[14] Theocharis, G.; Boechler, N.; Job, S.; Kevrekidis, P. G.; Porter, M. A.; Daraio, C., Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystal, Phys. Rev. E, 82, (2010)
[15] Aubry, S., Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103, 14, 201-250, (1997) · Zbl 1194.34059
[16] Flach, S.; Gorbach, A., Discrete breathers: advances in theory and applications, Phys. Rep., 467, 1-116, (2008)
[17] Hasan, M. A.; Cho, S.; Remick, K.; Vakakis, A. F.; McFarland, D. M.; Kriven, W. M., Experimental study of nonlinear acoustic bands and propagating breathers in ordered granular media embedded in matrix, Granular Matter, 17, 49-72, (2015)
[18] Kim, E.; Li, F.; Chong, C.; Theocharis, G.; Yang, J.; Kevrekidis, P., Highly nonlinear wave propagation in elastic woodpile periodic structures, Phys. Rev. Lett., 114, (2015)
[19] Xu, H.; Kevrekidis, P. G.; Stefanov, A., Traveling waves and their tails in locally resonant granular systems, J. Phys. A, 48, (2015) · Zbl 1312.74007
[20] Bonanomi, L.; Theocharis, G.; Daraio, C., Wave propagation in granular chains with local resonances, Phys. Rev. E, 91, (2015)
[21] Gantzounis, G.; Serra-Garcia, M.; Homma, K.; Mendoza, J. M.; Daraio, C., Granular metamaterials for vibration mitigation, J. Appl. Phys., 114, (2013)
[22] Kevrekidis, P. G.; Vainchtein, A.; Garcia, M. S.; Daraio, C., Interaction of traveling waves with mass-with-mass defects within a Hertzian chain, Phys. Rev. E, 87, (2013)
[23] Kim, E.; Yang, J., Wave propagation in single column woodpile phononic crystal: formation of turnable band gaps, J. Mech. Phys. Solids, 71, 33-45, (2014) · Zbl 1328.74052
[24] Liu, L.; James, G.; Kevrekidis, P.; Vainchtein, A., Nonlinear waves in a strongly nonlinear resonant granular chain, [nlin.PS]
[25] Fermi, E.; Pasta, J.; Ulam, S., Studies of nonlinear problems, tech. rep., I, report no. LA-1940, (1955), Los Alamos Scientific Laboratory
[26] Friesecke, G.; Wattis, J. A.D., Existence theorem for solitary waves on lattices, Comm. Math. Phys., 161, 2, 391-418, (1994) · Zbl 0807.35121
[27] Friesecke, G.; Pego, R. L., Solitary waves on FPU lattices: I. qualitative properties, renormalization and continuum limit, Nonlinearity, 12, 1601-1626, (1999) · Zbl 0962.82015
[28] Friesecke, G.; Pego, R. L., Solitary waves on FPU lattices: II. linear implies nonlinear stability, Nonlinearity, 15, 4, 1343-1359, (2002) · Zbl 1102.37311
[29] Iooss, G., Travelling waves in the Fermi-pasta-Ulam lattice, Nonlinearity, 13, 3, 849, (2000) · Zbl 0960.37038
[30] Iooss, G.; James, G., Localized waves in nonlinear oscillator chains, Chaos, 15, (2005) · Zbl 1080.37080
[31] Schneider, G., Bounds for the nonlinear Schrödinger approximation of the Fermi-pasta-Ulam system, Appl. Anal., 89, 9, 1523-1539, (2010) · Zbl 1194.35427
[32] James, G., Existence of breathers on FPU lattices, C. R. Acad. Sci. Ser. I Math., 332, 6, 581-586, (2001) · Zbl 1116.37303
[33] James, G., Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13, 1, 27-63, (2003) · Zbl 1185.37158
[34] Friesecke, G.; Pego, R. L., Solitary waves on Fermi-pasta-Ulam lattices: IV. proof of stability at low energy, Nonlinearity, 17, 1, 229-251, (2004) · Zbl 1103.37050
[35] Kevrekidis, P. G., Non-linear waves in lattices: past, present, future, IMA J. Appl. Math., 76, 3, 1-35, (2011) · Zbl 1230.37099
[36] Giannoulis, J.; Mielke, A., The nonlinear Schrödinger equation as a macroscopic limit for an osillator chain with cubic nonlinearities, Nonlinearity, 17, 551-565, (2004) · Zbl 1092.37050
[37] Giannoulis, J.; Mielke, A., Dispersive evolution of pulses in oscillator chains with general interaction potentials, Discrete Contin. Dyn. Syst. Ser. B, 6, 3, 493-523, (2006) · Zbl 1152.34035
[38] Osborne, A. R., (Nonlinear Ocean Waves and the Inverse Scattering Transform, International Geophysics Series, vol. 97, (2010), Academic Press Burlington)
[39] Gómez-Gardeñes, J.; Falo, F.; Floría, L. M., Mobile localization in nonlinear Schrödinger lattices, Phys. Lett. A, 332, 34, 213-219, (2004) · Zbl 1123.37322
[40] Hwang, G.; Akylas, T.; Yang, J., Gap solitons and their linear stability in one-dimensional periodic media, Physica D, 240, 1055-1068, (2011) · Zbl 1218.35224
[41] Cretegny, T.; Aubry, S., Spatially inhomogeneous time-periodic propagating waves in anharmonic systems, Phys. Rev. B, 55, 18, 929-932, (1997)
[42] Wattis, J. A.D.; James, L. M., Discrete breathers in honeycomb Fermi-pasta-Ulam lattices, J. Phys. A, 47, 34, (2014) · Zbl 1302.35330
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.