×

Nanoptera in nonlinear woodpile chains with zero precompression. (English) Zbl 1490.76223

Summary: We use exponential asymptotics to study travelling waves in woodpile systems modelled as singularly perturbed granular chains with zero precompression and small mass ratio. These systems are strongly nonlinear, and there is no analytic expression for their leading-order solution. We instead obtain an approximated leading-order solution using a hybrid numerical-analytic method. We show that travelling waves in these nonlinear woodpile systems are typically “nanoptera”, or travelling waves with exponentially small but non-decaying oscillatory tails which appear as a Stokes curve is crossed. We demonstrate that travelling wave solutions in the zero precompression regime contain two Stokes curves, and hence two sets of trailing oscillations in the solution. We calculate the behaviour of these oscillations explicitly, and show that there exist system configurations which cause the oscillations to cancel entirely, producing solitary wave behaviour. We then study the behaviour of travelling waves in woodpile chains as precompression is increased, and show that there exists a value of the precompression above which the two Stokes curves coalesce into a single curve, meaning that cancellation of the trailing oscillations no longer occurs. This is consistent with previous studies, which showed that cancellation does not occur in chains with strong precompression.

MSC:

76T25 Granular flows
34K11 Oscillation theory of functional-differential equations
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
35C08 Soliton solutions

Software:

AAA
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Deng, G.; Lustri, C. J.; Porter, M. A., Nanoptera in weakly nonlinear woodpile chains and diatomic granular chains, SIAM J. Appl. Dyn. Syst. (2021), in press · Zbl 1484.34061
[2] Kim, E.; Li, F.; Chong, C.; Theocharis, G.; Yang, J.; Kevrekidis, P. G., Highly nonlinear wave propagation in elastic woodpile periodic structures, Phys. Rev. Lett., 114, 11, Article 118002 pp. (2015)
[3] Xu, H.; Kevrekidis, P. G.; Stefanov, A., Traveling waves and their tails in locally resonant granular systems, J. Phys. A, 48, 19, Article 195204 pp. (2015) · Zbl 1312.74007
[4] Friesecke, G.; Wattis, J. A.D., Existence theorem for solitary waves on lattices, Comm. Math. Phys., 161, 2, 391-418 (1994) · Zbl 0807.35121
[5] Nesterenko, V. F., Propagation of nonlinear compression pulses in granular media, J. Appl. Mech. Tech. Phys., 24, 733-743 (1983)
[6] Nesterenko, V. F., Dynamics of Heterogeneous Materials (2001), Springer-Verlag: Springer-Verlag Heidelberg, Germany
[7] Sen, S.; Hong, J.; Bang, J.; Ávalos, E.; Doney, R., Solitary waves in the granular chain, Phys. Rep., 462, 2, 21-66 (2008)
[8] Porter, M. A.; Kevrekidis, P. G.; Daraio, C., Granular crystals: Nonlinear dynamics meets materials engineering, Phys. Today, 68, 11, 44-50 (2015)
[9] Chong, C.; Porter, M. A.; Kevrekidis, P. G.; Daraio, C., Nonlinear coherent structures in granular crystals, J. Phys.: Condens. Matter, 29, 41, Article 413003 pp. (2017)
[10] Deng, G.; Biondini, G.; Sen, S.; Kevrekidis, P. G., On the generation and propagation of solitary waves in integrable and nonintegrable nonlinear lattices, Eur. Phys. J. Plus, 135, 598 (2020)
[11] Lazaridi, A. N.; Nesterenko, V. F., Observation of a new type of solitary waves in a one-dimensional granular medium, J. Appl. Mech. Tech. Phys., 26, 405-408 (1985)
[12] Coste, C.; Falcon, E.; Fauve, S., Solitary waves in a chain of beads under Hertz contact, Phys. Rev. E, 56, 6104-6117 (1997)
[13] Daraio, C.; Nesterenko, V. F.; Herbold, E. B.; Jin, S., Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals, Phys. Rev. E, 73, Article 026610 pp. (2006)
[14] Hinch, E. J.; Saint-Jean, S., The fragmentation of a line of balls by an impact, Proc. R. Soc. A, 455, 3201-3220 (1999) · Zbl 0956.74035
[15] Manciu, F. S.; Sen, S., Secondary solitary wave formation in systems with generalized Hertz interactions, Phys. Rev. E, 66, Article 016616 pp. (2002)
[16] Job, S.; Melo, F.; Sokolow, A.; Sen, S., How Hertzian solitary waves interact with boundaries in a 1D granular medium, Phys. Rev. Lett., 94, Article 178002 pp. (2005)
[17] Ávalos, E.; Sen, S., How solitary waves collide in discrete granular alignments, Phys. Rev. E, 79, Article 046607 pp. (2009)
[18] Ávalos, E.; Sun, D.; Doney, R. L.; Sen, S., Sustained strong fluctuations in a nonlinear chain at acoustic vacuum: Beyond equilibrium, Phys. Rev. E, 84, Article 046610 pp. (2011)
[19] Ávalos, E.; Sen, S., Granular chain between asymmetric boundaries and the quasiequilibrium state, Phys. Rev. E, 89, Article 053202 pp. (2014)
[20] Deng, G.; Biondini, G.; Sen, S., Interactions of solitary waves in integrable and nonintegrable lattices, Chaos, 30, 4, Article 043101 pp. (2020) · Zbl 1450.37067
[21] Sen, S.; Krishna Mohan, T. R.; M. M. Pfannes, J., The quasi-equilibrium phase in nonlinear 1D systems, Physica A, 342, 1, 336-343 (2004), Proceedings of the VIII Latin American Workshop on Nonlinear Phenomena
[22] Przedborski, M.; Harroun, T. A.; Sen, S., Granular chains with soft boundaries: Slowing the transition to quasiequilibrium, Phys. Rev. E, 91, Article 042207 pp. (2015)
[23] Przedborski, M.; Sen, S.; Harroun, T. A., The equilibrium phase in heterogeneous Hertzian chains, J. Stat. Mech., 2017, 12, Article 123204 pp. (2017) · Zbl 1457.82145
[24] Sen, S.; Manciu, M.; Wright, J. D., Solitonlike pulses in perturbed and driven Hertzian chains and their possible applications in detecting buried impurities, Phys. Rev. E, 57, 2386-2397 (1998)
[25] Hascoët, E.; Herrmann, H. J., Shocks in non-loaded bead chains with impurities, Eur. Phys. J. B, 14, 1, 183-190 (2000)
[26] Martínez, A. J.; Yasuda, H.; Kim, E.; Kevrekidis, P. G.; Porter, M. A.; Yang, J., Scattering of waves by impurities in precompressed granular chains, Phys. Rev. E, 93, Article 052224 pp. (2016)
[27] Kim, E.; Martínez, A. J.; Phenisee, S. E.; Kevrekidis, P. G.; Porter, M. A.; Yang, J., Direct measurement of superdiffusive energy transport in disordered granular chain, Nature Commun., 9, 640 (2018)
[28] Martínez, A. J.; Kevrekidis, P. G.; Porter, M. A., Superdiffusive transport and energy localization in disordered granular crystals, Phys. Rev. E, 93, Article 022902 pp. (2016)
[29] Manciu, M.; Sen, S.; Hurd, A. J., Impulse propagation in dissipative and disordered chains with power-law repulsive potentials, Physica D, 157, 3, 226-240 (2001) · Zbl 0976.82049
[30] Harbola, U.; Rosas, A.; Romero, A. H.; Lindenberg, K., Pulse propagation in randomly decorated chains, Phys. Rev. E, 82, Article 011306 pp. (2010)
[31] Manjunath, M.; Awasthi, A. P.; Geubelle, P. H., Wave propagation in random granular chains, Phys. Rev. E, 85, Article 031308 pp. (2012)
[32] Theocharis, G.; Boechler, N.; Kevrekidis, P. G.; Job, S.; Porter, M. A.; Daraio, C., Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals, Phys. Rev. E, 82, Article 056604 pp. (2010)
[33] 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, Article 244302 pp. (2010)
[34] Molinari, A.; Daraio, C., Stationary shocks in periodic highly nonlinear granular chains, Phys. Rev. E, 80, Article 056602 pp. (2009)
[35] Ponson, L.; Boechler, N.; Lai, Y. M.; Porter, M. A.; Kevrekidis, P. G.; Daraio, C., Nonlinear waves in disordered diatomic granular chains, Phys. Rev. E, 82, Article 021301 pp. (2010)
[36] Porter, M. A.; Daraio, C.; Herbold, E. B.; Szelengowicz, I.; Kevrekidis, P. G., Highly nonlinear solitary waves in periodic dimer granular chains, Phys. Rev. E, 77, Article 015601 pp. (2008)
[37] Porter, M. A.; Daraio, C.; Szelengowicz, I.; Herbold, E. B.; Kevrekidis, P. G., Highly nonlinear solitary waves in heterogeneous periodic granular media, Physica D, 238, 6, 666-676 (2009) · Zbl 1160.37410
[38] Hoogeboom, C.; Man, Y.; Boechler, N.; Theocharis, G.; Kevrekidis, P. G.; Kevrekidis, I. G.; Daraio, C., Hysteresis loops and multi-stability: From periodic orbits to chaotic dynamics (and back) in diatomic granular crystals, Europhys. Lett., 101, 4, 44003 (2013)
[39] Herbold, E. B.; Kim, J.; Nesterenko, V. F.; Wang, S. Y.; Daraio, C., Pulse propagation in a linear and nonlinear diatomic periodic chain: Effects of acoustic frequency band-gap, Acta Mech., 205, 85-103 (2009) · Zbl 1167.74003
[40] Jayaprakash, K. R.; Starosvetsky, Y.; Vakakis, A. F., New family of solitary waves in granular dimer chains with no precompression, Phys. Rev. E, 83, 3, Article 036606 pp. (2011)
[41] Jayaprakash, K. R.; Starosvetsky, Y.; Vakakis, A. F.; Gendelman, O. V., Nonlinear resonances leading to strong pulse attenuation in granular dimer chains, J. Nonlinear Sci., 23, 3, 363-392 (2013) · Zbl 1319.70023
[42] Martínez, A. J.; Porter, M. A.; Kevrekidis, P. G., Quasiperiodic granular chains and Hofstadter butterflies, Phil. Trans. R. Soc. A, 376, Article 20170139 pp. (2018) · Zbl 1404.70056
[43] Vergara, L., Scattering of solitary waves from interfaces in granular media, Phys. Rev. Lett., 95, Article 108002 pp. (2005)
[44] Vergara, L., Delayed scattering of solitary waves from interfaces in a granular container, Phys. Rev. E, 73, Article 066623 pp. (2006)
[45] Nesterenko, V. F.; Daraio, C.; Herbold, E. B.; Jin, S., Anomalous wave reflection at the interface of two strongly nonlinear granular media, Phys. Rev. Lett., 95, 15, Article 158702 pp. (2005)
[46] Daraio, C.; Nesterenko, V. F.; Herbold, E. B.; Jin, S., Energy trapping and shock disintegration in a composite granular medium, Phys. Rev. Lett., 96, Article 058002 pp. (2006)
[47] Liu, L.; James, G.; Kevrekidis, P.; Vainchtein, A., Strongly nonlinear waves in locally resonant granular chains, Nonlinearity, 29, 11, 3496-3527 (2016) · Zbl 1352.37182
[48] Liu, L.; James, G.; Kevrekidis, P.; Vainchtein, A., Breathers in a locally resonant granular chain with precompression, Physica D, 331, 27-47 (2016) · Zbl 1364.74028
[49] Jiang, H.; Wang, Y.; Zhang, M.; Hu, Y.; Lan, D.; Zhang, Y.; Wei, B., Locally resonant phononic woodpile: A wide band anomalous underwater acoustic absorbing material, Appl. Phys. Lett., 95, 10, Article 104101 pp. (2009)
[50] Wu, L. Y.; Chen, L. W., Acoustic band gaps of the woodpile sonic crystal with the simple cubic lattice, J. Phys. D: Appl. Phys., 44, 4, Article 045402 pp. (2011)
[51] Kim, E.; Yang, J., Wave propagation in single column woodpile phononic crystals: Formation of tunable band gaps, J. Mech. Phys. Solids, 71, 33-45 (2014) · Zbl 1328.74052
[52] Feigel, A.; Veinger, M.; Sfez, B.; Arsh, A.; Klebanov, M.; Lyubin, V., Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses, Appl. Phys. Lett., 83, 22, 4480-4482 (2003)
[53] Liu, H.; Yao, J.; Xu, D.; Wang, P., Characteristics of photonic band gaps in woodpile three-dimensional terahertz photonic crystals, Opt. Express, 15, 2, 695-703 (2007)
[54] Boyd, J. P., A numerical calculation of a weakly non-local solitary wave: The \(\phi^4\) breather, Nonlinearity, 3, 1, 177-195 (1990) · Zbl 0743.65091
[55] Vainchtein, A.; Starosvetsky, Y.; Wright, J. D.; Perline, R., Solitary waves in diatomic chains, Phys. Rev. E, 93, 4, Article 042210 pp. (2016)
[56] Okada, Y.; Watanabe, S.; Tanaca, H., Solitary wave in periodic nonlinear lattice, J. Phys. Soc. Japan, 59, 8, 2647-2658 (1990)
[57] Tabata, Y., Stable solitary wave in diatomic Toda lattice, J. Phys. Soc. Japan, 65, 12, 3689-3691 (1996)
[58] Lustri, C. J.; Porter, M. A., Nanoptera in a period-2 Toda chain, SIAM J. Appl. Dyn. Syst., 17, 2, 1182-1212 (2018) · Zbl 1407.34020
[59] Faver, T. E., Nanopteron-stegoton traveling waves in spring dimer Fermi-Pasta-Ulam-Tsingou lattices, Q. Appl. Math., 78, 363-429 (2020) · Zbl 1458.37079
[60] Hoffman, A.; Wright, J. D., Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio, Physica D, 358, 33-59 (2017) · Zbl 1378.35067
[61] Lustri, C. J., Nanoptera and Stokes curves in the 2-periodic Fermi-Pasta-Ulam-Tsingou equation, Physica D, 402, Article 132239 pp. (2020) · Zbl 1453.82045
[62] Iooss, G.; Kirchgässner, K., Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys., 211, 2, 439-464 (2000) · Zbl 0956.37055
[63] Faver, T. E.; Hupkes, H. J., Micropterons, nanopterons and solitary wave solutions to the diatomic Fermi-Pasta-Ulam-Tsingou problem, Partial Differ. Equations Appl. Math., 4, Article 100128 pp. (2021)
[64] Kim, E.; Chaunsali, R.; Xu, H.; Jaworski, J.; Yang, J.; Kevrekidis, P. G.; Vakakis, A. F., Nonlinear low-to-high-frequency energy cascades in diatomic granular crystals, Phys. Rev. E, 92, Article 062201 pp. (2015)
[65] Potekin, R.; Jayaprakash, K. R.; McFarland, D. M.; Remick, K.; Bergman, L. A.; Vakakis, A. F., Experimental study of strongly nonlinear resonances and anti-resonances in granular dimer chains, Exp. Mech., 53, 861-870 (2013)
[66] Manjunath, M.; Awasthi, A. P.; Geubelle, P. H., Family of plane solitary waves in dimer granular crystals, Phys. Rev. E, 90, Article 032209 pp. (2014)
[67] Alfimov, G. L.; Korobeinikov, A. S.; Lustri, C. J.; Pelinovsky, D. E., Standing lattice solitons in the discrete NLS equation with saturation, Nonlinearity, 32, 9, 3445-3484 (2019) · Zbl 1423.34019
[68] Melvin, T. R.O.; Champneys, A. R.; Pelinovsky, D. E., Discrete traveling solitons in the Salerno model, SIAM J. Appl. Dyn. Syst., 8, 2, 689-709 (2009) · Zbl 1171.34041
[69] Oxtoby, O. F.; Barashenkov, I. V., Moving solitons in the discrete nonlinear Schrödinger equation, Phys. Rev. E, 76, Article 036603 pp. (2007)
[70] Oxtoby, O. F.; Pelinovsky, D. E.; Barashenkov, I. V., Travelling kinks in discrete \(\phi^4\) models, Nonlinearity, 19, 217-235 (2005) · Zbl 1101.82008
[71] Barashenkov, I. V.; Oxtoby, O. F.; Pelinovsky, D. E., Translationally invariant discrete kinks from one-dimensional maps, Phys. Rev. E, 72, Article 035602 pp. (2005)
[72] Barashenkov, I. V.; van Heerden, T. C., Exceptional discretizations of the sine-Gordon equation, Phys. Rev. E, 77, Article 036601 pp. (2008)
[73] King, J. R.; Chapman, S. J., Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations, European J. Appl. Math., 12, 4, 433-463 (2001) · Zbl 0999.34072
[74] Tovbis, A.; Tsuchiya, M.; Jaffé, C., Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Hénon map as an example, Chaos, 8, 3, 665-681 (1998) · Zbl 0987.37022
[75] Hunter, J. K.; Scheurle, J., Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32, 2, 253-268 (1988) · Zbl 0694.35204
[76] Boyd, J. P., Weakly non-local solitons for capillary-gravity waves: Fifth-degree Korteweg-de Vries equation, Physica D, 48, 129-146 (1991) · Zbl 0728.35100
[77] Benilov, E. S.; Grimshaw, R.; Kuznetsova, E. P., The generation of radiating waves in a singularly-perturbed Korteweg-de Vries equation, Physica D, 69, 3-4, 270-278 (1993) · Zbl 0791.35119
[78] Grimshaw, R.; Joshi, N., Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation, SIAM J. Appl. Math., 55, 1, 124-135 (1995) · Zbl 0814.34043
[79] Giardetti, N.; Shapiro, A.; Windle, S.; Wright, J. D., Metastability of solitary waves in diatomic FPUT lattices, Math. Eng., 1, 419-433 (2019) · Zbl 1432.37102
[80] Chapman, S. J.; King, J. R.; Adams, K. L., Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations, Proc. R. Soc. A, 454, 1978, 2733-2755 (1998) · Zbl 0916.34017
[81] Olde Daalhuis, A. B.; Chapman, S. J.; King, J. R.; Ockendon, J. R.; Tew, R. H., Stokes phenomenon and matched asymptotic expansions, SIAM J. Appl. Math., 55, 6, 1469-1483 (1995) · Zbl 0834.41026
[82] Shelton, J.; Milewski, P.; Trinh, P. H., On the structure of steady parasitic gravity-capillary waves in the small surface tension limit, J. Fluid Mech., 922, A16 (2021) · Zbl 1491.76016
[83] Chapman, S. J.; Trinh, P. H.; Witelski, T. P., Exponential asymptotics for thin film rupture, SIAM J. Appl. Math., 73, 1, 232-253 (2013) · Zbl 1272.34121
[84] Sen, S.; Manciu, M., Solitary wave dynamics in generalized Hertz chains: An improved solution of the equation of motion, Phys. Rev. E, 64, Article 056605 pp. (2001)
[85] Stokes, G. G., On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Cam. Phil. Soc., 10, 106-128 (1864)
[86] Dingle, R. B., Asymptotic Expansions: Their Derivation and Interpretation (1973), Academic Press: Academic Press New York, NY, USA · Zbl 0279.41030
[87] Boyd, J. P., Hyperasymptotics and the linear boundary layer problem: Why asymptotic series diverge, SIAM Rev., 47, 3, 553-575 (2005) · Zbl 1087.34032
[88] Boyd, J. P., The devil’s invention: Asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math., 56, 1, 1-98 (1999) · Zbl 0972.34044
[89] Berry, M. V., Stokes’ phenomenon; smoothing a victorian discontinuity, Pub. Math. L’IHÉS, 68, 211-221 (1988) · Zbl 0701.58012
[90] Berry, M. V., Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. R. Soc. A, 422, 1862, 7-21 (1989) · Zbl 0683.33004
[91] Berry, M. V.; Howls, C. J., Hyperasymptotics, Proc. R. Soc. A, 430, 1880, 653-668 (1990) · Zbl 0745.34052
[92] Berry, M. V., Asymptotics, superasymptotics, hyperasymptotics, (Segur, H.; Tanveer, S.; Levine, H., Asymptotics Beyond All Orders (1991), Plenum Publishing Corporation: Plenum Publishing Corporation Amsterdam, The Netherlands), 1-14
[93] Hinch, E. J., (Perturbation Methods. Perturbation Methods, Cambridge Texts in Applied Mathematics (1991), Cambridge University Press: Cambridge University Press Cambridge, UK) · Zbl 0746.34001
[94] Boyd, J. P., Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory, Mathematics and Its Applications (1998), Kluwer Publishers: Kluwer Publishers Amsterdam, The Netherlands · Zbl 0905.76001
[95] Baker, G. A., The Padé approximant method and some related generalizations, (Baker, G. A.; Gammel, J. L., The Padé Approximant in Theoretical Physics (1970), Academic Press: Academic Press New York and London), 1-39
[96] Nakatsukasa, Y.; Sète, O.; Trefethen, L. N., The AAA algorithm for rational approximation, SIAM J. Sci. Comput., 40, 1494-1522 (2018) · Zbl 1390.41015
[97] Verlet, L., Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev., 159, 98-103 (1967)
[98] Allen, M. P.; Tildesley, D. J., Computer Simulation of Liquids (1987), Clarendon Press: Clarendon Press Oxford, UK · Zbl 0703.68099
[99] Friesecke, G.; Pego, R. L., Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit, Nonlinearity, 12, 6, 1601-1627 (1999) · Zbl 0962.82015
[100] Friesecke, G.; Pego, R. L., Solitary waves on FPU lattices: II. Linear implies nonlinear stability, Nonlinearity, 15, 4, 1343-1359 (2002) · Zbl 1102.37311
[101] Friesecke, G.; Pego, R. L., Solitary waves on Fermi-Pasta-Ulam lattices: III. Howland-type Floquet theory, Nonlinearity, 17, 1, 207-227 (2003) · Zbl 1103.37049
[102] Friesecke, G.; Pego, R. L., Solitary waves on Fermi-Pasta-Ulam lattices: IV. Proof of stability at low energy, Nonlinearity, 17, 1, 229-251 (2003) · Zbl 1103.37050
[103] Molerón, M.; Chong, C.; Martínez, A. J.; Porter, M. A.; Kevrekidis, P. G.; Daraio, C., Nonlinear excitations in magnetic lattices with long-range interactions, New J. Phys., 21, 6, Article 063032 pp. (2019)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.