Citation: Nickolas Giardetti, Amy Shapiro, Stephen Windle, J. Douglas Wright. Metastability of solitary waves in diatomic FPUT lattices[J]. Mathematics in Engineering, 2019, 1(3): 419-433. doi: 10.3934/mine.2019.3.419
[1] | Faver TE, Wright JD (2018) Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity. SIAM J Math Anal 50: 182–250. doi: 10.1137/15M1046836 |
[2] | Lustri CJ, Porter MA (2018) Nanoptera in a period-2 Toda chain. SIAM J Appl Dyn Syst 17: 1182–1212. doi: 10.1137/16M108639X |
[3] | Porter M, Daraio C, Szelengowicz I, et al. (2009) Highly nonlinear solitary waves in heterogeneous periodic granular media. Phys D 238: 666–676. doi: 10.1016/j.physd.2008.12.010 |
[4] | Gaison J, Moskow S, Wright JD, et al. (2014) Approximation of polyatomic FPU lattices by KdV equations. Multiscale Model Simul 12: 953–995. doi: 10.1137/130941638 |
[5] | Qin WX (2015) Wave propagation in diatomic lattices. SIAM J Math Anal 47: 477–497. doi: 10.1137/130949609 |
[6] | Betti M, Pelinovsky DE (2013) Periodic traveling waves in diatomic granular chains. J Nonlinear Sci 23: 689–730. doi: 10.1007/s00332-013-9165-6 |
[7] | Chirilus-Bruckner M, Chong C, Prill O, et al. (2012) Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations. Discrete Contin Dyn Syst Ser S 5: 879–901. Available from: https://doi.org/10.3934/dcdss.2012.5.879. doi: 10.3934/dcdss.2012.5.879 |
[8] | Brillouin L (1953) Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices,2Eds., New York: Dover Publications, Inc. |
[9] | Tabata Y (1996) Stable solitary wave in diatomic Toda lattice. J Phys Soc Jpn 65: 3689–3691. doi: 10.1143/JPSJ.65.3689 |
[10] | Okada Y, Watanabe S, Tanaca H (1990) Solitary wave in periodic nonlinear lattice. J Phys Soc Jpn 59: 2647–2658. Available from: https://doi.org/10.1143/JPSJ.59.2647. doi: 10.1143/JPSJ.59.2647 |
[11] | Hoffman A, Wright JD (2017) Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio. Phys D 358: 33–59. doi: 10.1016/j.physd.2017.07.004 |
[12] | Vainchtein A, Starosvetsky Y, Wright JD, et al. (2016) Solitary waves in diatomic chains. Phys Rev E 93: 042210. doi: 10.1103/PhysRevE.93.042210 |
[13] | Schneider G, Wayne CE (1999) Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, In: International Conference on Differential Equations,Vol. 1, 2 (Berlin, 1999), 390–404, World Sci. Publ., River Edge, NJ, 2000. |
[14] | Friesecke G, Pego RL (1999) Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12: 1601–1627. |
[15] | Friesecke G, Pego RL (2002) Solitary waves on FPU lattices: II. Linear implies nonlinear stability. Nonlinearity 15: 1343–1359. |
[16] | Friesecke G, Pego RL (2004) Solitary waves on Fermi-Pasta-Ulam lattices: III. Howland-type Floquet theory. Nonlinearity 17: 207–227. |
[17] | Friesecke G, Pego RL (2004) Solitary waves on Fermi-Pasta-Ulam lattices: IV. Proof of stability at low energy. Nonlinearity 17: 229–251. |
[18] | Mizumachi T (2009) Asymptotic stability of lattice solitons in the energy space. Commun Math Phys 288: 125–144. doi: 10.1007/s00220-009-0768-6 |
[19] | Boyd JP (1998) Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics Generalized Solitons and Hyperasymptotic Perturbation Theory , In Series: Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers, vol. 442. Available from: https://doi.org/10.1007/978-1-4615-5825-5. |
[20] | Faver T (2017) Nanopteron-stegoton traveling waves in spring dimer Fermi-Pasta-Ulam-Tsingou lattices, in press. Available from: https://arxiv.org/abs/1710.07376. |
[21] | Lombardi E (2000) Oscillatory Integrals and Phenomena Beyond All Algebraic Orders with Applications to Homoclinic Orbits in Reversible Systems, In series: Lecture Notes in Mathematics. Berlin: Springer-Verlag, vol. 1741. Available from: https://doi.org/10.1007/BFb0104102. |
[22] | Sun SM (1999) Non-existence of truly solitary waves in water with small surface tension. Proc Math Phys Eng Sci 455: 2191–2228. doi: 10.1098/rspa.1999.0399 |
[23] | Martínez AJ, Kevrekidis PG, Porter MA (2016) Superdiffusive transport and energy localization in disordered granular crystals. Phys Rev E 93: 022902. Available from: https://doi.org/10.1103/physreve.93.022902. doi: 10.1103/PhysRevE.93.022902 |
[24] | Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2Eds., In series: Springer Series in Computational Mathematics. Springer, Heidelberg, 2010, vol. 31. |
[25] | Beyn WJ, Thümmler V (2004) Freezing solutions of equivariant evolution equations. SIAM J Appl Dyn Syst 3: 85–116. doi: 10.1137/030600515 |
[26] | Beyn WJ, Otten D, Rottmann-Matthes J (2018) Computation and stability of traveling waves in second order evolution equations. SIAM J Numer Anal 56: 1786–1817. doi: 10.1137/16M108286X |
[27] | Beale JT (1991) Exact solitary water waves with capillary ripples at infinity. Commun Pure Appl Math 44: 211–257. doi: 10.1002/cpa.3160440204 |
[28] | Sun SM (1991) Existence of a generalized solitary wave solution for water with positive Bond number less than 1/3. J Math Anal Appl 156: 471–504. doi: 10.1016/0022-247X(91)90410-2 |
[29] | LeVeque RJ, Yong DH (2003) Solitary waves in layered nonlinear media. SIAM J Appl Math 63: 1539–1560. doi: 10.1137/S0036139902408151 |
[30] | LeVeque RJ, Yong DH (2003) Phase plane behavior of solitary waves in nonlinear layered media, In: Hyperbolic Problems: Theory, Numerics, Applications. Berlin: Springer, 43–51. |
[31] | Kevrekidis PG, Stefanov AG, Xu H (2016) Traveling waves for the mass in mass model of granular chains. Lett Math Phys 106: 1067–1088. doi: 10.1007/s11005-016-0854-6 |
[32] | Pnevmatikos S, Flytzanis N, Remoissenet M (1986) Soliton dynamics of nonlinear diatomic lattices. Phys Rev B 33: 2308–2321. doi: 10.1103/PhysRevB.33.2308 |