Asymptotic approximations to travelling waves in the diatomic Fermi-Pasta-Ulam lattice

Jonathan A. D. Wattis; Jonathan A. D. Wattis

doi:10.3934/mine.2019.2.327

Mathematics in Engineering

2019, Volume 1, Issue 2: 327-342. doi: 10.3934/mine.2019.2.327

Previous Article Next Article

Research article Special Issues

Asymptotic approximations to travelling waves in the diatomic Fermi-Pasta-Ulam lattice

Jonathan A. D. Wattis ^,

School of Mathematical Sciences,University of Nottingham,University Park,Nottingham,NG7 2RD,UK

Received: 23 November 2018 Accepted: 14 February 2019 Published: 14 March 2019

We construct high-order approximate travelling waves solutions of the diatomic Fermi-Pasta-Ulam lattice using asymptotic techniques which are valid for arbitrary mass ratios. Separately small amplitude ansatzs are made for the motion of the lighter and heavier particles, which are coupled The Fredholm alternative is used to derive consistency conditions, whose solution generates small amplitude expansions for both sets of particles.
- solitary travelling waves,
- lattice dynamics,
- diatomic,
- FPU,
- asymptotic analysis
Citation: Jonathan A. D. Wattis. Asymptotic approximations to travelling waves in the diatomic Fermi-Pasta-Ulam lattice[J]. Mathematics in Engineering, 2019, 1(2): 327-342. doi: 10.3934/mine.2019.2.327

Related Papers:

Abstract

We construct high-order approximate travelling waves solutions of the diatomic Fermi-Pasta-Ulam lattice using asymptotic techniques which are valid for arbitrary mass ratios. Separately small amplitude ansatzs are made for the motion of the lighter and heavier particles, which are coupled The Fredholm alternative is used to derive consistency conditions, whose solution generates small amplitude expansions for both sets of particles.

References

[1]	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
[2]	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 Cont Dyn S 5: 879–901. doi: 10.3934/dcdss.2012.5.879
[3]	Collins MA (1981) A quasi-continuum approximation for solitons in an atomic chain Chem Phys Lett 77: 342–347.
[4]	Collins MA, Rice SA (1982) Some properties of large amplitude motion in an anharmonic chain with nearest neighbour interactions. J Chem Phys 77: 2607–2622. doi: 10.1063/1.444135
[5]	Collins MA (1985) Solitons in the diatomic chain. Phys Re. A 31: 1754–1762. doi: 10.1103/PhysRevA.31.1754
[6]	Faver TE, Wright JD (2015) Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity. arXiv: 1511.00942 [math.AP].
[7]	Fermi E, Pasta J, Ulam S (1955) Studies of nonlinear problems. Los Alamos report LA-1940, published later in Fermi E., Collected Papers (University of Chicago Press,Chicago), edited by Segre, E., (1965); also in Nonlinear Wave Motion, edited by Newell A. C., Lectures in Applied Mathematics, Vol. 15 (American Mathematical Society, Providence) (1974) p. 143.
[8]	Friesecke G, Wattis JAD (1994) Existence theorem for solitary waves on lattices. Comm Math Phys 161: 391–418. doi: 10.1007/BF02099784
[9]	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
[10]	Hoffman A, Wright JD (2017) Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio. Physica D 358: 33–59. doi: 10.1016/j.physd.2017.07.004
[11]	Huang G (1995) Soliton excitations in one-dimensional diatomic lattices. Phys Rev B 51: 12347– 12360. doi: 10.1103/PhysRevB.51.12347
[12]	Kevrekidis PG, Vainchtein A, Serra Garcia M et al. (2013) Interaction of traveling waves with mass-with-mass defects within a Hertzian chain. Phys Rev E 87: 042911. doi: 10.1103/PhysRevE.87.042911
[13]	Lustri CJ, Porter MA (2018) Nanoptera in a period-2 Toda chain. SIAM J Appl Dyn Syst 17: 1182–1212. doi: 10.1137/16M108639X
[14]	Ockendon JR, Howison SD, Lacey AA, et al. (1999) Applied Partial Differential Equations. Oxford: Oxford University Press, 43–44.
[15]	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
[16]	Ponson L, Boechler N, Lai YM, et al. (2010) Nonlinear waves in disordered diatomic granular chains. Phys Rev E 82: 021301. doi: 10.1103/PhysRevE.82.021301
[17]	Porubov AV, Andrianov IV (2013) Nonlinear waves in diatomic crystals. Wave Motion 50: 1153– 1160. doi: 10.1016/j.wavemoti.2013.03.009
[18]	Qin WX (2015) Wave propagation in diatomic lattices. SIAM J Math Anal 47: 477–497. doi: 10.1137/130949609
[19]	Rosenau P (1986) Dynamics of nonlinear mass spring chains near the Continuum limit. Phys Lett A 118: 222–227. doi: 10.1016/0375-9601(86)90170-2
[20]	Rosenau P (1987) Dynamics of dense lattices. Phys Rev B 36: 5868–5876. doi: 10.1103/PhysRevB.36.5868
[21]	Segur H, Kruskal MD (1987) Nonexistence of small amplitude breather solutions in $\phi^4$ theory. Phys Rev Lett 58: 747–750. doi: 10.1103/PhysRevLett.58.747
[22]	Tew RB, Wattis JAD (2001) Quasi-continuum approximations for travelling kinks in diatomic lattices. J Phys A: Math Gen 34: 7163–7180. doi: 10.1088/0305-4470/34/36/304
[23]	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
[24]	Wattis JAD (1993) Approximations to solitary waves on lattices, II: quasi-continuum approximations for fast and slow waves. J Phys A: Math Gen 26: 1193–1209. doi: 10.1088/0305-4470/26/5/036
[25]	Wattis JAD (1996) Approximations to solitary waves on lattices, III: monatomic lattice with second neighbour interactions. J Phys A: Math Gen 29: 8139–8157. doi: 10.1088/0305-4470/29/24/035
[26]	Wattis JAD (2001) Solitary waves in a diatomic lattice: Analytic approximations for a wide range of speeds by quasi-continuum methods. Phys Lett A 284: 16–22. doi: 10.1016/S0375-9601(01)00277-8
[27]	Wattis JAD, James LM (2014) Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices. J Phys A: Math Theor 47: 345101. doi: 10.1088/1751-8113/47/34/345101
[28]	Zabusky NJ, Kruskal MD (1965) Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15: 240–243.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)