Nanopteron-stegoton traveling waves in spring dimer Fermi-Pasta-Ulam-Tsingou lattices
Author:
Timothy E. Faver
Journal:
Quart. Appl. Math. 78 (2020), 363-429
MSC (2010):
Primary 35C07; Secondary 37K60, 35B25, 35Q53
DOI:
https://doi.org/10.1090/qam/1548
Published electronically:
August 2, 2019
MathSciNet review:
4100287
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Abstract: We study the existence of traveling waves in a spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. This is a one-dimensional lattice of identical particles connected by alternating nonlinear springs. As in the work of Faver and Wright on the mass dimer, or diatomic, lattice, we find that the lattice equations in the long wave scaling are singularly perturbed, and we apply a method of Beale to produce nanopteron traveling waves with wave speed slightly greater than the lattice’s speed of sound. The nanopteron wave profiles are the superposition of an exponentially decaying term (which itself is a small perturbation of a KdV $\operatorname {sech}^2$-type soliton) and a periodic term of very small amplitude. Further generalizing the spring forces from the mass dimer case, we allow the springs’ nonlinearity to contain higher order terms beyond the quadratic. This necessitates the use of composition operators to phrase the long wave problem, and these operators require delicate estimates due to the characteristic superposition of different function types from Beale’s ansatz. Unlike the diatomic case, the value of the leading order term in the traveling wave profiles alternates between particle sites, so that the spring dimer traveling waves are also “stegotons”, in the terminology of LeVeque and Yong. This behavior is absent in the mass dimer and confirms the approximation results of Gaison, Moskow, Wright, and Zhang for the spring dimer.
References
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- Eric Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders, Lecture Notes in Mathematics, vol. 1741, Springer-Verlag, Berlin, 2000. With applications to homoclinic orbits in reversible systems. MR 1770093
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- Randall J. LeVeque and Darryl H. Yong, Solitary waves in layered nonlinear media, SIAM J. Appl. Math. 63 (2003), no. 5, 1539–1560. MR 2001207, DOI https://doi.org/10.1137/S0036139902408151
- Raymond Mortini, The Faà di Bruno formula revisited, Elem. Math. 68 (2013), no. 1, 33–38 (English, with German summary). MR 3016464, DOI https://doi.org/10.4171/EM/216
- Jürgen Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 265–315. MR 199523
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- Wen-Xin Qin, Wave propagation in diatomic lattices, SIAM J. Math. Anal. 47 (2015), no. 1, 477–497. MR 3302592, DOI https://doi.org/10.1137/130949609
- S. M. Sun, Existence of a generalized solitary wave solution for water with positive Bond number less than $1/3$, J. Math. Anal. Appl. 156 (1991), no. 2, 471–504. MR 1103025, DOI https://doi.org/10.1016/0022-247X%2891%2990410-2
- Yuli Starosvetsky and Anna Vainchtein, Solitary waves in FPU lattices with alternating bond potentials, Mechanics Research Communications, 2017.
- Michael E. Taylor, Partial differential equations III. Nonlinear equations, 2nd ed., Applied Mathematical Sciences, vol. 117, Springer, New York, 2011. MR 2744149
- Anna Vainchtein, Yuli Starosvetsky, J. Douglas Wright, and Ron Perline, Solitary waves in diatomic chains, Phys. Rev. E, 93, 2016.
- Christine R. Venney and Johannes Zimmer, Persistence of supersonic periodic solutions for chains with anharmonic interaction potentials between neighbours and next to nearest neighbours, Dyn. Syst. 26 (2011), no. 4, 503–518. MR 2852938, DOI https://doi.org/10.1080/14689367.2011.620565
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- Eberhard Zeidler, Applied functional analysis, Applied Mathematical Sciences, vol. 109, Springer-Verlag, New York, 1995. Main principles and their applications. MR 1347692
References
- C. J. Amick and J. F. Toland, Solitary waves with surface tension. I. Trajectories homoclinic to periodic orbits in four dimensions, Arch. Rational Mech. Anal. 118 (1992), no. 1, 37–69. MR 1151926, DOI https://doi.org/10.1007/BF00375691
- J. Thomas Beale, Water waves generated by a pressure disturbance on a steady stream, Duke Math. J. 47 (1980), no. 2, 297–323. MR 575899
- J. Thomas Beale, Exact solitary water waves with capillary ripples at infinity, Comm. Pure Appl. Math. 44 (1991), no. 2, 211–257. MR 1085829, DOI https://doi.org/10.1002/cpa.3160440204
- Haïm Brezis and Petru Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, with Dedicated to the memory of Tosio Kato, J. Evol. Equ. 1 (2001), no. 4, 387–404. MR 1877265, DOI https://doi.org/10.1007/PL00001378
- John P. Boyd, New directions in solitons and nonlinear periodic waves: polycnoidal waves, imbricated solitons, weakly nonlocal solitary waves, and numerical boundary value algorithms, Advances in applied mechanics, Vol. 27, Adv. Appl. Mech., vol. 27, Academic Press, Boston, MA, 1990, pp. 1–82. MR 1096498, DOI https://doi.org/10.1016/S0065-2156%2808%2970194-7
- John P. Boyd, New directions in solitons and nonlinear periodic waves: polycnoidal waves, imbricated solitons, weakly nonlocal solitary waves, and numerical boundary value algorithms, Advances in applied mechanics, Vol. 27, Adv. Appl. Mech., vol. 27, Academic Press, Boston, MA, 1990, pp. 1–82. MR 1096498, DOI https://doi.org/10.1016/S0065-2156%2808%2970194-7
- John P. Boyd, Weakly nonlocal solitary waves and beyond-all-orders asymptotics: Generalized solitons and hyperasymptotic perturbation theory, Mathematics and its Applications, vol. 442, Kluwer Academic Publishers, Dordrecht, 1998. MR 1636975
- Matthew Betti and Dmitry E. Pelinovsky, Periodic traveling waves in diatomic granular chains, J. Nonlinear Sci. 23 (2013), no. 5, 689–730. MR 3101831, DOI https://doi.org/10.1007/s00332-013-9165-6
- Léon Brillouin, Wave Propagation in Periodic Structures, Dover Phoenix Editions, New York, NY, 1953.
- Martina Chirilus-Bruckner, Christopher Chong, Oskar Prill, and Guido Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 5, 879–901. MR 2877354, DOI https://doi.org/10.3934/dcdss.2012.5.879
- C. Chong, M. A. Porter, P. G. Kevrekidis, and C. Daraio, Nonlinear coherent structures in granular crystals, arXiv:1612.03977 (2016).
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI https://doi.org/10.1016/0022-1236%2871%2990015-2
- Thierry Dauxois, Fermi, Pasta, Ulam, and a mysterious lady, Physics Today, 61(1):55–57, 2008.
- Timothy E. Faver, Nanopteron-stegoton traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou lattices, Ph.D. thesis, Drexel University, Philadelphia, PA, May 2018.
- G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit, Nonlinearity 12 (1999), no. 6, 1601–1627. MR 1726667, DOI https://doi.org/10.1088/0951-7715/12/6/311
- G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. II. Linear implies nonlinear stability, Nonlinearity 15 (2002), no. 4, 1343–1359. MR 1912298, DOI https://doi.org/10.1088/0951-7715/15/4/317
- G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. III. Howland-type Floquet theory, Nonlinearity 17 (2004), no. 1, 207–227. MR 2023440, DOI https://doi.org/10.1088/0951-7715/17/1/013
- G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. IV. Proof of stability at low energy, Nonlinearity 17 (2004), no. 1, 229–251. MR 2023441, DOI https://doi.org/10.1088/0951-7715/17/1/014
- E. Fermi, J. Pasta, and S. Ulam, Studies of nonlinear problems, Lect. Appl. Math., 12:143–56, 1955.
- Gero Friesecke and Jonathan A. D. Wattis, Existence theorem for solitary waves on lattices, Comm. Math. Phys. 161 (1994), no. 2, 391–418. MR 1266490
- Timothy E. Faver and J. Douglas Wright, Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity, SIAM J. Math. Anal. 50 (2018), no. 1, 182–250. MR 3744993, DOI https://doi.org/10.1137/15M1046836
- Jeremy Gaison, Shari Moskow, J. Douglas Wright, and Qimin Zhang, Approximation of polyatomic FPU lattices by KdV equations, Multiscale Model. Simul. 12 (2014), no. 3, 953–995. MR 3226745, DOI https://doi.org/10.1137/130941638
- Aaron Hoffman and J. Douglas Wright, Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio, Phys. D 358 (2017), 33–59. MR 3694480, DOI https://doi.org/10.1016/j.physd.2017.07.004
- Gérard Iooss and Klaus Kirchgässner, Water waves for small surface tension: an approach via normal form, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 3-4, 267–299. MR 1200201, DOI https://doi.org/10.1017/S0308210500021119
- Mathew A. Johnson and J. Douglas Wright, Generalized solitary waves in the gravity-capillary Whitham equation, arXiv preprint arXiv:1807.11469 (2018).
- P. G. Kevrekidis, Non-linear waves in lattices: past, present, future, IMA J. Appl. Math. 76 (2011), no. 3, 389–423. MR 2806003, DOI https://doi.org/10.1093/imamat/hxr015
- Rainer Kress, Linear integral equations, Applied Mathematical Sciences, vol. 82, Springer-Verlag, Berlin, 1989. MR 1007594
- Eric Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders: With applications to homoclinic orbits in reversible systems, Lecture Notes in Mathematics, vol. 1741, Springer-Verlag, Berlin, 2000. MR 1770093
- Christopher J. Lustri and Mason A. Porter, Nanoptera in a period-2 Toda chain, SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, 1182–1212. MR 3788193, DOI https://doi.org/10.1137/16M108639X
- Randall J. LeVeque and Darryl H. Yong, Phase plane behavior of solitary waves in nonlinear layered media, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 43–51. MR 2053158
- Randall J. LeVeque and Darryl H. Yong, Solitary waves in layered nonlinear media, SIAM J. Appl. Math. 63 (2003), no. 5, 1539–1560. MR 2001207, DOI https://doi.org/10.1137/S0036139902408151
- Raymond Mortini, The Faà di Bruno formula revisited, Elem. Math. 68 (2013), no. 1, 33–38 (English, with German summary). MR 3016464, DOI https://doi.org/10.4171/EM/216
- Jürgen Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 265–315. MR 0199523
- Alexander Pankov, Travelling waves and periodic oscillations in Fermi-Pasta-Ulam lattices, Imperial College Press, London, 2005. MR 2156331
- Wen-Xin Qin, Wave propagation in diatomic lattices, SIAM J. Math. Anal. 47 (2015), no. 1, 477–497. MR 3302592, DOI https://doi.org/10.1137/130949609
- S. M. Sun, Existence of a generalized solitary wave solution for water with positive Bond number less than $1/3$, J. Math. Anal. Appl. 156 (1991), no. 2, 471–504. MR 1103025, DOI https://doi.org/10.1016/0022-247X%2891%2990410-2
- Yuli Starosvetsky and Anna Vainchtein, Solitary waves in FPU lattices with alternating bond potentials, Mechanics Research Communications, 2017.
- Michael E. Taylor, Partial differential equations III. Nonlinear equations, 2nd ed., Applied Mathematical Sciences, vol. 117, Springer, New York, 2011. MR 2744149
- Anna Vainchtein, Yuli Starosvetsky, J. Douglas Wright, and Ron Perline, Solitary waves in diatomic chains, Phys. Rev. E, 93, 2016.
- Christine R. Venney and Johannes Zimmer, Persistence of supersonic periodic solutions for chains with anharmonic interaction potentials between neighbours and next to nearest neighbours, Dyn. Syst. 26 (2011), no. 4, 503–518. MR 2852938, DOI https://doi.org/10.1080/14689367.2011.620565
- Christine R. Venney and Johannes Zimmer, Travelling lattice waves in a toy model of Lennard-Jones interaction, Quart. Appl. Math. 72 (2014), no. 1, 65–84. MR 3185132, DOI https://doi.org/10.1090/S0033-569X-2013-01320-4
- Eberhard Zeidler, Applied functional analysis, Applied Mathematical Sciences, vol. 109, Springer-Verlag, New York, 1995. Main principles and their applications. MR 1347692
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Additional Information
Timothy E. Faver
Affiliation:
Mathematics Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
MR Author ID:
1074839
Email:
t.e.faver@math.leidenuniv.nl
Keywords:
FPU,
FPUT,
nonlinear hetergeneous lattice,
dimer,
solitary traveling wave,
periodic traveling wave,
singular perturbation,
nanopteron,
stegoton,
composition operator
Received by editor(s):
August 7, 2018
Received by editor(s) in revised form:
May 23, 2019
Published electronically:
August 2, 2019
Article copyright:
© Copyright 2019
Brown University