Stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains

Herrmann, Michael; Matthies, Karsten

doi:10.1090/tran/7790

Stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains
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by Michael Herrmann and Karsten Matthies PDF

Trans. Amer. Math. Soc. 372 (2019), 3425-3486 Request permission

Abstract:

The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a long-standing open problem and has been solved so far only in a certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in which the waves propagate with near sonic speed, have large wave length, and carry low energy. In this paper we derive a similar result in a complementary asymptotic regime related to fast and strongly localized waves with high energy. In particular, we show that the spectrum of the linearized FPUT operator contains asymptotically no unstable eigenvalues except for the neutral ones that stem from the shift symmetry and the spatial discreteness. This ensures that high-energy waves are linearly stable in some orbital sense, and the corresponding nonlinear stability is granted by the general, non-asymptotic part of the seminal Friesecke-Pego result and the extension by Mizumachi.

Our analytical work splits into two principal parts. First we refine two-scale techniques that relate high-energy waves to a nonlinear asymptotic shape ODE and provide accurate approximation formulas. In this way we establish the existence, local uniqueness, smooth parameter dependence, and exponential localization of fast lattice waves for a wide class of interaction potentials with algebraic singularity. Afterwards we study the crucial eigenvalue problem in exponentially weighted spaces, so that there is no unstable essential spectrum. Our key argument is that all proper eigenfunctions can asymptotically be linked to the unique bounded and normalized solution of the linearized shape ODE, and this finally enables us to disprove the existence of unstable eigenfunctions in the symplectic complement of the neutral ones.

References

Jaime Angulo Pava, Nonlinear dispersive equations, Mathematical Surveys and Monographs, vol. 156, American Mathematical Society, Providence, RI, 2009. Existence and stability of solitary and periodic travelling wave solutions. MR 2567568, DOI 10.1090/surv/156
G. N. Benes, A. Hoffman, and C. E. Wayne, Asymptotic stability of the Toda $m$-soliton, J. Math. Anal. Appl. 386 (2012), no. 1, 445–460. MR 2834897, DOI 10.1016/j.jmaa.2011.08.007
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 10.3934/dcdss.2012.5.879
Jesús Cuevas-Maraver, Panayotis G. Kevrekidis, Anna Vainchtein, and Haitao Xu, Unifying perspective: solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability, Phys. Rev. E 96 (2017), no. 3, 032214, 9. MR 3816041, DOI 10.1103/physreve.96.032214
W. Dreyer, M. Herrmann, and A. Mielke, Micro-macro transition in the atomic chain via Whitham’s modulation equation, Nonlinearity 19 (2006), no. 2, 471–500. MR 2199399, DOI 10.1088/0951-7715/19/2/013
Gero Friesecke and Karsten Matthies, Atomic-scale localization of high-energy solitary waves on lattices, Phys. D 171 (2002), no. 4, 211–220. MR 1945738, DOI 10.1016/S0167-2789(02)00604-8
G. Friesecke and A. Mikikits-Leitner, Cnoidal waves on Fermi-Pasta-Ulam lattices, J. Dynam. Differential Equations 27 (2015), no. 3-4, 627–652. MR 3435125, DOI 10.1007/s10884-013-9343-0
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 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 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 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 10.1088/0951-7715/17/1/014
Daniel C. Mattis (ed.), The many-body problem, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. An encyclopedia of exactly solved models in one dimension. MR 1241902, DOI 10.1142/9789812796523
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 10.1137/130941638
Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. MR 901236, DOI 10.1016/0022-1236(87)90044-9
Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), no. 2, 308–348. MR 1081647, DOI 10.1016/0022-1236(90)90016-E
Michael Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 4, 753–785. MR 2672069, DOI 10.1017/S0308210509000146
Michael Herrmann, High-energy waves in superpolynomial FPU-type chains, J. Nonlinear Sci. 27 (2017), no. 1, 213–240. MR 3600830, DOI 10.1007/s00332-016-9331-8
Michael Herrmann and Karsten Matthies, Asymptotic formulas for solitary waves in the high-energy limit of FPU-type chains, Nonlinearity 28 (2015), no. 8, 2767–2789. MR 3382585, DOI 10.1088/0951-7715/28/8/2767
Michael Herrmann and Karsten Matthies, Uniqueness of solitary waves in the high-energy limit of FPU-type chains, Patterns of dynamics, Springer Proc. Math. Stat., vol. 205, Springer, Cham, 2017, pp. 3–15. MR 3775401, DOI 10.1007/978-3-319-64173-7_{1}
Michael Herrmann and Alice Mikikits-Leitner, KdV waves in atomic chains with nonlocal interactions, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 2047–2067. MR 3411552, DOI 10.3934/dcds.2016.36.2047
A. Hoffman and C. E. Wayne, Counter-propagating two-soliton solutions in the Fermi-Pasta-Ulam lattice, Nonlinearity 21 (2008), no. 12, 2911–2947. MR 2461047, DOI 10.1088/0951-7715/21/12/011
Aaron Hoffman and C. E. Wayne, Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model, J. Dynam. Differential Equations 21 (2009), no. 2, 343–351. MR 2506667, DOI 10.1007/s10884-009-9134-9
A. Hoffman and C. E. Wayne, Orbital stability of localized structures via Bäcklund transformations, Differential Integral Equations 26 (2013), no. 3-4, 303–320. MR 3059166
A. Hoffman and C. Eugene Wayne, A simple proof of the stability of solitary waves in the Fermi-Pasta-Ulam model near the KdV limit, Infinite dimensional dynamical systems, Fields Inst. Commun., vol. 64, Springer, New York, 2013, pp. 185–192. MR 2986936, DOI 10.1007/978-1-4614-4523-4_{7}
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 10.1016/j.physd.2017.07.004
Amjad Khan and Dmitry E. Pelinovsky, Long-time stability of small FPU solitary waves, Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 2065–2075. MR 3640588, DOI 10.3934/dcds.2017088
Tetsu Mizumachi, Asymptotic stability of lattice solitons in the energy space, Comm. Math. Phys. 288 (2009), no. 1, 125–144. MR 2491620, DOI 10.1007/s00220-009-0768-6
Tetsu Mizumachi, $N$-soliton states of the Fermi-Pasta-Ulam lattices, SIAM J. Math. Anal. 43 (2011), no. 5, 2170–2210. MR 2837499, DOI 10.1137/100792457
Tetsu Mizumachi, Asymptotic stability of $N$-solitary waves of the FPU lattices, Arch. Ration. Mech. Anal. 207 (2013), no. 2, 393–457. MR 3005321, DOI 10.1007/s00205-012-0564-x
Yvan Martel and Frank Merle, Asymptotic stability of solitons of the subcritical gKdV equations revisited, Nonlinearity 18 (2005), no. 1, 55–80. MR 2109467, DOI 10.1088/0951-7715/18/1/004
Tetsu Mizumachi and Robert L. Pego, Asymptotic stability of Toda lattice solitons, Nonlinearity 21 (2008), no. 9, 2099–2111. MR 2430663, DOI 10.1088/0951-7715/21/9/011
Robert L. Pego and Michael I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys. 164 (1994), no. 2, 305–349. MR 1289328
Guido Schneider and C. Eugene Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 390–404. MR 1870156
Gerald Teschl, Almost everything you always wanted to know about the Toda equation, Jahresber. Deutsch. Math.-Verein. 103 (2001), no. 4, 149–162. MR 1879178
Dmitry Treschev, Travelling waves in FPU lattices, Discrete Contin. Dyn. Syst. 11 (2004), no. 4, 867–880. MR 2112708, DOI 10.3934/dcds.2004.11.867
M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity 12 (1999), no. 3, 673–691. MR 1690199, DOI 10.1088/0951-7715/12/3/314
Haitao Xu, Jesús Cuevas-Maraver, Panayotis G. Kevrekidis, and Anna Vainchtein, An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices, Philos. Trans. Roy. Soc. A 376 (2018), no. 2117, 20170192, 26. MR 3789551, DOI 10.1098/rsta.2017.0192
N. J. Zabusky and M. D. Kruskal, Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240–243.