Some breathers and multi-breathers for FPU-type chains. (English) Zbl 07136926

Summary: We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is spectrally stable. Symmetry properties are considered as well. In addition, we construct solutions that are close to (possibly infinite) sums of breather solutions.


37K60 Lattice dynamics; integrable lattice equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
82C22 Interacting particle systems in time-dependent statistical mechanics


Ada95; MPFR
Full Text: DOI arXiv


[1] Arioli, G., Gazzola, F.: Existence and numerical approximation of periodic motions of an infinite lattice of particles. ZAMP 46, 898-912 (1995) · Zbl 0838.34046
[2] Arioli, G.; Gazzola, F., Periodic motions of an infinite lattice of particles with nearest neighbour interaction, Nonlin. Anal. TMA, 26, 1103-1114 (1996) · Zbl 0867.70004
[3] Arioli, G.; Gazzola, F.; Terracini, S., Multibump periodic motions of an infinite lattice of particles, Math. Zeit., 223, 627-642 (1996) · Zbl 0871.34028
[4] Arioli, G.; Koch, H.; Terracini, S., Two novel methods and multi-mode periodic solutions for the Fermi-Pasta-Ulam model, Commun. Math. Phys., 255, 1-19 (2004) · Zbl 1076.70008
[5] Arioli, G.; Koch, H., Spectral stability for the wave equation with periodic forcing, J. Differ. Equ., 265, 2470-2501 (2018) · Zbl 1402.35039
[6] Ada Reference Manual, ISO/IEC 8652:2012(E). Available e.g. at http://www.ada-auth.org/arm.html
[7] A free-software compiler for the Ada programming language, which is part of the GNU Compiler Collection; see http://gnu.org/software/gnat/. Accessed 12 Sept 2016
[8] Berman, G. P.; Izraileva, F. M., The Fermi-Pasta-Ulam problem: fifty years of progress, Chaos, 15, 015104 (2005) · Zbl 1080.37077
[9] Braun O.M., Kivshar Y.S.: The Frenkel-Kontorova model. Springer, Berlin (2004) · Zbl 1140.82001
[10] Fontich, E.; Llave, R.; Sire, Y., Construction of invariant whiskered tori by a parameterization method. Part II: quasi-periodic and almost periodic breathers in coupled map lattices, J. Differ. Equ., 259, 2180-2279 (2015) · Zbl 1351.37258
[11] Fontich, E.; Llave, R.; Martin, P., Dynamical systems on lattices with decaying interaction II: hyperbolic sets and their invariant manifolds, J. Differ. Equ., 250, 2887-2926 (2011) · Zbl 1213.37115
[12] Gorbach, A. V.; Flach, S., Discrete breathers—advances in theory and applications, Phys. Rep., 467, 1-116 (2008)
[13] Gallavotti G. (ed.): The Fermi-Pasta-Ulam problem. A status report, Lecture Notes in Physics 728. Springer, Berlin (2008) · Zbl 1138.81004
[14] Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976) · Zbl 0342.47009
[15] Kevrekidis, P.; Cuevas-Maraver, J.; Pelinovsky, D., Energy Criterion for the spectral stability of discrete breathers, Phys. Rev. Lett., 117, 094101 (2016)
[16] Koukouloyannis, V.; Kevrekidis, P., On the stability of multibreathers in Klein-Gordon chains, Nonlinearity, 22, 2269-2285 (2009) · Zbl 1179.37103
[17] MacKay, R., Discrete breathers: classical and quantum, Phys. A, 288, 174-198 (2000)
[18] Pankov A.: Traveling Waves And Periodic Oscillations in Fermi-Pasta-Ulam Lattices. Imperial College Press, UK (2005) · Zbl 1088.35001
[19] Pelinovsky, D.; Sakovich, A., Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations, Nonlinearity, 25, 3423-3451 (2012) · Zbl 1323.37044
[20] Rabinowitz, P. H., Multibump solutions of differential equations: an overview, Chin. J. Math., 24, 1-36 (1996) · Zbl 0968.37019
[21] The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754-2008
[22] The MPFR library for multiple-precision floating-point computations with correct rounding. GNU MPFR version 4.0.2 (2019). http://www.mpfr.org/
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