Uniqueness of Solitary Waves in the High-Energy Limit of FPU-Type Chains

  • Michael Herrmann
  • Karsten Matthies
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 205)


Recent asymptotic results in [12] provided detailed information on the shape of solitary high-energy travelling waves in FPU atomic chains. In this note we use and extend the methods to understand the linearisation of the travelling wave equation. We show that there are not any other zero eigenvalues than those created by the translation symmetry and this implies a local uniqueness result. The key argument in our asymptotic analysis is to replace the linear advance-delay-differential equation for the eigenfunctions by an approximate ODE.


Lattice waves High-energy limit FPU-type chain Uniqueness of solitary waves Asymptotic analysis 

Mathematics Subject Classification:

37K60 37K40 74H10 



The authors are grateful for the support by the Deutsche Forschungsgemeinschaft (DFG individual grant HE 6853/2-1) and the London Mathematical Society (LMS Scheme 4 Grant, Ref 41326). KM would like to thank for the hospitality during a sabbatical stay at the University of Münster.


  1. 1.
    Archilla, J.F.R., Kosevich, Y.A., Jiménez, N., Sánchez-Morcillo, V.J., García-Raffi, L.M.: Ultradiscrete kinks with supersonic speed in a layered crystal with realistic potentials. Phys. Rev. E 91, 022912 (Feb 2015)Google Scholar
  2. 2.
    Friesecke, G., Matthies, K.: Atomic-scale localization of high-energy solitary waves on lattices. Phys. D 171(4), 211–220 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Friesecke, G., Pego, R.L.: Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12(6), 1601–1627 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Friesecke, G., Pego, R.L.: Solitary waves on FPU lattices. II. Linear implies nonlinear stability. Nonlinearity 15(4), 1343–1359 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Friesecke, G., Pego, R.L.: Solitary waves on Fermi-Pasta-Ulam lattices. III. Howland-type Floquet theory. Nonlinearity 17(1), 207–227 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Friesecke, G., Pego, R.L.: Solitary waves on Fermi-Pasta-Ulam lattices. IV. Proof of stability at low energy. Nonlinearity 17(1), 229–251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fraternali, F., Senatore, L., Daraio, C.: Solitary waves on tensegrity lattices. J. Mech. Phys. Solids 60(6), 1137–1144 (2012)CrossRefGoogle Scholar
  8. 8.
    Filip, A.-M., Venakides, S.: Existence and modulation of traveling waves in particle chains. Comm. Pure Appl. Math. 51(6), 693–735 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Friesecke, G., Wattis, J.A.D.: Existence theorem for solitary waves on lattices. Comm. Math. Phys. 161(2), 391–418 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Herrmann, M.: Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains. Proc. Roy. Soc. Edinburgh Sect. A 140(4), 753–785 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Herrmann, M.: High-energy waves in superpolynomial FPU-type chains. J. Nonlinear Sci. 27(1), 213–240 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Herrmann, M., Matthies, K.: Asymptotic formulas for solitary waves in the high-energy limit of FPU-type chains. Nonlinearity 28(8), 2767–2789 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Herrmann, M., Rademacher, J.D.M.: Heteroclinic travelling waves in convex FPU-type chains. SIAM J. Math. Anal. 42(4), 1483–1504 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Iooss, G., James, G.: Localized waves in nonlinear oscillator chains. Chaos 15, 015113 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pankov, A.: Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices. Imperial College Press, London (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Schwetlick, H., Zimmer, J.: Existence of dynamic phase transitions in a one-dimensional lattice model with piecewise quadratic interaction potential. SIAM J. Math. Anal. 41(3), 1231–1271 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Teschl, G.: Almost everything you always wanted to know about the Toda equation. Jahresber. Deutsch. Math.-Verein. 103(4), 149–162 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Treschev, D.: Travelling waves in FPU lattices. Discrete Contin. Dyn. Syst. 11(4), 867–880 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Truskinovsky, L., Vainchtein, A.: Solitary waves in a nonintegrable Fermi-Pasta-Ulam chain. Phys. Rev. E 90(042903), 1–8 (2014)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für Numerische und Angewandte MathematikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

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