Uniqueness of Solitary Waves in the High-Energy Limit of FPU-Type Chains
Recent asymptotic results in  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.
KeywordsLattice 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.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
- 19.Truskinovsky, L., Vainchtein, A.: Solitary waves in a nonintegrable Fermi-Pasta-Ulam chain. Phys. Rev. E 90(042903), 1–8 (2014)Google Scholar