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Branching patterns of wave trains in mass-in-mass lattices

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  11 January 2024

Ling Zhang  and
Shangjiang Guo
Ling Zhang
Affiliation:
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, People's Republic of China (zhangling@cug.edu.cn)
Shangjiang Guo
Affiliation:
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, People's Republic of China Center for Mathematical Sciences, China University of Geosciences, Wuhan, Hubei 430074, People's Republic of China (guosj@cug.edu.cn)
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Abstract

We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near $\mu =0$ in the nonresonance case and in the resonance case $p:q$ where $q$ is not an integer multiple of $p$. Furthermore, we obtain the multiplicity of bichromatic wave trains in $p:q$ resonance where $q$ is an integer multiple of $p$, based on the singular theorem.

Keywords

Wave trainsbifurcationresonancesingularity theory

MSC classification

Secondary: 34K13: Periodic solutions 34K18: Bifurcation theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 27
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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