Solitary waves in atomic chains and peridynamical media

Michael Herrmann; Karsten Matthies; Michael Herrmann; Karsten Matthies

doi:10.3934/mine.2019.2.281

Mathematics in Engineering

2019, Volume 1, Issue 2: 281-308. doi: 10.3934/mine.2019.2.281

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Solitary waves in atomic chains and peridynamical media

Michael Herrmann ¹,
Karsten Matthies ^{2
,
,}

1.
Technische Universität Braunschweig, Institute of Computational Mathematics, Universitätsplatz 2, 38106 Braunschweig, Germany
2.
University of Bath, Department of Mathematical Sciences, Bath, BA2 7AY, United Kingdom

Received: 23 September 2018 Accepted: 23 January 2019 Published: 07 March 2019

Peridynamics describes the nonlinear interactions in spatially extended Hamiltonian systems by nonlocal integro-di erential equations, which can be regarded as the natural generalization of lattice models. We prove the existence of solitary traveling waves for super-quadratic potentials by maximizing the potential energy subject to both a norm and a shape constraint. We also discuss the numerical computation of waves and study several asymptotic regimes.
- peridynamics,
- Hamiltonian lattices,
- solitary waves,
- variational,
- asymptotic analysis
Citation: Michael Herrmann, Karsten Matthies. Solitary waves in atomic chains and peridynamical media[J]. Mathematics in Engineering, 2019, 1(2): 281-308. doi: 10.3934/mine.2019.2.281

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Abstract

Peridynamics describes the nonlinear interactions in spatially extended Hamiltonian systems by nonlocal integro-di erential equations, which can be regarded as the natural generalization of lattice models. We prove the existence of solitary traveling waves for super-quadratic potentials by maximizing the potential energy subject to both a norm and a shape constraint. We also discuss the numerical computation of waves and study several asymptotic regimes.

References

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