Existence of Solitary Waves in One Dimensional Peridynamics

@article{Pego2018ExistenceOS,
  title={Existence of Solitary Waves in One Dimensional Peridynamics},
  author={Robert L. Pego and Truong-Son Van},
  journal={Journal of Elasticity},
  year={2018},
  pages={1-30}
}
We give a rigorous proof of existence for solitary waves of a peridynamics model in one space dimension recently investigated by Silling (J. Mech. Phys. Solids 96:121–132, 2016). We adapt the variational framework developed by Friesecke and Wattis (Commun. Math. Phys. 161:391–418, 1994) for the Fermi-Pasta-Ulam-Tsingou lattice equations to treat a truncated problem which cuts off short-range interactions, then pass to the limit. 
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References

SHOWING 1-7 OF 7 REFERENCES
Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit
This paper is the first in a series to address questions of qualitative behaviour, stability and rigorous passage to a continuum limit for solitary waves in one-dimensional non-integrable lattices
Unimodal wavetrains and solitons in convex Fermi–Pasta–Ulam chains
  • M. Herrmann
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2010
We consider atomic chains with nearest neighbour interactions and study periodic travelling waves and homoclinic travelling waves, which are called wavetrains and solitons, respectively. Our main
Existence theorem for solitary waves on lattices
AbstractIn this article we give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian $$H = \sum\limits_{n \in \mathbb{Z}} {\left(
Studies of nonlinear problems i
Their studies were first described in Los Alamos Report LA-1940 May 1955 LA-1940 Page 4 " We have, therefore, a dynamical system of 64 particles with forces acting between neighbors with fixed end