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arXiv · 1607.08534

Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential

Boris Buffoni·Hartmut Schwetlick·Johannes Zimmer
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Abstract

The Frenkel-Kontorova model for dislocation dynamics from 1938 is given by a chain of atoms, where neighbouring atoms interact through a linear spring and are exposed to a smooth periodic on-site potential. A dislocation moving with constant speed corresponds to a heteroclinic travelling wave, making a transition from one well of the on-site potential to another. The ensuing system is nonlocal, nonlinear and nonconvex. We present an existence result for a class of smooth nonconvex on-site potentials. Previous results in mathematics and mechanics have been limited to on-site potentials with harmonic wells. To overcome this restriction, we first develop a global centre manifold theory for anharmonic wave trains, then parametrise the centre manifold to obtain asymptotically correct approximations to the solution sought, and finally obtain the heteroclinic wave via a fixed point argument.

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Boris Buffoni, Hartmut Schwetlick, Johannes Zimmer. 2016-07-28. Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential. https://arxiv.org/abs/1607.08534

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