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Existence and Uniqueness of Traveling Wave for Accelerated Frenkel–Kontorova Model

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Abstract

In this paper, we study the existence and uniqueness of traveling wave solution for the accelerated Frenkel–Kontorova model. This model consists in a system of ODE that describes the motion particles in interaction. The most important applications we have in mind is the motion of crystal defects called dislocations. For this model, we prove the existence of traveling wave solutions under very weak assumptions. The uniqueness of the velocity is also studied as well as the uniqueness of the profile which used different types of strong maximum principle. As far as we know, this is the first result concerning traveling waves for accelerated, spatially discrete system.

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Acknowledgments

The first author was partially supported by ANR AMAM (ANR 10-JCJC 0106), ANR IDEE (ANR-2010-0112-01). This work was partially supported by ANR HJNet (ANR-12-BS01-0008-01) and PHC Utique N30608WB.

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Authors and Affiliations

  1. Labo. de Mathématiques de l’INSA - LMI (EA 3226 - FR CNRS 3335), INSA de Rouen, Normandie Université, 685 Avenue de l’Université, 76801, St Etienne du Rouvray Cedex, France

    N. Forcadel & S. Walha

  2. Laboratory of “Stability and control of systems, nonlinear PDEs”, LR 13 ES 21, Faculty of Sciences, University of Sfax, Sfax, Tunisia

    A. Ghorbel & S. Walha

  3. Higher Institute of Business Administration of Sfax, University of Sfax, Airport Road Km 4, PB 1013, 3018, Sfax, Tunisia

    A. Ghorbel

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  1. N. Forcadel
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  2. A. Ghorbel
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  3. S. Walha
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Correspondence to N. Forcadel.

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Forcadel, N., Ghorbel, A. & Walha, S. Existence and Uniqueness of Traveling Wave for Accelerated Frenkel–Kontorova Model. J Dyn Diff Equat 26, 1133–1169 (2014). https://doi.org/10.1007/s10884-014-9403-0

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  • DOI: https://doi.org/10.1007/s10884-014-9403-0

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