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Breather Solutions in Periodic Media

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Abstract

For nonlinear wave equations existence proofs for breathers are very rare. In the spatially homogeneous case up to rescaling the sine-Gordon equation \({\partial^2_t u = \partial^2_x u - \sin (u)}\) is the only nonlinear wave equation which is known to possess breather solutions. For nonlinear wave equations in periodic media no examples of breather solutions have been known so far. Using spatial dynamics, center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation

$$ s(x) \partial^2_t u(x,t) = \partial^2_x u(x,t) - q(x) u(x,t)+ r(x)u(x,t)^3, $$

with spatially periodic coefficients s, q, and r on the real axis. Such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage.

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Authors and Affiliations

  1. Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Carsten Blank, Martina Chirilus-Bruckner, Vincent Lescarret & Guido Schneider

Authors
  1. Carsten Blank
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  2. Martina Chirilus-Bruckner
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  3. Vincent Lescarret
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  4. Guido Schneider
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Correspondence to Guido Schneider.

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Communicated by P. Constantin

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Blank, C., Chirilus-Bruckner, M., Lescarret, V. et al. Breather Solutions in Periodic Media. Commun. Math. Phys. 302, 815–841 (2011). https://doi.org/10.1007/s00220-011-1191-3

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  • DOI: https://doi.org/10.1007/s00220-011-1191-3

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