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Modulating Pulse Solutions to Quadratic Quasilinear Wave Equations over Exponentially Long Length Scales

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Abstract

This paper presents an existence proof for modulating pulse solutions to a wide class of quadratic quasilinear Klein-Gordon equations of the form

$$\partial_t^2 u = \partial_x^2 u - u + f_1(u, \partial_x u, \partial_t u)\partial_x^2 u + f_2(u, \partial_x u, \partial_t u).$$

Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave-train; they are also referred to as ‘moving breathers’ since they are time-periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with three stable, three unstable and infinitely many neutral directions. By transforming part of the equation into a normal form with an exponentially small remainder term and using a generalisation of local invariant-manifold theory to the quasilinear setting, we prove the existence of small-amplitude modulating pulses on domains in space whose length is exponentially large compared to the magnitude of the pulse.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

    M. D. Groves

  2. Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    G. Schneider

Authors
  1. M. D. Groves
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  2. G. Schneider
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Correspondence to M. D. Groves.

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Communicated by A. Kupiainen

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Groves, M.D., Schneider, G. Modulating Pulse Solutions to Quadratic Quasilinear Wave Equations over Exponentially Long Length Scales. Commun. Math. Phys. 278, 567–625 (2008). https://doi.org/10.1007/s00220-007-0400-6

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  • DOI: https://doi.org/10.1007/s00220-007-0400-6

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