Research Article

Moving gap solitons in periodic potentials

  1. Dmitry Pelinovsky1,*,
  2. Guido Schneider2

Article first published online: 25 MAR 2008

DOI: 10.1002/mma.1002

Issue

Mathematical Methods in the Applied Sciences

Mathematical Methods in the Applied Sciences

Volume 31, Issue 14, pages 1739–1760, 25 September 2008

Additional Information

How to Cite

Pelinovsky, D. and Schneider, G. (2008), Moving gap solitons in periodic potentials. Math. Meth. Appl. Sci., 31: 1739–1760. doi: 10.1002/mma.1002

Author Information

  1. 1

    Department of Mathematics, McMaster University, Hamilton, Ont., Canada L8S 4K1

  2. 2

    Institut für Analysis, Dynamik und Modellierung, Fakultät für Mathematik und Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

*Department of Mathematics, McMaster University, Hamilton, Ont., Canada L8S 4K1

Publication History

  1. Issue published online: 7 AUG 2008
  2. Article first published online: 25 MAR 2008
  3. Manuscript Received: 30 APR 2007

Funded by

  • Humboldt Research Foundation
  • Deutsche Forschungsgemeinschaft (DFG)
  • Land Baden-Württemberg

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Keywords:

  • spatial dynamics;
  • homoclinic orbits;
  • moving gap solitons;
  • Gross–Pitaevskii equation;
  • periodic potentials

Abstract

We address the existence of moving gap solitons (traveling localized solutions) in the Gross–Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit solutions of the coupled-mode system. We show, however, that exponentially decaying traveling solutions of the Gross–Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled-mode formalism and are estimated by using techniques of spatial dynamics and local center-stable manifold reductions. Existence of bounded traveling solutions of the Gross–Pitaevskii equation with a single bump surrounded by oscillatory tails on a large interval of the spatial scale is proven by using these techniques. Copyright © 2008 John Wiley & Sons, Ltd.

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