Breather Solutions of the Discrete p-Schrödinger Equation

  • Guillaume JamesEmail author
  • Yuli Starosvetsky
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 7)


We consider the discrete p-Schrödinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order \(\alpha = p - 1> 1\). Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even- or odd-parity reflectional symmetries. We derive in addition analytical approximations for the breather profiles and the corresponding intersecting stable and unstable manifolds, valid on a whole range of nonlinearity orders α. In the limit of weak nonlinearity (α → 1+), we introduce a continuum limit connecting the stationary DpS and logarithmic nonlinear Schrödinger equations. In this limit, breathers correspond asymptotically to Gaussian homoclinic solutions. We numerically analyze the stability properties of breather solutions depending on their even- or odd-parity symmetry. A perturbation of an unstable breather generally results in a translational motion (traveling breather) when α is close to unity, whereas pinning becomes predominant for larger values of α.


Continuum Limit Invariant Manifold Unstable Manifold Homoclinic Orbit Stable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



G.J. acknowledges financial support from the Rhône-Alpes Complex Systems Institute (IXXI). Y.S. is grateful to Israel Science Foundation (Grant 484 /12) for financial support.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire Jean KuntzmannUniversité de Grenoble and CNRSGrenoble Cedex 9France
  2. 2.INRIA, Bipop Team-Project, ZIRST MontbonnotSaint IsmierFrance
  3. 3.Faculty of Mechanical EngineeringTechnion – Israel Institute of TechnologyTechnion City, HaifaIsrael

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