Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Continuation and bifurcations of breathers in a finite discrete NLS equation

Pages: 1227 - 1245, Volume 4, Issue 5, October 2011      doi:10.3934/dcdss.2011.4.1227

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Panayotis Panayotaros - Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F., Mexico (email)

Abstract: We present results on the continuation of breathers in the discrete cubic nonlinear Schrödinger equation in a finite one-dimensional lattice with Dirichlet boundary conditions. In the limit of small inter-site coupling the equation has a finite number of breather solutions and as we increase the coupling we see numerically that all breather branches undergo either fold or pitchfork bifurcations. We also see branches that persist for arbitrarily large coupling and converge to the linear normal modes of the system. The stability of the breathers that persist generally changes as the coupling is varied, although there are at least two branches that preserve their linear and nonlinear stability properties throughout the continuation.

Keywords:  Discrete NLS equations, breathers, bifurcations.
Mathematics Subject Classification:  Primary: 37J20, 35Q55; Secondary: 37J45, 37M20, 37L60.

Received: April 2009;      Revised: October 2009;      Available Online: December 2010.