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

Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation

Pages: 225 - 246, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.225

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Roy H. Goodman - Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, United States (email)
Jeremy L. Marzuola - Mathematics Department, University of North Carolina, Phillips Hall, CB#3250, Chapel Hill, NC 27599, United States (email)
Michael I. Weinstein - Department of Applied Physics and Applied Mathematics, Department of Mathematics, Columbia University, New York City, NY 10024, United States (email)

Abstract: We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption, as well as the behavior of Bose-Einstein condensates. For small $L^2$ norm (low power), the solution executes beating oscillations between the two wells. There is a power threshold at which a symmetry breaking bifurcation occurs. The set of guided mode solutions splits into two families of solutions. One type of solution is concentrated in either well of the potential, but not both. Solutions in the second family undergo tunneling oscillations between the two wells. A finite dimensional reduction (system of ODEs) derived in [17] is expected to well-approximate the PDE dynamics on long time scales. In particular, we revisit this reduction, find a class of exact solutions and shadow them in the (NLS/GP) system by applying the approach of [17].

Keywords:  Nonlinear Schrödinger equation, tunneling, effective dynamics, Hamiltonian systems, shadowing.
Mathematics Subject Classification:  Primary: 35Q55, 35Q60.

Received: November 2013;      Revised: April 2014;      Available Online: August 2014.