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

Roy H. Goodman; Jeremy L. Marzuola; Michael I. Weinstein

doi:10.3934/dcds.2015.35.225

Article Contents

2015, Volume 35, Issue 1: 225-246. Doi: 10.3934/dcds.2015.35.225

This issue Previous Article Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case Next Article From compact semi-toric systems to Hamiltonian $S^1$-spaces

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

1.
Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102
2.
Mathematics Department, University of North Carolina, Phillips Hall, CB#3250, Chapel Hill, NC 27599
3.
Department of Applied Physics and Applied Mathematics, Department of Mathematics, Columbia University, New York City, NY 10024

Received: November 2013

Revised: April 2014

Published: January 2015

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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].

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Mathematics Subject Classification: Primary: 35Q55, 35Q60.

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