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

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

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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 L2 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].

Original languageEnglish (US)
Pages (from-to)225-246
Number of pages22
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume35
Issue number1
DOIs
StatePublished - Jan 1 2015

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Keywords

  • Effective dynamics
  • Hamiltonian systems
  • Nonlinear Schrödinger equation
  • Shadowing
  • Tunneling

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