Wave Modes Trapped in Rotating Nonlinear Potentials
We study modes trapped in a rotating ring with the local strength of the nonlinearity modulated as \(\cos \left (2\theta \right )\) , where θ is the azimuthal angle. This modulation pattern may be of three different types: self-focusing (SF), self-defocus
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Abstract We study modes trapped in a rotating ring with the local strength of the nonlinearity modulated as cos .2/, where is the azimuthal angle. This modulation pattern may be of three different types: self-focusing (SF), self-defocusing (SDF), and alternating SF-SDF. The model, based on the nonlinear Schrödinger (NLS) equation with periodic boundary conditions, applies to the light propagation in a twisted pipe waveguide, and to a Bose-Einstein condensate (BEC) loaded into a toroidal trap, under the action of the rotating nonlinear pseudopotential induced by means of the Feshbach resonance in an inhomogeneous external field. This is the difference from the recently considered similar setting with the rotating linear potential. In the SF, SDF, and alternating regimes, four, three, and five different types of stable trapped modes are identified, respectively: even, odd, second-harmonic (2H), symmetry-breaking, and 2H-breaking ones. The shapes and stability of these modes, together with transitions between them, are investigated in the first rotational Brillouin zone. Ground-state modes are identified in each regime. Boundaries between symmetric and asymmetric modes are also found in an analytical form, by means of a two-mode approximation.
Y. Li Faculty of Engineering, Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel Department of Applied Physics, South China Agricultural University, Guangzhou 510642, China e-mail: [email protected] W. Pang Department of Experiment Teaching, Guangdong University of Technology, Guangzhou 510006, China e-mail: [email protected] B.A. Malomed () Faculty of Engineering, Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel e-mail: [email protected] R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__8, © Springer International Publishing Switzerland 2014
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1 Introduction Optical and matter waves exhibit a plenty of dynamical scenarios under the action of effective nonlinear potentials (which may sometimes be combined with usual linear potentials) [1]. The dynamics of such systems is governed by the nonlinear Schrödinger equation (NLSE) in optical media, or Gross-Pitaevskii equation (GPE) in the context of Bose-Einstein condensates (BECs). In either case, the nonlinear pseudopotential [2] may be induced by a regular [3, 4] or singular [5] spatial modulation of the local nonlinearity. These systems have been studied theoretically in a variety of one- [3,5] and two-dimensional (1D and 2D) [4] settings, and recently reviewed in Ref. [1]. To the same general class belong models which predict that, in any dimension D, stable fundamental and vortex solitons can be supported by a purely self-defocusing (SDF) nonlinearity growing towards the periphery (r ! 1) at any rate faster than r D [6]. In optics, such nonlinear potentials may
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