Exact states in waveguides with periodically modulated nonlinearity

¹ Department of Mathematics and Physics, Azusa Pacific University - Azusa, CA 91702-7000, USA
² Department of Mechanical Engineering, University of Hong Kong - Pokfulam Road, Hong Kong
³ School of Engineering, Fraser Noble Building, King's College, University of Aberdeen - Aberdeen, AB24 3UE, UK
⁴ Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University Tel Aviv 69978, Israel
⁵ Laboratory of Nonlinear Optical Informatics, ITMO University - St. Petersburg 197101, Russia

Received: 27 July 2017
Accepted: 3 November 2017

Abstract

We introduce a one-dimensional model based on the nonlinear Schrödinger/Gross-Pitaevskii equation where the local nonlinearity is subject to spatially periodic modulation in terms of the Jacobi function, with three free parameters including the period, amplitude, and internal form-factor. An exact periodic solution is found for each set of parameters and, which is more important for physical realizations, we solve the inverse problem and predict the period and amplitude of the modulation that yields a particular exact spatially periodic state. A numerical stability analysis demonstrates that the periodic states become modulationally unstable for large periods, and regain stability in the limit of an infinite period, which corresponds to a bright soliton pinned to a localized nonlinearity-modulation pattern. The exact dark-bright soliton complex in a coupled system with a localized modulation structure is also briefly considered. The system can be realized in planar optical waveguides and cigar-shaped atomic Bose-Einstein condensates.