Exact States in Waveguides With Periodically Modulated Nonlinearity

TL;DR

This paper presents exact solutions for waveguides with spatially periodic nonlinearities, analyzes their stability, and explores potential physical realizations in optics and Bose-Einstein condensates.

Contribution

It introduces a model with exact solutions for periodically modulated nonlinearities and solves the inverse problem to design specific spatial states.

Findings

Exact periodic solutions are derived for the nonlinear Schrödinger equation with Jacobi dn modulation.

Stability analysis shows states are modulationally unstable at large periods but stable as period approaches infinity.

The model can be implemented in optical waveguides and Bose-Einstein condensates.

Abstract

We introduce a one-dimensional model based on the nonlinear Schrodinger/Gross-Pitaevskii equation where the local nonlinearity is subject to spatially periodic modulation in terms of the Jacobi dn 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. 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. 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.