A PT -Symmetric Dual-Core System with the Sine-Gordon Nonlinearity and Derivative Coupling

Jesús Cuevas-Maraver,Boris Malomed,Panayotis Kevrekidis

doi:10.3390/sym8060039

Abstract

As an extension of the class of nonlinear PT -symmetric models, we propose a system of sine-Gordon equations, with the PT symmetry represented by balanced gain and loss in them. The equations are coupled by sine-field terms and first-order derivatives. The sinusoidal coupling stems from local interaction between adjacent particles in coupled Frenkel–Kontorova (FK) chains, while the cross-derivative coupling, which was not considered before, is induced by three-particle interactions, provided that the particles in the parallel FK chains move in different directions. Nonlinear modes are then studied in this system. In particular, kink-kink (KK) and kink-anti-kink (KA) complexes are explored by means of analytical and numerical methods. It is predicted analytically and confirmed numerically that the complexes are unstable for one sign of the sinusoidal coupling and stable for another. Stability regions are delineated in the underlying parameter space. Unstable complexes split into free kinks and anti-kinks that may propagate or become quiescent, depending on whether they are subject to gain or loss, respectively.

Highlights

Dual-core waveguides with intrinsic nonlinearity carried by each core offer a convenient setting for the creation of stable dissipative solitons, by application of linear gain to one core and leaving the parallel-coupled mate one lossy
We are interested in solutions for KK and KA complexes interpolating between different flat states, i.e., fixed points (FPs) of Equations (25) and (26), φ0 and ψ0
For the KK complexes, we display the central part of the φ and ψ components and compare them to the perturbative prediction given by Equation (50); for the KA modes, we display both components only in numerical form, as an analytical solution of perturbative Equation (51) is not available

Summary

Introduction

Dual-core waveguides with intrinsic nonlinearity carried by each core offer a convenient setting for the creation of stable dissipative solitons, by application of linear gain to one core and leaving the parallel-coupled mate one lossy. This possibility was first proposed in the context of nonlinear fiber optics in [1,2]; see a review in [3]. Nonlinear conservative models, including those originating from optics [13,14], give rise to continuous families of soliton solutions, rather than isolated ones

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