Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials.

Yong Chen,Zhenya Yan

doi:10.1038/srep23478

Abstract

Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields.

Highlights

The complex gain-or-loss term has been verified to admit optical rogue waves[51]
We focus on the generalized form of the third-order nonlinear Schrödinger (NLS) equation[52,53,54] in non-Hermitian potentials, that is, the NLS equation with third-order dispersion (TOD) and complex PT -symmetry potentials i ∂ψ ∂z
G ψ 2ψ where Raman effect, nonlinear dispersion terms, and higher-order dispersion terms are neglected[49,50,55,56], ψ ≡ ψ(x, z) is a complex wave function of x, z, z denotes the propagation distance, the real parameter β stands for the coefficient of TOD, the PT -symmetric potential requires that V(x) = V(− x) and W(x) = − W(− x) describing the real-valued external potential and gain-and-loss distribution, respectively, and g > 0 is real-valued inhomogeneous self-focusing nonlinearity

Summary

Introduction

The NLS equation with only third-order dispersion was used to numerically confirm the experimental observation of the spectral signature of the collision between a soliton and the dispersive wave[52]. The PT -symmetric linear and nonlinear modes in the third-order NLS equation were not studied before. Our aim in this paper is to investigate the linear and nonlinear modes of the third-order NLS equation in the presence of physically interesting PT -symmetric potentials, e.g., Scarff-II-like potential and harmonic-Gaussian potential. We find that some parameters can modulate the stable nonlinear modes even if the linear PT -symmetric phases are broken. In Section of Results, we introduce the NLS equation with third-order dispersion in the presence of complex PT -symmetric potentials. We consider the nonlinear modes and their stability in the PT -symmetric Scarff-II-like and harmonic-Gaussian potentials.

Methods

Results

Conclusion