Three-dimensional structures of the spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in PT-symmetric potentials.

Chao-Qing Dai,Yan Wang

doi:10.1371/journal.pone.0100484

Chao-Qing Dai, Yan Wang

Open Access

https://doi.org/10.1371/journal.pone.0100484

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Abstract

The spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in -symmetric potentials is investigated, and two families of analytical three-dimensional spatiotemporal structure solutions are obtained. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results indicate that solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the self-focusing medium, while they are always unstable for all parameters in the self-defocusing medium. Moreover, some dynamical properties of these solutions are discussed, such as the phase switch, power and transverse power-flow density. The span of phase switch gradually enlarges with the decrease of the competing parameter k in -symmetric potentials. The power and power-flow density are all positive, which implies that the power flow and exchange from the gain toward the loss domains in the cell.

Highlights

In the last few decades, there has been a surge of interest in obtaining exact analytical solutions of nonlinear partial differential equations (NPDEs) to describe the natural physical phenomena in numerous branches from mathematical physics, engineering sciences, chemistry to biology [1,2,3]
The aim of this paper is to present 3D spatiotemporal structures of 3DNLSE with power-law nonlinearity in PT -symmetric potentials
Linear stability analysis of analytical solutions We study the linear stability of solutions (2) with (6) and (8)

Summary

Introduction

In the last few decades, there has been a surge of interest in obtaining exact analytical solutions of nonlinear partial differential equations (NPDEs) to describe the natural physical phenomena in numerous branches from mathematical physics, engineering sciences, chemistry to biology [1,2,3]. The nonlinear Schrodinger equation (NLSE), as one of important nonlinear models, has become an intensely studied subjects due to its potential applications in physics, biology and other fields. Abundant mathematical solutions and physical localized structures for various NLSEs have been reported. Two-dimensional accessible solitons [10] and nonautonomous solitons [11] for NLSE in parity-time (PT )-symmetric potentials have been reported. Various nonlinear localized structures in PT -symmetric potentials have been extensively studied. Nonlinear localized modes in PT symmetric optical media with competing gain and loss were studied [14]. Three-dimensional (3D) spatiotemporal structures in PT -symmetric potentials are less studied. 3D spatiotemporal structures in PT -symmetric potentials with power-law nonlinearities are hardly reported

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