On Solutions of an Extended Nonlocal Nonlinear Schrödinger Equation in Plasmas

Yehui Huang,Zhenjun Ye,Hongqing Jing,Min Li,Yuqin Yao

doi:10.3390/math8071099

Abstract

The parity-time symmetric nonlocal nonlinear Schrödinger equation with self-consistent sources (PTNNLSESCS) is used to describe the interaction between an high-frequency electrostatic wave and an ion-acoustic wave in plasmas. In this paper, the soliton solutions, rational soliton solutions and rogue wave solutions are derived for the PTNNLSESCS via the generalized Darboux transformation. We find that the soliton solutions can exhibit the elastic interactions of different type of solutions such as antidark-antidark, dark-antidark, and dark-dark soliton pairs on a continuous wave background. Also, we discuss the degenerate case in which only one antidark or dark soliton remains. The rogue wave solution is derived in some specially chosen situations.

Highlights

In 1998, the parity-time (PT) symmetry firstly appeared in quantum mechanics since Bender and Boettcher pointed out that non-Hermitian Hamiltonians exhibit entirely real spectra, provided that they respect both the parity and time-reversal symmetries [1]
Xu et al studied the nonsingular localized wave solutions of the partially PT-symmetric nonlocal Davey Stewartson I equation with zero background via the elementary Darboux transformation [14]
This paper is organized as follows: In Section 2, we review elementary Darboux transformation of the PT-symmetric nonlinear Schrödinger (NLS) equation and derive the generalized Darboux transformation of the PTNNLSESCS

Summary

Introduction

In 1998, the parity-time (PT) symmetry firstly appeared in quantum mechanics since Bender and Boettcher pointed out that non-Hermitian Hamiltonians exhibit entirely real spectra, provided that they respect both the parity and time-reversal symmetries (usually called the parity-time symmetry) [1]. M. Li et al derived the rational solutions of the PT-symmetric nonlocal NLS equation by generalized Darboux transformation [9]. Studied novel higher-order rational solitons of the integrable nonlocal nonlinear Schrödinger equation with the self-induced PT-symmetric potential by the generalized perturbation N-1 fold Darboux transformation [12]. T. Xu et al studied the nonsingular localized wave solutions of the partially PT-symmetric nonlocal Davey Stewartson I equation with zero background via the elementary Darboux transformation [14]. We construct the generalized Darboux transformation for the parity-time symmetric nonlocal nonlinear Schrödinger equation with self-consistent sources (PTNNLSESCS) and further reveal the soliton solution and the rational soliton phenomena on the cw background.

Elementary Darboux Transformation for the PTNNLSESCS

Soliton Solutions of the PTNNLSESCS

Rational Solutions of the PTNNLSESCS

Conclusions