Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrödinger and Resonant Nonlinear Schrödinger Equation with the Parabolic Law

Wei Gao,Haci Mehmet Baskonus,Ahmad M Husien,Hajar Farhan Ismael,Hasan Bulut

doi:10.3390/app10010219

Abstract

In this paper, the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation in parabolic law media are investigated to obtain the dark, singular, bright-singular combo and periodic soliton solutions. Two powerful methods, the m + G ′ G improved expansion method and the exp − φ ξ expansion method are utilized to construct some novel solutions of the governing equations. The obtained optical soliton solutions are presented graphically to clarify their physical parameters. Moreover, to verify the existence solutions, the constraint conditions are utilized.

Highlights

In the current century, many entropy problems have been expressed by using mathematical models that are nonlinear partial differential equations
Featured Application: The optical soliton solutions obtained in this research paper may be of concern and useful in many fields of science, such as mathematical physics, applied physics, nonlinear science, and engineering
Many entropy problems have been expressed by using mathematical models that are nonlinear partial differential equations

Summary

Introduction

Many entropy problems have been expressed by using mathematical models that are nonlinear partial differential equations. Nonlinear models of the celebrated Klein–Gordon and Dirac equations have been found to admit accurate time dependent soliton-like solutions with the shapes of the so-called q-plane waves. The nonlinear Schrödinger equation, involving cubic and quartic-order dispersion terms, has been investigated to seek the exact optical soliton solutions via the undetermined coefficients method [36], the modified Kudryashov approach [37], the complete discrimination system method [38], the generalized tanh function method [39], the sin-cosine method, as well as the Bernoulli equation approach [40], the semi-inverse variation method [41], a simple equation method [3], and the extended sinh-Gordon expansion method [42]. We use two methods to investigate soliton solutions of the cubic-quartic nonlinear Schrödinger equation and cubic-quartic resonant nonlinear Schrödinger equation with the parabolic law, namely Equations (3) and (4)

Instructions for the Methods

The Cubic-Quartic Nonlinear Schrödinger Equation

The Cubic-Quartic Resonant Nonlinear Schrödinger Equation

Conclusions