Eur. Phys. J. Plus (2017) 132: 371
https://doi.org/10.1140/epjp/i2017-11655-9

Regular Article

Elliptic function and solitary wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability

¹ Faculty of Science, Jiangsu University, 212013, Zhenjiang, Jiangsu, China
² Mathematics Department, Faculty of Science, Taibah University, Al-Ula, Saudi Arabia
³ Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

^* e-mail: aly742001@yahoo.com

Received: 18 March 2017
Accepted: 12 July 2017
Published online: 30 August 2017

Abstract

The higher-order nonlinear Schrödinger equation (NLSE) with fourth-order dispersion, cubic-quintic terms, self-steepening and nonlinear dispersive terms describes the propagation of extremely short pulses in optical fibers. In this paper, the elliptic function, bright and dark solitons and solitary wave solutions of higher-order NLSE are constructed by employing a modified extended direct algebraic method, which has important applications in applied mathematics and physics. Furthermore, we also present the formation conditions of the bright and dark solitons for this equation. The modulation instability is utilized to discuss the stability of these solutions, which shows that all solutions are exact and stable. Many other higher-order nonlinear evolution equations arising in applied sciences can also be solved by this powerful, effective and reliable method.