Optik 158 (2018) 434–450 Contents lists available at ScienceDirect Optik journal homepage: www.elsevier.de/ijleo Original research article Optical soliton and rogue wave solutions of the ultra-short femto-second pulses in an optical fiber via two different methods and its applications Mostafa M.A. Khater a , Aly R. Seadawy b,c,∗ , Dianchen Lu a,∗∗ a Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China b Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia c Mathematics Department, Faculty of Science, Beni-Suef University, Egypt a r t i c l e i n f o Article history: Received 21 November 2017 Accepted 22 December 2017 Keywords: Ultra-short femtosecond pulses in an optical fiber Nonlinear complex fractional Kundu–Eckhaus equation New auxiliary equation method Novel ( G ′ G ) -expansion method Optical solitary traveling wave solutions a b s t r a c t In this research, we study exact and solitary traveling solutions of the Kundu–Eckhaus equation with variable coefficients and its relation with famous Schrödinger equation and the complex Burgers equation. Nonlinear complex fractional Kundu–Eckhaus equation has vital role in description of the propagation of the ultra-short femtosecond pulses in an optical fiber. This model has a vital role in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. We apply a new auxiliary equation method and novel ( G ′ G ) -expansion method on this model to get new exact and solitary traveling wave solutions. The aim of these solutions is showing the slowly varying amplitude of the pulse envelope and discover new physical properties of ultra-short femto second pulses and that happened by using the new from of soliton solutions which obtained by mentioned methods. © 2017 Elsevier GmbH. All rights reserved. 1. Introduction The beginning of the twentieth century was the beginning of the electronic, scientific, industrial revolution and even the atomic revolution. Many of the mathematical, physical and chemical models that humans have been able to understand and study their behavior have emerged, allowing it to use in this scientific progress that has arrived, but it has not been enough, and has begun to look forward to new horizons of development, progress and sophistication. In order to keep pace with the progress made by mankind, the human person was in dire need of a tool that would enable it to build structures with complex specifications very and also very precise devices so that the human being study and learn until he got to femtosecond lasers. State-of-the-art laser processing techniques with ultrashort light pulses can be used to structure materials with a sub-micrometre resolution and for examples of the application of this techniques (micro-optical components, direct laser writing (DLW) of suitable photoresists and other transparent media can create intricate three-dimensional photonic crystals (PhC), optical waveguides, tissue engineering (TE) scaffolds, etc.) and not just that but also there exist many of optical devices that specially design for ultrashort pulses is the pulse compressor. The pulse compressor can be used to control the spectral phase of ultrashort pulses. With the advent of all these capabilities and possibilities, scientists have studied an ultrashort ∗ Corresponding author at: Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia. ∗∗ Corresponding author at: Faculty of Science, Jiangsu University, R. Zhenjiang, Jiangsu 212013, PR China. E-mail addresses: Aly742001@yahoo.com (A.R. Seadawy), dclu@ujs.edu.cn (D. Lu). https://doi.org/10.1016/j.ijleo.2017.12.120 0030-4026/© 2017 Elsevier GmbH. All rights reserved.