Vol.:(0123456789) Optical and Quantum Electronics ( 2018) 50:314 https://doi.org/10.1007/s11082-018-1583-0 1 3 W‑shaped optical solitons of Chen–Lee–Liu equation by Laplace–Adomian decomposition method O. González‑Gaxiola 1  · Anjan Biswas 2,3 Received: 9 June 2018 / Accepted: 27 July 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract This paper studies Chen–Lee–Liu equation in optical fbers by the aid of Laplace Adomian decomposition method. The search is for W-shaped solitons numerically. The numerical results together with high level accuracy plots are exhibited. Keywords Nonlinear Schrödinger equation · Chen–Lee–Liu equation · W-shaped optical solitons · Adomian decomposition method · Laplace transform 1 Introduction The dynamics of optical solitons provide cutting edge technology to fber-optic communication system. There are several models that govern this dynamics for polarization-preserving fbers as well as birefringent fbers. One of the most visible equations is the nonlinear Schrödinger’s equation (NLSE). While this model is very commonly studied in nonlinear optics, there are three models that are a byproduct of this equation. They are referred to as derivative NLSE (DNLSE) and are commonly designated as DNLSE-I, DNLSE-II and DNLSE-III. This paper will be studying DNLSE-II model that is alternatively referred to as Chen–Lee–Liu (CLL) equation. There have been considerable studies with regards to this equation in the past (Triki et al. 2017, 2018a; Biswas et al. 2018a, b; Banaja et al. 2017; Bakodah et al. 2017; Asma et al. 2018). However, the results that have been reported thus far are all outcomes of ana- lytical fndings. This paper will address CLL equation using a purely numerical scheme. It is the Laplace Adomian decomposition (LADM) method. This algorithm will retrieve numeri- cal simulations to the model to reveal W-shaped optical soliton solutions to the model. After a quick re-visitation of the model, the numerical scheme will be introduced and fnally the * O. González-Gaxiola ogonzalez@correo.cua.uam.mx 1 Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana- Cuajimalpa, Vasco de Quiroga 4871, 05348 Mexico City, Mexico 2 Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762, USA 3 Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa