Exact solutions of perturbed nonlinear Schrödinger’ s equation with Kerr law nonlinearity by improved tan /n ðÞ 2  -expansion method Naveed Ahmed 1 · Amna Irshad 1 · Syed Tauseef Mohyud-Din 2 · Umar Khan 3 Received: 2 October 2017 / Accepted: 27 December 2017 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract This paper carries out exact solutions of the perturbed nonlinear Scho ¨dinger’s equation withy Kerr law nonlinearity by using the improved tan /n ðÞ 2  -expansion method. The exact solutions contain four types: hyperbolic function solution, trigonometric func- tion solution, exponential solution, and rational solution. The method appears to be easier and faster by means of symbolic computational system and can be applied to the other nonlinear evolution equations in mathematical physics. Keywords Improved tan /n ðÞ 2  -expansion method · Hyperbolic function solution · Trigonometric function solution · Rational solution the perturbed nonlinear Scho ¨dinger’s equation with Kerr law nonlinearity 1 Introduction “The most incomprehensible thing about the world is that it is at all comprehensible” (Albert Einstein), but the question is how do we fully understand incomprehensible things? Nonlinear Science provides some clues in the regard (He 2009). Since the world around us is inherently nonlinear. Solitary wave solutions of nonlinear evolution equations plays an important role in the study of nonlinear complex physical phenomena and becomes one of the most exciting and extremely active area of research investigation. Which appears in various branches of mathematical-physical sciences such & Naveed Ahmed nidojan@gmail.com 1 Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan 2 University of Islamabad (UoI), Islamabad, Pakistan 3 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 123 Opt Quant Electron (2018)50:45 https://doi.org/10.1007/s11082-017-1314-y