Fractional optical solitons for the conformable space– time nonlinear Schro ¨dinger equation with Kerr law nonlinearity Mustafa Inc 1 • Abdullahi Yusuf 1,2 • Aliyu Isa Aliyu 1,2 • Dumitru Baleanu 3,4 Received: 18 January 2018 / Accepted: 22 February 2018 Ó Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this work, we obtain new soliton solutions for the conformable space–time nonlinear Schro ¨dinger equation (CSTNLSE) with Kerr law nonlinearity. Two integration schemes which are projective Ricatti and extended Jacobi elliptic function methods are applied to reach such solutions. The constraints conditions for the existence of soliton solutions are reported. Numerical simulations for some of the obtained solutions are presented. Keywords CSTNLSE  PR method  EJEF method and conformable derivative 1 Introduction In recent time, fractional calculus appear in different fields of science and engineering such as physics, biology, rheology, viscoelasticity, control theory, signal processing, systems identification and electrochemistry. In order to describe nonlinear physical phenomena, obtaining exact solutions for fractional differential equation is one of the key aspect. This & Abdullahi Yusuf yusufabdullahi@fud.edu.ng Mustafa Inc minc@firat.edu.tr Aliyu Isa Aliyu aliyu.isa@fud.edu.ng Dumitru Baleanu dumitru@cankaya.edu.tr 1 Department of Mathematics, Science Faculty, Firat University, 23119 Elazig, Turkey 2 Department of Mathematics, Science Faculty, Federal University Dutse, Jigawa 7156, Nigeria 3 Department of Mathematics, Cankaya University, o ¨g ˇretmenler Cad, 1406530 Ankara, Turkey 4 Institute of Space Sciences, Magurele, Bucharest, Romania 123 Opt Quant Electron (2018) 50:139 https://doi.org/10.1007/s11082-018-1410-7