The G 0 G ; 1 G  -expansion method and its applications for constructing many new exact solutions of the higher- order nonlinear Schro ¨dinger equation and the quantum Zakharov–Kuznetsov equation Elsayed M. E. Zayed 1 • Ayad M. Shahoot 2 • Khaled A. E. Alurrfi 3 Received: 20 October 2017 / Accepted: 15 January 2018 Ó Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this article, we apply the two variable G 0 G ; 1 G  -expansion method with the aid of symbolic computation to construct many new exact solutions for two higher-order nonlinear partial differential equatuions (PDEs) namely, the higher-order nonlinear Schro ¨ dinger (NLS) equation with derivative non-kerr nonlinear terms describing pulse of the propagation beyond ultrashort range in optical communication systems and the higher- order nonlinear quantum Zakharov–Kuznetsov (QZK) equation which arises in quantum magneto plasma . Also, based on Lie ´nard equation, we find many other diffrent new soliton solutions of the above NLS equation. Soliton solutions, periodic solutions, rational func- tions solutions and Jacobi elliptic functions solutions are obtained. Comparing our new solutions obtained in this article with the well-known solutions are given. Keywords The G 0 G ; 1 G  -expansion method  Exact solutions  Soliton solutions  Periodic solutions  Higher-order nonlinear Schro ¨dinger equations  Nonlinear quantum Zakharov–Kuznetsov equation & Elsayed M. E. Zayed eme_zayed@yahoo.com Ayad M. Shahoot drshahoot@yahoo.com Khaled A. E. Alurrfi alurrfi@yahoo.com 1 Department of Mathematics, Faculty of Sciences, Zagazig University, Zagazig, Egypt 2 Department of Physics, Faculty of Science, Mergib University, Khoms, Libya 3 Department of Mathematics, Faculty of Arts and Science, Mergib University, Msallata, Libya 123 Opt Quant Electron (2018) 50:96 https://doi.org/10.1007/s11082-018-1337-z