Int J Theor Phys (2009) 48: 1872–1876 DOI 10.1007/s10773-009-9957-5 1-Soliton Solution of the Nonlinear Schrödinger’s Equation with Kerr Law Nonlinearity Using Lie Symmetry Analysis C. Masood Khalique · Anjan Biswas Received: 26 November 2008 / Accepted: 27 January 2009 / Published online: 6 February 2009 © Springer Science+Business Media, LLC 2009 Abstract This paper obtains the 1-soliton solution of the nonlinear Schrödinger’s equation in a Kerr law media. The technique that is used to carry out the integration of this equation is the Lie symmetry analysis. Keywords Solitons · Lie symmetry · Kerr law 1 Introduction The nonlinear Schrödinger’s equation (NLSE) is a very important equation in the area of Applied Mathematics, Theoretical Physics, Engineering Sciences and Biological Sci- ences [2, 6]. In particular, NLSE appears in the study of Fiber Optics and Bose-Einstein condensates. There are various kinds of solutions that are known for this equation. These include the periodic waves, doubly periodic waves, cnoidal waves, solitary waves and many more. There are various techniques that are used to integrate NLSE and obtain these kinds of solutions. The common methods that are frequently seen in various text books and re- search papers are the classical method of Inverse Scattering Transform (IST), Hirota’s bilin- ear method and others. In this paper, however, a fairly less common method of integrability will be discussed to carry out the integration of the NLSE with Kerr law nonlinearity. The method is the usage of Lie symmetries. The basic idea of Lie symmetries is to study the invariance property of a given differential equation under continuous group of transforma- tions. Lately, the symmetry analysis technique has been widely used to carry out the in- tegration of many equations including the Lane-Emden equation [5] and quintic nonlinear C.M. Khalique · A. Biswas ( ) International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, Republic of South Africa e-mail: biswas.anjan@gmail.com A. Biswas Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA