Exact solutions of nonlinear Schrödinger equation

In the first chapter, of this thesis,we present a brief information about nonlinear partial differantial equations (NPDEs) which play an important role in nonlinear phenomena. In order to better understanding these nonlinear phenomena, many mathematicians as well as physicists have been made big efforts to seek more exact solutions to NPDEs. Therefore, several powerful methods have been proposed to obtain exact solutions , such as inverse scattering method (Zakharov and Shabat 1972) and Hirota direct method (Hirota 2004). In this chapter, we also give a brief information about Nonlinear Schrödinger Equation(NLS) which is one of the most important NPDEs. In the second chapter, historical developments of NLS are given. We also review the history of soliton, since the first recorded observation of the 'great solitary wave' by Russell in 1834, as means of developing the mathematical properties of a large class of solvable nonlinear equations such as the KdV and the NLS. In the third chapter ,the basic definitions about differantial equations are given. In addition, we present the Complex Fourier Transform (CFT) which is one of the most important tool when solving ODEs and in particular PDEs. We solve the wave equation which is an example of using the CFT. In this chapter, we also construct the travelling wave solutions for some nonlinear evolution equations such as the Burgers equation and KdV equation. Furthermore, in this chapter, we present Hirota's direct method of constructing multi-soliton solutions to integrable nonlinear evolution equations such as the KdV and NLS. Chapter four is devoted to NLS Equation. Firstly, we construct a travelling wave solution for the NLS equation. Furthermore, by using Hirota's direct method we present one-soliton and two soliton solutions for the NLS. Results and discussion are given in chapter five.