Finite difference methods for numerical solution of the generalized Schrödinger Equation

This thesis explores the numerical solutions of the generalized Schrödinger (GNLS) equation using the finite difference method. In the first chapter, general information about the thesis and its objectives is provided. The second chapter reviews previous studies on the GNLS equation. In the third chapter, test problems are addressed, including the propagation of solitary waves and the collision of soliton waves. It also explores time-discretization methods with both second and fourth-order accuracy for the GNLS equation under specified initial and boundary conditions. Chapters four through nine investigate the approximate solutions of the GNLS equation. The presented method primarily uses time-discretization schemes with second and fourth-order accuracy. Subsequently, space discretization with finite difference approximations using five, six, seven, eight, nine, and ten points is proposed for each time discretization schemes. At the end of each chapter, the accuracy of the proposed methods is evaluated using two test problems. The tenth chapter presents a discussion of the results obtained, while the final chapter offers recommendations for future research.