Wave solutions of nonlinear evolution equations modeling physical phenomena

Abstract

This thesis aims to present some aspects of a few methods that have been introduced recently to solve nonlinear partial differential equations (PDEs) which represent some physically relevant systems in applied physics, applied mathematics, especially in optics, fluids. This thesis is organized in twelve chapters as listed below: Chapter-1 is introductory and consists of four sections. In first section, the importance of nonlinear partial differential equation in modeling many physical phenomena is discussed. Some different types of traveling wave solutions are introduced in the second section. In third section, we study some properties of wave phenomena. In Fourth section, we review some important methods for solving nonlinear partial differential equations. In Chapter-2, we utilize the modified Jacobi elliptic function method and the modified expansion method to get explicit exact solutions for the fourth order non-linear partial differential equation that describe the pulses dynamics in the optical fibers. With the aid of these methods, we get many exact solutions like bright and dark solitons, periodic, hyperbolic, and rational type solutions. In Chapter-3, the improved modified extended tanh function method is applied to secure optical soliton solutions in magneto-optic waveguides with parabolic non-local law of refractive index. Dark, bright and singular solitons are obtained. Also, periodic wave solutions, Jacobi elliptic function solutions, Weierstrass elliptic solutions, exponential solutions, hyperbolic solutions and other solutions are extracted. In Chapter-4, the improved modified extended tanh scheme is implemented to extract exact travelling wave solutions to cubic–quartic perturbed nonlinear Schrödinger's equation with sextic-power law of refractive index. Various types of solutions are extracted such as bright soliton, singular soliton, dark soliton, singular periodic wave solution, periodic wave solution, Jacobi elliptic functions, plane wave and hyperbolic wave solutions. In Chapter-5, the improved modified extended tanh function method is applied to study the perturbed Gerdjikov-Ivanov equation (PGI) which describe the dynamics of the soliton in optical fibers. New exact solutions for the PGI equation will be introduced including weierstrass elliptic, Jacobi elliptic, dark and bright solitons, singular periodic and rational type solutions. The graphical representation of some solutions is illustrated.