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Abstract Partial differential equations (PDEs) are clearly important in characterizing, simulating and predicting nonlinear phenomena across a wide range of physical and engineering science disciplines. As a result, we became interested in finding exact solutions to these equations and used three main methods to find these solutions. These methods are the modified extended tanh function method, the modified expanded direct algebraic method, and the modified expanded mapping method. These methods have been applied to many models such as Kundu-Eckhaus, Davy-Stewartson equation, complex Ginzburg-Landau equation, and nonlinear Kurteweg-de Vries-Kadomtsev-Petviashvili equation. This thesis aims to find isolated wave solutions and other exact solutions of some nonlinear partial differential equations (NLPDEs). The thesis is divided into nine chapters, which are as follows: Chapter (1): It is an introductory chapter that includes seven sections. The first section explains the importance of nonlinear partial differential equations in modeling many physical phenomena. The second section provides the historical background of the soliton. The third section explains the classifications of nonlinear effects that can be applied to partial differential equations. In the fourth section, we discuss the various types of traveling wave solutions. In the fifth section, we discuss the principle of homogeneous balance. The importance and applications of solitons in life are discussed in the sixth section. We discuss some main analytical methods for solving nonlinear partial differential equations in the seventh section. Chapter (2): In this chapter, the improved modified extended tanh function method is applied for the purpose of finding new analytical solitons and several other solutions of the Kundu–Eckhaus equation describing the propagation of ultrashort light pulses in optical fibers taking into account pentagonal nonlinearity and Raman effect in the study. Moreover, some of the solutions obtained are presented graphically in 2D and 3D graphs and contour plots. Chapter (3): We discuss the generalized Kundu–Eckhaus equation with additive dispersion via the improved modified extended tanh function technique which has led us to several new and innovative exact and soliton solutions in optical fibers in this chapter. Furthermore, some of the solutions obtained were illustrated graphically in the form of 2D, 3D and contour plots. Chapter (4): This chapter is concerned with applying the modified extended mapping method to obtain new optical solitons of magnetic waves using Kudryashov’s law of nonlinear refraction of a coupled system from the generalized nonlinear Schr¨odinger equation. In addition, some of the derived solutions are graphically illustrated in the form of 2D, 3D, and contour plots. Chapter (5): This chapter discusses the Davey-Stewartson equation (DSE), which explains how waves move through water at a finite depth while being affected by gravity and surface tension, and is analyzed using a modified extended mapping method to find many different soliton solutions. In addition, some of the solutions obtained are graphically illustrated in the form of 2D, 3D, and contour graphs. Chapter (6): In this chapter, the modified extended direct algebraic method is applied to search for the derivation of new solitons and other exact wave solutions of a coupled system from the highly dispersive complex Ginzburg-Landau equation in birefringent fibers with the polynomial nonlinearity law. Moreover, some of the solutions obtained were represented in 2D and 3D graphic form and contour plots. Chapter (7): In this chapter, the Nonlinear Schr¨odinger Equation (NLSE) with dimensions (2+1) with nonlinearity and fourth-order dispersion was studied. Several photonic soliton solutions are suitable for the present problem are analyzed using a modified extended direct algebraic method. Moreover, some of the solutions obtained were represented graphically in the form of 2D, 3D, and contour plots. Chapter (8): This chapter represents a study of solitary wave solutions of the nonlinear Kortewegde Vries-Kadomtsev-Petviashvili equation arising in water waves using the improved modified extended tanh function technique. Moreover, some of the solutions obtained were represented graphically. Chapter (9): In this chapter, we studied the generalized nonlinear Schr¨odinger equation (NLSE) using the improved modified extended tanh function method and obtained explicit exact solutions by applying the improved modified extended tanh method. The results of the study have important implications for how solitons propagate in nonlinear optics. Moreover, to better understand the behaviors of some of these found solutions, we showed their outlines in 2D, 3D, and contour graphs. |