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Abstract Nonlinear partial differential equations are of great importance in interpreting the phenomena that arise in studying various topics in engineering and scientific fields, such as chemical kinetics, solid-state physics, plasma physics, population models, fluid mechanics, and nonlinear optics. This thesis aims to obtain, soliton and other numerous kinds of analytic solutions, which play a vital role in studying the dynamical behavior of many physical phenomena. Several integration techniques have been reported to get analytic solutions for nonlinear partial differential equations such as the improved modified extended tanh-function method, the modified extended direct algebraic method, and the modified extended mapping method. This thesis consists of seven chapters organized as follows : Chapter (1): Preliminaries and basic concepts which are introduced in two sections. Section (1), explores a brief introduction to clarify definitions, applications, and properties of ( partial differential equations, stochastic differential equations, wave solutions, and solitary wave solutions of partial differential equations ). Section (2), introduces some important methods to solve nonlinear partial differential equations. Chapter (2): In this chapter, we introduce the new stochastic Schr¨odinger-Hirota model which lies in many physical systems that are subject to random fluctuations or noise such as atmospheric systems, ii iii optics, biology, plasma physics, nonlinear fiber optics, fluid dynamics, and others. We used the improved modified extended tanh function method to obtain new stochastic wave solutions for the proposed model such as bright, singular, periodic, singular periodic, exponential, rational, and Jacobi elliptic doubly periodic solutions. Moreover, we plotted two-dimensional and threedimensional graphs to explain the effect of noise on some of the obtained solutions. Chapter (3): In this chapter, the improved modified extended tanh-function method is applied for nonlinear Schr¨odinger equation with Kudryashov’s quintuple power-law of refractive index having nonlinear chromatic dispersion. Various types of solutions are extracted such as bright, dark, singular, periodic, singular periodic, rational, exponential, hyperbolic, and Weierstrass elliptic doubly periodic type solutions. Moreover, for the physical illustration, some of the obtained solutions are represented graphically. Chapter (4): In this chapter, new stochastic (3+1) dimensional nonlinear Schr¨odinger equation is introduced. A modified extended mapping method is used to obtain different new optical solitons and exact solutions like bright, dark, triangular periodic, hyperbolic, rational, Jacobi elliptic functions, and Weierstrass elliptic functions solutions. The impact of the noise is illustrated graphically using examples of some of the retrieved solutions with various noise strengths. Chapter (5): In this chapter, the new stochastic fourth-order (2+1) dimensional Schr¨odinger equation with higher-order odd and even terms equation is introduced for the first time. By using the improved modified extended tanh function method, we introduced new optical stochastic soliton and exact stochastic soliton solutions such as bright, dark, triangular periodic, hyperbolic, singular periodic, singular soliton, Jacobi elliptic functions, and Weierstrass elliptic functions solutions. Also, we introduce graphic illustrations for the impact of the noise by using examples of some of the retrieved solutions with various noise strengths. Chapter (6): iii iv In this chapter, the optical soliton and exact solutions of the stochastic concatenation model are obtained by using the modified extended direct algebraic method. This method provides a wide variety of solutions, including bright, dark, dark combo, singular combo, triangular periodic, rational, Jacobi elliptic functions, and Weierstrass elliptic functions solutions. The impact of the noise is illustrated graphically using examples of some of the retrieved solutions with various noise strengths. Chapter (7): In this chapter, we discuss the analytic solutions for the new stochastic fourth-order perturbed nonlinear Schr¨odinger equation. Our Study mainly depends on applying the improved modified extended tanh function method to get the soliton wave solutions and other exact wave solutions of our model of interest. These solutions such as (bright, dark, singular) solitons, hyperbolic solutions, periodic solutions, singular periodic solutions, Jacobi elliptic functions solutions, and Weierstrass elliptic functions solutions. Graphical representations of some of the extracted solutions using different noise intensities are shown to demonstrate the influence of the noise |