الفهرس 
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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,
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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):
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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