الفهرس 
AcknowledgementOnly 14 pages are availabe for public view
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Abstract
During the last three decades much attention has been paid to the nonlinear studies including the famous Korteweg-de Vries equation which model the propagation of long waves in shallow water.
The present thesis is mainly devoted to study the nonlinear problems with some applications in shallow water.
The thesis consists of a general introduction, three chapters, list of references, three figures and two appendices which are briefly summarized as follows:
In the introduction the subject is defined and reviewed from the time of Scott Russell up to the recent literature. The importance of the problem in various domains of mathematics is reported.
In chapter I : The basic equations of fluid mechanics is briefly written, together with the boundary conditions and used to derive the nonlinear KdV equation modeling the shallow water waves in a chanal of homogeneous depth.
In chapter II ; The main objective of this chapter is the studying of some analytical solution methods for the KdV equa¬tion. This chapter contains introduction and three sections.
The first section is devoted to the direct algebraic method. This section divided into two parts, the first deals with the series solution method. This method is the basis of the direct algebraic method employing a series in real
texponential functions for solving the NL Kdv equation in terms of the general theory of autonomous ordinary differential \ equation (ODE). This method has been used to obtain a solution
If or an initial value problem of KdV type. In the second section , the general steady state solution for the same initial value problem treated in section 1 by using the Jacobian-elliptic functions.
The last section is devoted to generate new solutions from old once for the KdV equation by using Backlund Btransformations technique. The problem of obtaining explicitly and exact solutions of soliton equations of the AKNS class is considered. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system. The method of characteristics used and Backlund transformations (BTs) are employed to generate new solutions from the old ones. Thus, the family of new solution for the KdV equation is obtained.
In chapter III : This chapter is mainly concerned with the numerical treatment of some initial value problems of the KdV equation. Two numerical methods are explained and used to obtain approximate solutions. In the first, we used the Hopscotch method of Greig and Morris [35] which lead into a solution presented graphically in Figures 1 and 2. In the second, we used the Crank-Nicelson method [8 8] to obtain a solution presented graphically in Figure 3.
Finally the computer programs used in the last two numerical experiments are reported in appendices A and B.