الفهرس 
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Abstract
There is no doubt that the Korteweg–de Vries equation has applicable to a wider range of problems, including internal waves and waves in sheared flows in addition to surface water waves. We introduce a new equation G-KdV class (3+1) equation, N.G-KdV class (3+1) equation and DNA equation (3+1) dimensions, which have some remarkable mathematical properties. Furthermore, they also appear as a useful model in a great many physical situations. In this thesis we look at some properties of those equations. The general question which are considering is whether the properties of unique solutions of those equations, we introduce in this thesis anew physical mathematical methods to get the solitary and soliton solution for those equations, these equations can be considered as homologues to KdV equation in (3+1) dimensions. Thus, it is necessary to establish well-Posedness , the existence of solitary waves and other elementary solutions and the existence of other properties such as, Conservation laws and Bäcklund transformation. These are the specific questions that we consider in this thesis. The thesis is organized as follows, it consists of five chapters; chapter 1, to make the thesis self-contained, we begin a compressive review of KdV, KP and KdV in (3+1) dimensions equations. In chapter 2, we introduce G-KdV class (3+1) equation and the direct algebraic method to KdV equations in (3+1) dimensions to get the exact solutions and we verify this solution by using ( Mathematica program ) then applied numerical method (HAM) to G-KdV class (3+1) equation , study their classifications and make an experiment by use special data then the comparison shows that the obtained solutions in excellent 2 agreement and the graphs illustrate the different between the exact solutions and numerical solutions, the numerical solutions has a rapid or a fast convergence of solutions. Chapter 3 includes the well-posedness of N.G-KdV class (3+1) equation. This chapter begins with the introduction, the second section introduce the reduction of a semi linear system of partial differential equations, third section introduce characteristics of the system , the section four introduce a normal form of the first order, the section five introduce well posedness of N.G-KdV class (3+1) equation and uniqueness and in last section six introduce the existence of the equation. This part has been published in ”Theoretical Mathematics & Applications” . Vol. 3, no. 4, pp. 105-137, 2013. Chapter 4 is devoted to study conservation laws and exact solution of G-KdV class (3+1) equation, in first section study the preservation of conservation laws, the second section introduce new function method tan cot-tanh coth to get the exact travelling wave solutions for G-KdV class (3+1) equation, this application has been published in ”International Journal of Applied Computational Science & Mathematics”. Vol. 4, no. 1, pp. 77-92, (2014). and in last section three introduce another applications of DNA equation has been accepted to publish in ” International Journal of Applied Mathematics and Mechanics ” , 2014. Chapter 5, introduces new physical mathematical methods, auto-Bäcklund transformation and the Soliton Solution for G-KdV class (3+1) equation. In first section, we introduce a new Modulation (G G′/ )-expansion Method, this part has been accepted to publish in 3 ” Journal of Applied Mathematics and Bioinformatics”, 2014. in the second section we get the exact solution of G-KdV class (3+1) equation by using Modulation (G G′/ )-expansion Method, then we introduce the generalized extended mapping method, in section four, we get the exact solution of G-KdV class (3+1) equation by using The generalized extended mapping method, in section five, we study a modified Auto-Bӓcklund transformation and in last section six, we get the solution G-KdV class (3+1) equation by using a modified Auto-Bӓcklund transformation, this part has been accepted in ”ITALIAN JOURNAL PURE AND APPLIED MATHEMATICS”, 2014.