الفهرس 
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Abstract
During the last four decades much attention has been paid to the nonlinear studies on the Backlund transformations (BTs), pseudo-spherical surfaces (pss), conservation laws and exact solutions for Yang-Mills equations.
In this thesis we implemented the BTs, inverse scattering problem (ISP), conserva¬tion laws and exact solutions for some nonlinear evolution equations (NLEEs) which describe pss. We obtained exact soliton solutions for some nonlinear partial differential equations (NLPDEs), consequently we find exact solutions for self-dual Yang Mills equations. The thesis consists of an introduction and ten chapters, together with an Arabic and an English summaries and organized as follows:
Introduction:
The introduction includes a short historical discussion of the early geometers’ ideas on ”integrable geometric constructions” followed by a quick hint, for the importance of BTs and conservation laws, we give a brief survey of the Yang-Mills theories. Chapter 1:
This chapter explains the basis of pss, BTs, conservation laws for some NLEEs and Yang-Mills equations, the general basic consideration for the models considered in this thesis together with the necessary preliminaries.
Chapter 2:
In this chapter, we obtained the BTs for some NLEEs that describe pss, based on a geometrical property of these surfaces. We point out here that the results of this chapter are submitted to (J. of differential Geometry).
Chapter 3:
In this chapter, we find the conservation laws for some NLEEs that describe pss. based on a geometrical property of these surfaces. We point out here that the results of this chapter arc published in (J. of Geometry and Physics, 51, 332(2004)).
Chapter 4:
In this chapter, A generalized inverse scattering method (ISM) and the funda¬mental equations of pss are given by extending the results of Konno, Wadati (1975) and Sasaki (1979) respectively. An infinite number of conserved quantities are also obtained by solving a set of coupled Riccati equations. We obtained the inverse scattering method for the 3x3— dimensional Lax operators. Consequently we developed the Gelfand-Levitan-Maxchenko equation for a 3— dimensional Lax pair. We point out here that the results of this chapter are submitted to (Inverse Problems). Chapter 5:
In this chapter, we obtained all local conservation laws for some NLPDEs. The Conservation laws, does not depend on the system having a Lagrangian formulation, in contrast to Noerther’s theorem, which requires a Lagrangian. Different methods to construct new exact solution classes for the same NLPDEs are also presented, which are named hyperbolic function method and the BTs. On the other hand other methods and transformations are developed to obtain exact solutions for the same NLPDEs. We point out here that the results of this chapter are submitted to (II Nuovo Cimento B).
Chapter 6:
In this chapter, A new exact soliton solution classes are generated from a known solutions either the seed solution is constant or a travelling wave for some NLEEs which describe pss. We point out here that the results of this chapter are accepted for publication in (J. of Computational and Applied Mathematics).
Chapter 7:
In this chapter, we present a set of exact solutions for sine-Gordon and Liouville’s equations in 2 dimensions by applying the BTs. Other transformations are used to obtain new classes of exact solutions for sine-Gordon in (2+1) and (3+1) dimensions as well. Consequently we find exact solutions for self-dual Yang Mills equations. We point out here that the results of this chapter are published in (Chaos, Solitons & Fractals 10, 1309(1999)).
Chapter 8:
In this chapter, we give several classes of exact solutions for Burgers’ type and sine-Gordon equations in two dimensions by applying the BTs. Other transformations are used to obtain several classes of exact solutions for two dimensional generalized korteweg-de Vries type, 2+1 and the original 3 + 1 dimensional Liouville’s equations. Consequently we find exact solutions for self-dual Yang Mills equations. We point out here that the results of this chapter are submitted to (II Nuovo Cimento B).
Chapter 9:
In this chapter, we present a set of exact solutions for nonlinear schrodinger and Korteweg - de Vries equations in two dimensions by applying the BTs. Consequently we find exact solutions for self-dual SU(2) and SU(3) Yang Mills equations. We point out here that the results of this chapter are published in (International J. of Theoretical Physics, 41. 409(2002 )).
Chapter 10:
In this chapter, we present a new representation and exact solutions for the self-duality equations. We point out here that the results of this chapter are submitted to (International J. of Theoretical Physics).