Reduction of the Self-dual Yang-Mills Equations to Sinh-Poisson Equation and Exact Solutions GHARIB .M.GHARIB, RANIA SAADEH Mathematics Department, College of Science Zarqa University, JORDAN Abstract: - The geometric properties of differential systems are used to demonstrate how the sinh-poisson equation describes a surface with a constant negative curvature in this paper. The canonical reduction of 4- dimensional self dual Yang Mills theorem is the sinh-poisson equation, which explains pseudo spherical surfaces. We derive the B¨ acklund transformations and the travelling wave solution for the sinh-poisson equation in specific. As a result, we discover exact solutions to the self-dual Yang-Mills equations. KeyWords :- B¨ acklund transformations; Nonlinear evolution equations; Yang-Mills theory; Pseudo- spherical surfaces. Received: May 13, 2021. Revised: September 24, 2021. Accepted: October 6, 2021. Published: October 26, 2021. 1 Introduction The concept of integrable structures has been a contentious issue in mathematics for the last three decades. The focus includes but not limited into mechanics, dynamics, mathematics, algebra, physics, evaluation, and geometry. Many partial differential equations, which are still studied due to their significance in physics and mathematics, have relations with the geometry of surfaces embedded in 3D space [1]. This has long been known that there is a relationship between surfaces in Euclidean three-space with a constant negative Gaussian curvature, the SineGordon Formula, and Bäcklund transformations that are applicable to the equation given [2]. Furthermore, including pseudo spherical surfaces, the main Bäcklund transformation for something like the SineGordon Formula appears to have been a normal linear structure (pss) [3-5]. The Yang-Mills (YM) behavior is one of the key components of the standard model, which has been phenomenological very popular up to this point. A Yang-Mills field adds to the curvature of space-time in the same way as every other field does, according to General Relativity. There are some physically important circumstances in which gravitational fields are exceptionally strong, and the impact of curvature on the propagation of matter fields, as well as the Yang-Mills fields' back-reaction, cannot be overlooked [6]. Once Yang-Mills concept was proposed in the mid- twentieth century, it was understood that the quantum model of the Maxwell system, known as Quantum Electrodynamics (QED), provided a detailed explanation of the quantum structure of electro_magnetic fields and forces. The issue of whether or not the nonabelian analog was needed to represent other natural forces, especially the weak force, which is concerned with certain types of radioactivity, and the powerful or nuclear force, which is concerned with the mixing of protons and neutrons around nuclei, among other things, arose. Yang-Mills theories, which describe fundamental laws of interactions, are central to primary particle physics.Topological solitons such as instantons, monopoles, vortices, calorons, and merons played important roles in the study of non-perturbative properties, duality structures, and quark con nements, among other aspects, in these theories. Since the weak and nuclear forces are unrelated to long-range fields or particles of low mass [7-9]. The masslessness of classical Yang-Mills waves was a major impediment to applying Yang-Mills theory to them. In the 1960s and 1970s, these obstacles to physical applications of non-abelian gauge theory were overcome. The self-dual Yang-Mills equations (SDYM) are well-known for having a large number of integrable systems. The Painleve equations and classical soliton equations in 1+1 dimensions are finite- dimensional Lie algebra reductions of the SDYM equations (LG). It is important to use infinite dimensional algebras to minimize the SDYM equations [10-12]. Anti-self-dual Yang-Mills(ASDYM) equation, on the other side, has a close relationship with lower- WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2021.20.57 Gharib. M. Gharib, Rania Saadeh E-ISSN: 2224-2880 540 Volume 20, 2021