EXACT SOLUTION OF THE CLASSICAL SU(2) YANG-MILLS FIELD EQUATIONS

In this paper we find a new representation for self-duality equations. In addition exact solution class of the classical SU(2) Yang-Mills field equations in four-dimensional Euclidean space and two exact solution classes for SU(2) YangMills equations when is ρ a complex analytic function are also obtained. AMS Subject Classification: 35, 53C, 58J, 58Z05


Introduction
The self-dual Yang-Mills equations (a system of equations for Lie algebra valued functions of C 4 ) play a central role in the field of integrable systems and also play a fundamental role in several other areas of mathematics and physics, see [1]- [4].In addition the self-dual Yang-Mills equations are of great importance in their own right and have found a remarkable number of applications in physics and mathematics as well.These equations arise in the context of gauge theory (see [5]), in classical general relativity (see [6], [7]), and can be used as a powerful tool in the analysis of 4-manifolds, see [8].The Yang-Mills equations are a set of coupled, second-order partial differential equations in four dimensions for the Lie algebra-valued gauge potential functions A µ , and are extremely difficult to solve in general.The self-dual Yang-Mills equations describe a connection for a bundle over the Grassmannian of two-dimensional subspaces of the twistor space, see [9], [10].
In this paper we found a new representation for self-duality equations.In addition exact solution class of the classical SU (2) Yang-Mills field equations in four-dimensional Euclidean space and two exact solution classes for SU (2) Yang-Mills equations when ρ is a complex analytic function are also obtained.This paper is organized as follows: This introduction followed by the new representation of the self-duality equations in Section 2. In Section 3 we found an exact solution class of the classical SU (2) Yang-Mills field equations.Moreover two exact solution classes for self-dual SU (2) gauge fields on Euclidean space when ρ is a complex analytic function are given in Section 4. Finally, we give some conclusions in Section 5.

New Representation of the Self-Duality Equations
The essential idea of Yang and Mills (1954) [11] is to consider an analytic continua-tion of the gauge potential A µ into complex space where x 1 , x 2 , x 3 and x 4 are complex.The self-duality equations F µν = * F µν are then valid also in complex space , in a region containing real space where the x , s are real.Now consider four new complex variables y, ȳ ,z and z defined by it is simple to check that the self-duality equations F µν = * F µν reduces to Equations ( 2) can be immediately integrated, since they are pure gauge, to give [12]-[14] where D and D are arbitrary 2 × 2 complex matrix functions of y, ȳ ,z and z , and with determinant = 1 (for SU(2) gauge group) and D y = ∂ y D , etc.For real gauge fields A µ .= −A + µ (the symbol . = is used for equations valid only for real values of x 1 , x 2 , x 3 and x 4 ), we require Gauge transformations are the transformations where U is a 2 × 2 matrix function of y, ȳ ,z , z with determined = 1.Under transformation (5), equation ( 4) remains unchanged.We now define the hermitian matrix  [15]- [17] as  has the very important property of being invariant under the gauge transformation (5).The only non vanishing field strengths in terms of  becomes (u , v = y , z) and the remaining self-duality equation (2) takes the form The action density in terms of  [18]is where where F µν are the gauge field strengths.Our construction begins by explicit parametrization of the matrix and for real gauge fields A µ .= −A + µ ,we require φ .
= real ,ρ .= ρ * (ρ * ≡ complex conjugate of ρ).The self-duality equations (8) take the form where The positive definite Hermitian matrix  = DD + can be factored into a product upper and lower (or vice versa) triangular matrices as follows It is evident from (15) that one can choose a gauge so that D = R or D = R I and it is easy to check that in both gauges the self-duality equations ( 12)-( 14) (in the case of D = R I all the φ, ρ, ρ are replaced by φ I , ρ I , ρI ).
From equation (15) we see that R −1 R I is a unitary matrix so that we can always make a gauge transformation from the gauge R to the R I gauge.Theorem 1.If (φ, ρ, ρ) satisfy equations ( 12)-( 14) then so do (φ I , ρ I , ρI ) defined by (see [19]) Proof.By equating equation (15) ,we obtain the following equations We solve the system of equation ( 18) we obtain the relations in equation ( 17).

Exact Solution Class of the Classical SU(2) Yang-Mills Field Equations
To obtain an exact solution class of the classical SU(2) Yang-Mills field equations in four-dimensional Euclidean space, consider the system.
Let us make the ansatz Where g = g(x 1 , x 2 , x 3 , x 4 ) is a real function of x µ , µ = 1, 2, 3, 4 , φ and σare real functions of g and a is a real constant.Then equations ( 19), (20) give the relations Where the prime means differentiation with respect to g.The above relations imply that the determinant of the coefficients of (g y ȳ + g z z) and (g y g ȳ + g z g z ) is zero i.e. ( We shall determine φ and σ from the above equation (24), let (φσ) = c ,where c is a constant, then (φσ) ′ = 0, We suppose Applying theorem (1) to φ and ρ of equation ( 26), then we get Equations ( 26) and ( 27) is a new class of solutions of Yang-Mills equations for self-dual SU(2)gauge fields.
Equations ( 45) and ( 46) is a new class of solutions of Yang-Mills equations for self-dual SU (2) gauge fields.

Conclusions
A new class of solutions of Yang-Mills equations for self-dual SU (2)gauge fields are investigated.In this paper we found a new representation for self-duality equations.In addition exact solution class of the classical SU (2) Yang-Mills field equations in four-dimensional Euclidean space and two exact solution classes for SU (2) Yang-Mills equations when ρ is a complex analytic function are also obtained.