EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 3, 2017, 563-573 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Canonical reduction of the Self-Dual Yang Mills equations to complex Ginzburg-Landau equations and exact solutions A.R. Shehata1,∗, J.F.Alzaidy 2 1 Mathematics Department, Faculty of Science, Minia University, Egypt 2 Mathematics Department, Faculty of Science, King Abdulaziz University, Saudi Arabia Abstract. The (constrained) canonical reduction of four-dimensional self-dual Yang-Mills (SDYM) theory to two-dimensional complex Ginzburg-Landau equation are considered. On the other hand, other methods and transformations are developed to obtain exact solutions for the original two dimensional complex Ginzburg-Landau equation. The corresponding gauge potential Aµ and the gauge field strengths Fµν are also obtained. For these nonlinear evolution equations (NLEEs) which describe pseudo-spherical surfaces (pss) two new exact solution classes are generated from known solutions by using the Bäcklund transformations with the aid of Mathematica , either the seed solution is constant or a traveling wave. 2010 Mathematics Subject Classifications: 35, 53C, 58J, 58Z05 Key Words and Phrases: SDYM, complex Ginzburg-Landau equation, Bäcklund transforma- tions 1. Introduction The self-dual Yang-Mills (SDYM) equations (a system of equations for Lie algebra- valued functions of C4) play a central role in the field of integrable systems and also play a fundamental role in several other areas of mathematics and physics [17, 14]. It arises in relativity [22, 13] and in field theory [6]. The SDYM equations describe a connection for a bundle over the Grassmannian of two-dimensional subspaces of the twistor space. Inte- grability for a SDYM connection means that its curvature vanishes on certain two-planes in the tangent space of the Grassmannian. As shown in [18, 21]. This allows one to char- acterize SDYM connections in terms of the splitting problem for a transition function in a holomorphic bundle over the Riemann sphere, i.e. the trivialization of the bundle [15, 16]. ∗Corresponding author. Email addresses: shehata1433@yahoo.com (A.R. Shehata), j-f-h-z@hotmail.com (J.F.Alzaidy) http://www.ejpam.com 563 c© 2017 EJPAM All rights reserved. A.R. Shehata , J.F.Alzaidy / Eur. J. Pure Appl. Math, 10 (3) (2017), 563-573 564 The theory of integrable systems has been an active area of mathematics for the past thirty years. Different aspects of the subject have fundamental relations with mechan- ics and dynamics, applied mathematics, algebraic structures, theoretical physics, analysis including spectral theory and geometry. In recent decades, a class of transformations having their origin in the work by Bäcklund in the late nineteenth century has provided a basis for remarkable advances in the study of nonlinear partial differential equations (NLPDEs)[17, 14, 22, 13, 6, 18, 21, 15, 16, 12, 10].The importance of Bäcklund transfor- mations (BTs) and their generalizations is basically twofold. Thus, on one hand, invariance under a BT may be used to generate an infinite sequence of solutions for certain NLPDEs by purely algebraic superposition principles. On the other hand, BTs may also be used to link certain NLPDEs[9, 24, 25, 19, 14, 28, 11, 7] (particularly NLEEs modelling nonlinear waves) to canonical forms whose properties are well known [8, 20, 1]. Nonlinear wave phenomena have attracted the attention of physicists for a long time. Investigation of a certain kind of NLPDEs has made great progress in the last decades. These equations have a wide range of physical applications and share several remarkable properties [3, 2, 4, 5]: (i) the initial value problem can be solved exactly in terms of lin- ear procedures, the so-called ” inverse scattering method (ISM)”; (ii) they have an infinite number of ” conservation laws ”; (iii) they have ”BTs”; (iv) they describe pseudo-spherical surfaces (pss), and hence one may interpret the other properties (i -iii) from a geometrical point of view; (v) they are completely integrable [22, 3, 4]. Non-Abelian gauge theories first appeared in the seminal work of Yang and Mills [29] as a non-Abelian generalization of Maxwell’s equations. Let G be a Lie group (referred to as the gauge group) with Lie algebra (LG) and let {xµ}µ=1,2,3,4be coordinates on a four- dimensional manifold M which can be R4, R1,3 or R2,2. Given the gauge potential Aµ(x) ∈ LG, we introduce the covariant derivatives Dµ = ∂µ −Aµ (1) and their commutators Fµν = −[Dµ, Dν ] = ∂µAν − ∂νAµ − [Aµ, Aν ], (2) where Fµν are the gauge field strengths. The Yang- Mills equations are a set of coupled, second-order NLPDEs in four dimen- sions for the LG-valued gauge potential functionsAµ’s, and are extremely difficult to solve in general. It is however possible to obtain a special class of first-order reductions of the full Yang-Mills equations by noting that any Fµν that satisfies λFµν = ∗Fµν , λ = { ±1 on R4, R2,2, ±i on R3,1. (3) A.R. Shehata , J.F.Alzaidy / Eur. J. Pure Appl. Math, 10 (3) (2017), 563-573 565 All real solutions of the equations ∗Fµν = ±iFµν are trivial. On R4 and R2,2, the equations ∗Fµν = (−)Fµν are called the (anti) SDYM equations. Now consider four complex variables y, ȳ, z and z̄ defined in [27] √ 2y = x1 + ix2, √ 2ȳ = x1 − ix2, √ 2z = x3 − ix4, √ 2z̄ = x3 + ix4, (4) it is simple to check that the self-duality equations Fµν = ∗Fµν reduces to Fyz = 0, Fȳz̄ = 0, Fyȳ + Fzz̄ = 0. (5) Equations (5) are the compatibility condition of the linear problem [29] (ψy + iζψz̄) = (Ay + iζAz̄)ψ, (6) (ψz − iζψȳ) = (Az − iζAȳ)ψ, (7) where ζ is a parameter, independent of y, ȳ, z and z̄. The compatibility condition is simply (∂z − iζ∂ȳ)(∂y + iζ∂z̄)ψ = (∂y + iζ∂z̄)(∂z − iζ∂ȳ)ψ. (8) On using equations (6) and (7), this gives [Fyz − iζ(Fyȳ + Fzz̄)− ζ2Fȳz̄]ψ = 0., (9) Equations (5) can be immediately integrated, since they are pure gauge, to give Ay = D−1Dy, Az = D−1Dz, Aȳ = D̄−1D̄ȳ, Az̄ = D̄−1D̄z̄, (10) where D and D̄ are arbitrary 2×2 complex matrix functions of y, ȳ, z and z̄ with determi- nant = 1 (for SU(2) gauge group) and Dy = ∂yD, etc. For real gauge fields Aµ=̇−A+ M (the symbol =̇ is used for equations valid only for real values of x1, x2, x3and x4), we require D̄=̇(D+)−1. (11) Gauge transformations are the transformations D → DU, D̄ → D̄U, U+U=̇I, (12) where U is a 2×2 matrix function of y, ȳ, z, z̄ with determined = 1. Under transformation (12), equation (11) remains unchanged. We now define the hermitian matrix J as J = DD̄−1=̇DD+. (13) J has the very important property of being invariant under the gauge transformation (12). The only non vanishing field strengths in terms of J becomes Fuv̄ = −D̄−1(J −1Ju)v̄D̄. (14) A.R. Shehata , J.F.Alzaidy / Eur. J. Pure Appl. Math, 10 (3) (2017), 563-573 566 (u, v = y, z) and the remaining self-duality equation (5) takes the form (J −1Jy)ȳ + (J −1Jz)z̄ = 0. (15) The action density in terms of J is φ(J ) = −1 2 TrFµνFµν = −2Tr(FyȳFzz̄ + Fyz̄Fȳz) = −2Tr{(J −1Jy)ȳ(J −1Jz)z̄ − (J −1Jy)z̄(J −1Jz)ȳ}. (16) In this paper, the canonical reduction of four dimensional self-dual Yang-Mills theory to two dimensional complex Ginzburg-Landau (cGL) equation are considered. We give a new of exact solution for the cGL equation by applying the BTs method with the aid of Mathematica [28, 11, 7, 8, 20, 1, 3, 2, 4, 5, 27, 29]. Consequently we find exact solutions for self-dual Yang Mills equations. In addition the corresponding gauge potential Aµ and the gauge field strengths Fµν are also obtained. The paper is organized as follows: On one hand the reduction of Yang-Mills theory to cGL equation , and exact solutions are presented in sections 2 and 3 respectively. Moreover the gauge potential Aµ and the gauge field strengths Fµν are also obtained. Section 4 contains the conclusion. 2. The canonical reduction of four-dimensional SDYM theory to two dimensional cGL equation Suppose that Aµ’s depend on x = y + ȳ and t = z only. If we use a gauge in which Aȳ = 0, in terms of the matrix-valued functions P := Ay, Q := Az, R := Az̄, the SDYM equations (5) are Rx = 0 (17) Qx − Pt − [P,Q] = 0, (18) Rt − Px − [Q,P ] = 0. (19) Let R take the canonical form R = ( −i 2a 0 0 i 2a ) . (20) We then find that P = ( 0 ue−iµt −u∗eiµt 0 ) , (21) Q = ( ia|u|2 aiuxe −iµt aiu∗xe iµt −ia|u|2 ) , (22) from Eq. (18), we obtain the cGL equation iut + auxx + 2a|u|2u+ µu = 0, (23) where a is a complex constant and µ is real constant. A.R. Shehata , J.F.Alzaidy / Eur. J. Pure Appl. Math, 10 (3) (2017), 563-573 567 3. The BTs and exact solution for cGL equation We recall the definition [14, 3] of a differential equation (DE) that describes a pss. Let M2 be a two dimensional differentiable manifold with coordinates (x, t). A DE for a real function u(x, t) describes a pss if it is a necessary and sufficient condition for the existence of differentiable functions fij , 1 ≤ i ≤ 3, 1 ≤ j ≤ 2, (24) depending on u and its derivatives such that the one-forms ω1 = f11dx+ f12dt, ω2 = f21dx+ f22dt, ω3 = f31dx+ f32dt, (25) satisfy the structure equations of a pss, i.e., dω1 = ω3 ∧ ω2, dω2 = ω1 ∧ ω3, dω3 = ω1 ∧ ω2. (26) As a consequence, each solution of the DE provides a local metric on M2, whose Gaussian curvature is constant, equal to -1. Moreover, the above definition is equivalent to saying that DE for u is the integrability condition for the problem [25, 4]: dφ = Ωφ, φ = ( φ1 φ2 ) , (27) where d denotes exterior differentiation, φ is a column vector and the 2 × 2 matrix Ω (Ωij , i, j = 1, 2) is traceless Ω = 1 2 ( ω2 ω1 − ω3 ω1 + ω3 −ω2 ) . Take Ω = ( ηdx+Adt qdx+Bdt rdx+ Cdt −ηdx−Adt ) = Sdx+ Tdt, (28) from Eqs. (27) and (28), we obtain φx = Sφ, φt = Tφ, (29) where S and T are two 2× 2 null-trace matrices S = ( η q r −η ) , (30) T = ( A B C −A ) . (31) A.R. Shehata , J.F.Alzaidy / Eur. J. Pure Appl. Math, 10 (3) (2017), 563-573 568 Here η is a parameter, independent of x and t , while q and r are functions of x and t. Now 0 = d2φ = dΩφ− Ω ∧ dφ = (dΩ− Ω ∧ Ω)φ, which requires the vanishing of the two form Θ ≡ dΩ− Ω ∧ Ω = 0, (32) or in component form −Ax + qC − rB = 0 qt − 2Aq −Bx + 2ηB = 0 rt − Cx + 2Ar − 2ηC = 0. (33) Chern and Tenenblat [12] obtained Eq. (33) directly from the structure equations (26). By suitably choosing r,A,B and C in (33), we shall obtain various cGL equation which q must satisfy. Konno and Wadati introduced the function [23] Γ = φ1 φ2 , (34) this function first appeared used and explained in the geometric context of pss equations in [10, 24], and see also the classical papers by Sasaki [26] and Chern-Tenenblat [12]. Then Eq. (29) is reduced to the Riccati equations: ∂Γ ∂x = ηΓ− rΓ2 + q, (35) ∂Γ ∂t = 2AΓ− CΓ2 +B. (36) Our procedure in the following is that we construct a transformation Γ′ satisfying the same equation as (35) and (36) with a potential u′ where u′ = u+ f(Γ, η), (37) Chern and Tenenblat [12] introduced several examples of (37) for pss equations. For use in the sequel, we list the cGL equation and their corresponding BT in the following. The cGL equation For any solution u(x, t) of the cGL equation (23), the matrices S and T are S = ( η ue−iµt −u∗eiµt −η ) , (38) T = ( 2iη2a+ ia|u|2 (2iηau+ aiux)e−iµt (−2iηau∗ + aiu∗x)eiµt −2iη2a− ia|u|2 ) , (39) A.R. Shehata , J.F.Alzaidy / Eur. J. Pure Appl. Math, 10 (3) (2017), 563-573 569 the above matrices S, T satisfy Eqs. (33). Then Eq. (35) becomes ∂Γ ∂x = ηΓ + ue−iµt + u∗eiµtΓ2. (40) If we choose Γ′ and u′ as [3] Γ′ = 1 Γ∗ , (41) u′ = u− 4η Γeiµt 1 + |Γ|2 . (42) Now we shall choose some known solution of the cGL equation and substitute this solution into the corresponding matrices S and T . Next, we solve Eqs. (29) for φ1 and φ2. Then, by (34) and the corresponding BT we shall obtain the new solution for the cGL equation. Substitute u = 0 into the matrices S and T in (38) and (39), then by (29) we have dφ = φxdx+ φtdt = Sφdρ, (43) where S = ( η 0 0 −η ) , (44) ρ = x+ bt, b = 2iaη. (45) The solution of Eq. (43) is φ = eSρφ0 = ( 1 + ρS + ρ2S2 2! + ρ3S3 3! + · · · ) φ0, (46) where φ0 is a constant column vector. The solution of Eq. (46) is φ = ( cosh ηρ+ sinh ηρ 0 0 cosh ηρ− sinh ηρ ) φ0. (47) Now, we choose φ0 = (1, 1)T in (47), then we have φ = ( eηρ e−ηρ ) . (48) Substitute (48) into (34), then by (42), we obtain the new solutions of the cGL equation(23) u′ = −2ηei(2aη 2)tsech(2ηx). (49) We can calculate the gauge potential Aµ and the gauge field strengths Fµν from equations (6)-(10) and (20)-(22), then Ay = ( 0 −2ηea1tsech(2ηx) 2ηe−a1tsech(2ηx) 0 ) , Aȳ = 0 A.R. Shehata , J.F.Alzaidy / Eur. J. Pure Appl. Math, 10 (3) (2017), 563-573 570 Ay = ( a2 b2 b3 −a2 ) , Az̄ = ( −i 2a 0 0 i 2a ) , (50) where a1 = 2iaη2, a2 = 2a1sech2(2ηx), b2 = 2a1e a1tsech(2ηx) tanh(2ηx), b3 = 2a1e −a1tsech(2ηx) tanh(2ηx). Consequently, we obtain the gauge field strengths Fµν as follows: Fyȳ = −∂xAy, Fzz̄ = ∂tAz̄ − [Az, Az̄] Fyz = ∂xAz − ∂tAy − [Ay, Az], Fȳz̄ = ∂xAz̄ − [Aȳ, Az̄], (51) we note that the self-dual SU(2) Yang Mills equations holds. There are several points to be made: (i) The classical solutions to nonlinear field equations give us insight into the bound state behavior of field theories, particular interest are new class of classical solutions of the Yang Mills equations. (ii) We chose the group SU(2) because we wanted to include explicit time-dependence in the classical soltions of the Yang Mills equations, this does not exclude eventually using higher groups. (iii) We solved the nonlinear Yang -Mills equation by using the time -dependent solutions of electrodynamics, the physical picture is as follows: Due to the influence of an external field, this particle, because it is accelerating, produces a gauge potential Aµ obeying (50). This gauge potential, in turn, can be used to form the ansatz (50), which exactly solves the nonlinear Yang-Mills equation. 4. Conclusions A soliton is a localized pulse-like nonlinear wave that possesses remarkable stability properties. Typically, problems that admit soliton solutions are in the form of evolution equations that describe how some variable or set of variables evolve in time from a given state. 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