The symmetry structures of ASD manifolds

Transcription

1 The symmetry structures of ASD manifolds 1 1 School of Mathematics University of the Witatersrand, Johannesburg South Africa African Institute of Mathematics

2 Outline 1 Introduction 2 3

3 The dispersionless integrable systems in 3+1 dimensions do not admit soliton solutions and there is no associated Riemann Hilbert problem here the corresponding Lie group is finite dimensional. Hoever, such systems may be described in terms of anti-self-duality (ASD) - Quaternion-Kahler four-manifolds, or equivalently ASD Einstein manifolds, are locally determined by one scalar function subject to Przanoski s equation (see [1] and [2] and references therein). In this case the unknon in the equations is a metric on some four-manifold (Penrose [3]) - this makes the dispersionless systems more geometric and may be of importance in the description of shock formations (see Manakov and Santini [4] and Dunajski [5]).

4 If the Ricci-flat condition is imposed on top of the anti-self-duality, the ork of Pleba nski [6] implies the existence of a local coordinate system (x, y, z, ) and a function u on an open set such that any ASD Ricci-flat metric, g, is locally of the form ds 2 = 2(dzd + ddx u xx dz 2 u yy d 2 + 2u xy ddz) (1.1) here u(x, y, z, ) satisfies the second heavenly equation (she) u x u zy + u xx u yy u 2 xy = 0. (1.2)

5 In [5], the above equations are studied subject to the existence of a conformal Killing vector X satisfying L X g = pg (1.3) for some function p and L is the Lie derivative. The relationship beteen the HE and the Monge-Ampere equation is discussed in, inter alia, Husain [7] and Malykh & Sheftel [8]. Recently, Doubrov and Ferapontov [9], introduced the general HE arising in the study of normal forms of the integrable four-dimensional Monge-Ampere equation. In [10], Manakov and Santini describe reductions of the HE based on certain self similar transformations.

6 Here, e study the symmetries, viz., Noether and Lie symmetries, that arise from the Euler-Lagrange equations, i.e., the geodesic equations, related to the metric (1.1). We sho the relationship beteen these and the Killing vectors admitted by the metric. As is ell knon, the Noether or the Lie symmetries can be used to successively reduce the geodesic equations and in the former case, the symmetries allo for double reduction for each Noether symmetry as each symmetry corresponds to a knon conservation la via Noether s theorem.

7 Notations and definitions We present some salient features of an Euler Lagrange system of differential equations. Consider an r th-order system of partial differential equations of n independent and m dependent variables, viz.[11, 12, 13], E β (t, v, v (1),..., v (r) ) = 0, β = 1,..., m. (1.4) A conservation la of (1.4) is a solution of the equation given by, D i T i = 0, (1.5) on the solutions of (1.4), here D i is the total derivative operator given by, D i = t i + vi α α + v v α ij vj α +, i = 1,..., n and T = (T 1,..., T n ) a conserved vector of (1.4).

8 Notations and definitions If A is the universal space of differential functions, then a Lie-Bäcklund operator (vector field) is given by X = ξ i t i + η α v α + ζα i v α i + ζ α i 1 i 2 v α i 1 i 2 +, here ξ i, η α A and the additional coefficients are ζ α i = D i (W α ) + ξ j v α ij, ζ α i 1 i 2 = D i1 D i2 (W α ) + ξ j v α ji 1 i 2, (1.6). and W α is the Lie characteristic function defined by W α = η α ξ j v α j, here vi α represent the first derivatives of v α (ith respect to the ith independent variable t i, v α represents all the second ij

9 Notations and definitions In this ork, e assume that X is a Lie point symmetry operator, i.e., ξ and η are functions of t and v and are independent of derivatives of v. The Euler-Lagrange equations, if they exist, associated ith (1.4) is the system δl/δv α = 0, α = 1,..., m, here the Euler-Lagrange operator δ/δv α is given by δ δv α = v α + s 1 ( 1) s D i1 D is v α i 1 i s, α = 1,..., m. L is referred to as a Lagrangian and a Noether symmetry operator X of L arises from a study of the invariance properties of the associated functional L = L(t, v, v (1),..., v (r) )dt Ω defined over Ω. If e include point dependent gauge terms f,..., f n, the Noether symmetries Symmetries X are -given ASD manifolds by a Killing-type

10 Heavenly equation We, firstly, present a procedure to determine exact solutions of the heavenly equation. To this end, e resort to the algebra of Lie point symmetry generators of the equation, i.e., the one parameter Lie group of transformations that leave invariant the equation. For the procedure and details, e refer the reader to [11, 13]. It can be shon, that a basis of the Lie point symmetry algebra of (1.2) is X 1 = x u, X 2 = f 3 (, z) y, X 3 = (yf 4 (, z) + x f 4 d) u, X 4 = xf 5 (z) u, X 5 = 2 y xy u, X 6 = + y y + u u X 7 = 2 x x + y y u u, X 8 = 2f 6 (z) y + x 2 fz 6 u, X 9 = 2f 1 (z, ) x 2 fz 1 d + (x 2 fzzd 1 + y(f 1 + 2xfz 1 )) u, X 10 = 6f 7 (z) + 6fz 7 y x 2 fzz 7 u, X 11 = 6f 2 (z, ) z 6(x fzzd 2 + yfz 2 ) y +6(yf 2 + xfz 2 ) x 6 fz 2 d +(x 3 fzzz 2 + y(y 2 f 2 + 3x(yfz 2 + xfzz))) 2 u

11 (a). In X 11, if f 2 = 1 6z, e get the scaling symmetry generator X = x x + y y z z leading to the invariants and transformed variables X = x, Y = y/, Z = z, U = u, U = U(X, Y, Z ) so that (1.2) becomes U X + XU XX YU XY + ZU XZ U YZ + U XX U YY U 2 XY = 0, (1.8) hich admits, inter alia, the symmetry generator X 1 11 = 2 X + Y 2 U and the scaling generator X 2 11 = X X + Y Y + 3U U.

12 Via X11 1, it can be shon that (1.8) reduces to the canonical equation Ū YZ Y 2 = 0, here Ū = U 1 2 XY 2 (Ū = Ū(Y, Z )). It can be shon, thus, that a solution of the heavenly equation is u = 1 zy xy 2 2. (1.9) Similarly, (1.4) may be transformed using X11 2, i.e., α = Y /X, Z = Z and Ū = U/X 3 ith Ū = Ū(α, Z ). That is, 2(α 2 αūα+3ū)ūαα (2α+1)ŪαZ (Ūα+6α)Ūα+6(ŪZ +Ū) = 0.

13 (b). For an alternative reduction of (1.3), one may take a linear combination of X 10 and X 11 ith F 7 = 1 6 z and 1 6t, respectively. (c). X 6 + X 7 leads to similarity variables α = tx 3, β = yx 2, z = z and U = u ith U = U(α, β, z). (d). A travelling ave reduction takes the form cu αα u βγ + u αα u ββ u 2 αβ = 0, here u = u(α, β, γ), α = x ct, β = y kt and γ = z mt, c, k and m are constants ( ave speeds in the direction of x, y and z, respectively). The reduced PDE may then be analysed further using a Lie symmetry reduction. Thus, a large class of reductions and invariant, exact solutions of (1.2) are obtainable.

14 Geodesic equations In general, the Lagrangian of the geodesic equations is given by L = z y + x u xx z 2 u yy 2 + 2u xy z, here is the derivative ith respect to the arclength variable s. The Euler-Lagrange equations are: 2 u xyy + 2 z u xxy z 2 u xxx = 0, z 2 u yyy + 2 z u xyy z 2 u xxy = 0, y 2 u xy 2 (u yyz + 2u xy ) + 2z u xx + z 2 u xxz + 2y z u xxy 2 (y u xyy z u xx + x u xxy ) + 2x z u xxx = 0, x + 2 u yy + 2 u yy + 2 z u yyz + 2 y u yyy 2z u xy 2z 2 u +2 x u xyy 2y z u xyy z 2 u xx 2x z u xxy = 0. (2.1)

15 For u = 1 zy xy given in (1.9), the Euler-Lagrange (geodesic) equations are given by 2 = 0, z 2 2 z 2 z = 0, y 2 2 y 2y + 2 y = 0, 2yz 2 + ( x 2 2 (z y + y z + yz ) ) 3 x 2 2 ( x y z + x yz ) = 0. (2.2)

16 The Lie point symmetry generators that leave invariant the system (2.2) are: X 1 = ( 2 1)y 2+ 2 x z, X 2 = y 2 2 (1 + 2) x z, X 3 = s, X 4 = s s, X 5 = ( 1 + 2)z 2 x y, X 6 = z 2 (1 + 2) x y, X 7 = s x, X 8 = ln x, X 9 = 1 x, X 10 = ln s, X 11 = x x, X 12 = x x + y y, X 13 = x x + z z. (2.3)

17 The above algebra contain the algebra of Noether symmetry generators X in hich L X L = f. Alternatively X satisfies (1.7) hich in this context becomes the equation XL + LDσ = Df, (2.4) here X = σ s + A 1 + A 2 x + A 3 y + A 4 z, σ, A i are functions of (s, t, x, y, z), D is the total derivative ith respect to s.

18 Equation (2.4) becomes zη 2 2 xτ 2 2yzτ 2 φ 2 yς η z 2 + 2yτ z / 2 +z η + z η + x z η x + y z η y + z 2 η z 2x 2 σ 2yz 2 σ 2 2 x σ + 4y z σ 2y z σ 2x 3 σ 2 2 x σ + 4y 2 z σ + 4y x z σ x 2yz 2 y σ y 2 2yz 2 z σ z 2 +x τ 2yz τ + 2yz x τ x 2 2yy z τ y 2 y z σ 2x 2 x σ x 2x y z σ x 2x 2 y σ y 2 x y σ y + 4y y z σ y 2yz 3 σ 2 2yz 2 x σ x 2 2 x 2 σ x 2y 2 z σ y 2x 2 z σ z 2 x z σ z + 4y z 2 σ z 2y z 2 σ z + 2x τ + 2yz τ 2 + 2x 2 τ + x 2 τ x 2yx z τ x + 2x z τ z + 2yz 2 τ 2 + 2yz z τ z 2 + 2x y τ y + x φ x + y φ y + z φ z 2y ς +x y ς x 2y y ς y + y 2 ς y 2y z ς z + x τ 2y z τ + 2yz y τ y 2 + x z τ z 2yz 2 τ z + y ς 2y 2 ς + y z ς z + x y τ y + 2x x τ x + φ + 2 φ + y ς 2y + ( y z + x ( zy x ) ( y ) z ) (σ + σ + x σ +

19 For the gauge f = 0, the separation by monomials and subsequent tedious calculations lead to the folloing vector fields, X 1 = ( 2 1)y 2+ 2 x z, X 2 = y 2 2 (1 + X 3 = s, X 4 = s s + + z z, X 5 = ( 1 + 2)z 2 x y, X 6 = z 2 (1 + 2) x y, X 7 = + x x, X 8 = s s + + y y, X 9 = 1 x. (2.6) Let X p = X 4 X 8 = y y + z z.

20 Noether first integrals Each of these or a linear combination lead to a conservation la for the geodesic equations by Noether s theorem above. Also, the non zero gauge may also be investigated. We list some for illustrative purposes. i. X 3 : T 3 = yz 2 + (x 2yz )+ 2 ( x +y z ) 2 ii. X 7 : T 7 = x + 2yz ( ( + x 2yz iii. X 6 : T 6 = ) ) 2 z z iv. X = 2s s + + ( x x + y y + ) z z : 2syz 2 T = + x 3 + 2s 2 x + 2s x zy + yz 4sy z + 2sy z

21 Killing vectors/isometries The algebra of Killing vectors based on the equivalent metric is a subalgebra of the algebra above, i.e., L X g = 0. These are generated by X 1 X 2, X p, X 7, X 9. Whilst this is proved to be alays the case (see Bokhari et al [14]-[17]), e note that X 5 and X 6 are independent of the arclength variable s and yet are not Killing vectors.

22 We have studied the Noether and Lie symmetries, that arise from the Euler-Lagrange equations, i.e., the geodesic equations, related to the ASD Ricci-flat metric hich depends on the second heavenly equation. It as shon that the Killing vectors are contained in the Noether symmetries generated by the Lagrangian of the geodesic equations. Interestingly, a number of symmetries hich are Noether and not Killing vectors are independent of the arclength variable s. One could further explore various equations on the manifold like the ave equation and also the nature of the general heavenly equation.

23 George: This is not Goodbye...Thank you for your contributions and Good luck!

24 For Further Reading I M. F. Hogner, J. Math. Phys. 53, (2012) S. Chakravarty, L. Mason and E. T. Neman, J. Math. Phys. 32, 1458 (1991) R. Penrose, Gen. Rel. Grav. 7, 31, (1976) S. V. Manakov and P. M. Santini, J. Phys. A: Math. Theor. 42, (2009) M. Dunajski, J. Phys. A: Math. Theor. 41, (2008) J. F. Plebanski, J. Math. Phys. 16, 2395 (1975) V. Husain, Class. Quantum Gmv. 11, 921 (1994)

25 For Further Reading II A. A. Malykh and M. B. Sheftel, J. Phys. A.: Math. Theor (2011) B. Doubrov and E. V. Ferapontov, J. Geometry and Physics (2010) S. V. Manakov and P. M. Santini, J. Phys. A: Math. Theor (2011) Springer-Verlag classics by G. Bluman & S. Kumei and P. Olver, on the Application of Lie groups to differential equations

26 For Further Reading III E. Noether, Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, 2, 235 (1918) (English translation in Transport Theory and Statistical Physics, 1(3), 186 (1971)) H. Stephani, Differential Equations: their solution using symmetries Cambridge University Press, Cambridge (1989) A.H. Bokhari, A.Y. Al-Deik, A.H. Kara, M.Karim, and F.D. Zaman, Journal of Mathematical Physics,, 52 (6), (2011) S. Jamal, A.H. Kara and Ashfaque H. Bokhari, J. Phys., (2012)

27 For Further Reading IV A. H. Bokhari and A. H. Kara, Gen. Relativ. Gravit., 39, 2053 (2007) A. H. Bokhari, A. H. Kara, A. R. Kashif and F. D. Zaman, Int J Th. Physics 45(6), 1029 (2006)