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The Symmetry Structures of ASD Manifolds

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The Symmetry Structures of ASD Manifolds Introduction Lie & Noether symmetries and Killing vectors Summary The symmetry structures of ASD manifolds A H Kara1 1School of Mathematics University of the Witwatersrand, Johannesburg South Africa African Institute of Mathematics A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Outline 1 Introduction 2 Lie & Noether symmetries and Killing vectors 3 Summary A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary The dispersionless integrable systems in 3+1 dimensions do not admit soliton solutions and there is no associated Riemann–Hilbert problem where the corresponding Lie group is finite dimensional. However, 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 Przanowski’s equation (see [1] and [2] and references therein). In this case the unknown 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]). A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary If the Ricci-flat condition is imposed on top of the anti-self-duality, the work of Pleba´nski [6] implies the existence of a local coordinate system (x; y; z; w) and a function u on an open set such that any ASD Ricci-flat metric, g, is locally of the form 2 2 2 ds = 2(dzdw + dwdx − uxx dz − uyy dw + 2uxy dwdz) (1.1) where u(x; y; z; w) satisfies the second heavenly equation (sHE) 2 uwx − uzy + uxx uyy − uxy = 0: (1.2) A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary In [5], the above equations are studied subject to the existence of a conformal Killing vector X satisfying LX g = pg (1.3) for some function p and L is the Lie derivative. The relationship between 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. A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Here, we 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 show the relationship between these and the Killing vectors admitted by the metric. As is well known, the Noether or the Lie symmetries can be used to successively reduce the geodesic equations and in the former case, the symmetries allow for double reduction for each Noether symmetry as each symmetry corresponds to a known conservation law via Noether’s theorem. A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Notations and definitions We present some salient features of an Euler Lagrange system of differential equations. Consider an rth-order system of partial differential equations of n independent and m dependent variables, viz.[11, 12, 13], E (t; v; v(1);:::; v(r)) = 0 ; = 1;:::; m : (1.4) A conservation law of (1.4) is a solution of the equation given by, i Di T = 0 ; (1.5) on the solutions of (1.4), where Di is the total derivative operator given by, @ @ @ = + + + ⋅ ⋅ ⋅ ; = ;:::; Di i vi vij i 1 n @t @v @vj and T = (T 1;:::; T n) a conserved vector of (1.4). A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Notations and definitions If A is the universal space of differential functions, then a Lie-Bäcklund operator (vector field) is given by @ @ @ @ X = i + + + + ⋅ ⋅ ⋅ ; @ti @v i @v i1i2 @v i i1i2 where i ; 2 A and the additional coefficients are j i = Di (W ) + vij ; = D D (W ) + j v ; i1i2 i1 i2 ji1i2 (1.6) . and W is the Lie characteristic function defined by j W = − vj ; where vi represent the first derivatives of v (with respect to the ith independent variableA H Karati , vij Symmetriesrepresents - ASD manifolds all the second derivatives of v and so on. Introduction Lie & Noether symmetries and Killing vectors Summary Notations and definitions In this work, we assume that X is a Lie point symmetry operator, i.e., and are functions of t and v and are independent of derivatives of v. The Euler-Lagrange equations, if they exist, associated with (1.4) is the system L/v = 0, = 1;:::; m, where the Euler-Lagrange operator /v is given by @ X @ = + (−1)sD ⋅ ⋅ ⋅ D ; = 1;:::; m: v @v i1 is @v s≥1 i1⋅⋅⋅is 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 Z L = L(t; v; v(1);:::; v(r))dt Ω defined over Ω. If we include point dependent gauge terms f1;:::; fn, the Noether symmetriesA H Kara SymmetriesX are -given ASD manifolds by a Killing-type equation XL + LDi i = Di fi : (1.7) Corresponding to each X, a conserved vector T = (T 1;:::; T n) is obtained via Noether’s Theorem. Introduction Lie & Noether symmetries and Killing vectors Summary Heavenly equation We, firstly, present a procedure to determine exact solutions of the heavenly equation. To this end, we 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, we refer the reader to [11, 13]. It can be shown, that a basis of the Lie point symmetry algebra of (1.2) is 3 4 R 4 X1 = x@u; X2 = f (w; z)@y ; X3 = (yf (w; z) + x f dw)@u; 5 X4 = xf (z)@u; X5 = 2w@y − xy@u; X6 = w@w + y@y + u@u; 6 2 6 X7 = 2w@w − x@x + y@y − u@u; X8 = −2f (z)@y + x fz @u; 1 R 1 2 R 1 1 1 X9 = 2f (z; w)@x − 2 fz dw + (x fzz dw + y(fw + 2xfz ))@u; 7 7 2 7 X10 = 6f (z)@w + 6fz @y − x fzz @u; 2 R 2 2 X11 = 6f (z; w)@z − 6(x fzz dw + yfz )@y 2 2 R 2 +6(yfw + xfz )@x − 6 fz dw@w 3 R 2 2 2 2 2 +(x fzzz + y(y fww + 3x(yfzw + xfzz )))@u A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary 2 1 (a). In X11, if f = − 6 z, we get the scaling symmetry generator X = w@w − x@x + y@y − z@z leading to the invariants and transformed variables X = xw; Y = y=w; Z = zw; U = u; U = U(X; Y ; Z) so that (1.2) becomes 2 UX + XUXX − YUXY + ZUXZ − UYZ + UXX UYY − UXY = 0; (1.8) which admits, inter alia, the symmetry generator 1 2 X11 = 2@X + Y @U and the scaling generator 2 X11 = X@X + Y @Y + 3U@U . A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary 1 Via X11, it can be shown that (1.8) reduces to the canonical equation 1 U¯ + Y 2 = 0; YZ 2 ¯ 1 2 ¯ ¯ where U = U − 2 XY (U = U(Y ; Z)). It can be shown, thus, that a solution of the heavenly equation is 1 zy 3 1 xy 2 u = − − : (1.9) 6 w 2 2 w 2 Similarly, (1.4) may be transformed using X11, i.e., = Y =X, Z = Z and U¯ = U=X 3 with U¯ = U¯ ( ; Z ). That is, 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 2( − U +3U)U −(2 +1)U Z −(U +6 )U +6(UZ +U) = 0: A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary (b). For an alternative reduction of (1.3), one may take a linear 7 1 1 combination of X10 and X11 with F = 6 z and − 6 t, respectively. 3 2 (c). X6 + X7 leads to similarity variables = tx , = yx , z = z and U = u with U = U( ; ; z). (d). A ‘travelling wave’ reduction takes the form 2 −cu − u + u u − u = 0; where u = u( ; ; ), = x − ct, = y − kt and = z − mt, c, k and m are constants (‘wave 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. A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Geodesic equations In general, the Lagrangian of the geodesic equations is given by 0 0 0 0 02 02 0 0 L = z y + w x − uxx z − uyy w + 2uxy w z ; where 0 is the derivative with respect to the arclength variable s. The Euler-Lagrange equations are: 00 02 0 0 02 −w − w uxyy + 2w z uxxy − z uxxx = 0; 00 02 0 0 02 −z − w uyyy + 2w z uxyy − z uxxy = 0; 00 00 02 00 02 0 0 −y − 2w uxy − w (uyyz + 2uxyw ) + 2z uxx + z uxxz + 2y z uxxy 0 0 0 0 0 0 −2w (y uxyy − z uxxw + x uxxy ) + 2x z uxxx = 0; 00 00 02 0 0 0 0 00 02 −x + 2w uyy + w uyyw + 2w z uyyz + 2w y uyyy − 2z uxy − 2z uxyz 0 0 0 0 02 0 0 +2w x uxyy − 2y z uxyy − z uxxw − 2x z uxxy = 0: (2.1) A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary 3 2 = − 1 zy − 1 xy For u 6 w 2 2 w given in (1.9), the Euler-Lagrange (geodesic) equations are given by w 02 00 w − w = 0; 02 0 0 zw − 2w z − 00 = ; w 2 w z 0 yw 02 − 2ww 0y 0 − 2wyw 00 + w 2y 00 = 0; 2yzw 02 + w xw 02 − 2 (zw 0y 0 + yw 0z0 + yzw 00) − w 3x00 −2w 2 (w 0x0 − y 0z0 + xw 00 − yz00) = 0: (2.2) A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary The Lie point symmetry generators that leave invariant the system (2.2) are: p p p X = −( 2 − 1)yw −2+ 2@ + w −1+ 2@ ; 1 p p x p z X = yw −2− 2(1 + 2)@ + w −1− 2@ ; 2 x z p p p X = @ ; X = s@ ; X = −(−1 + 2)zw 2@ + w 1+ 2@ ; 3 s p 4 p s 5 p x y − 2 1− 2 s X6 = zw (1 + 2)@x + w @y ; X7 = w @x ; ln w 1 X8 = w @x ; X9 = w @x ; X10 = ln w@s; X11 = w@w − x@x ; X12 = x@x + y@y ; X13 = x@x + z@z : (2.3) A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary The above algebra contain the algebra of Noether symmetry generators X in which LX L = f .
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