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 , ∈ 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
′ ′ ′ ′ ′2 ′2 ′ ′ L = z y + w x − uxx z − uyy w + 2uxy w z ,
where ′ is the derivative with respect to the arclength variable s. The Euler-Lagrange equations are:
′′ ′2 ′ ′ ′2 −w − w uxyy + 2w z uxxy − z uxxx = 0, ′′ ′2 ′ ′ ′2 −z − w uyyy + 2w z uxyy − z uxxy = 0, ′′ ′′ ′2 ′′ ′2 ′ ′ −y − 2w uxy − w (uyyz + 2uxyw ) + 2z uxx + z uxxz + 2y z uxxy ′ ′ ′ ′ ′ ′ −2w (y uxyy − z uxxw + x uxxy ) + 2x z uxxx = 0, ′′ ′′ ′2 ′ ′ ′ ′ ′′ ′2 −x + 2w uyy + w uyyw + 2w z uyyz + 2w y uyyy − 2z uxy − 2z uxyz ′ ′ ′ ′ ′2 ′ ′ +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 ′2 ′′ w − w = 0, ′2 ′ ′ zw − 2w z − ′′ = , w 2 w z 0 yw ′2 − 2ww ′y ′ − 2wyw ′′ + w 2y ′′ = 0, 2yzw ′2 + w xw ′2 − 2 (zw ′y ′ + yw ′z′ + yzw ′′) − w 3x′′ −2w 2 (w ′x′ − y ′z′ + xw ′′ − yz′′) = 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:
√ √ √ X = −( 2 − 1)yw −2+ 2∂ + w −1+ 2∂ , 1 √ √ x √ z X = yw −2− 2(1 + 2)∂ + w −1− 2∂ , 2 x z √ √ √ X = ∂ , X = s∂ , X = −(−1 + 2)zw 2∂ + w 1+ 2∂ , 3 s √ 4 √ s 5 √ 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 . Alternatively X satisfies (1.7) which in this context becomes the equation
XL + LD = Df , (2.4) where X = ∂s + A1∂w + A2∂x + A3∂y + A4∂z , , Ai are functions of (s, t, x, y, z), D is the total derivative with respect to s.
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Equation (2.4) becomes
′2 ′2 ′2 ′2 ′ ′ ′ ′ ′2 2 − x w − 2yz w + w + y&w − 2w z + 2y w z 2 z w w w 2 w 3 w w 2 w / w ′2 ′ ′2 ′ + ′′ + ′ ′ + ′ ′ + ′ ′ + ′2 − 2xw − 2yzw z w z w x z x y z y z z w w 2 ′ ′ ′ ′3 ′3 − ′ ′′ + 4yw z − ′ ′′ − 2xw w − 2yzw w − ′2 ′ 2w x w 2y z w w 2 2w x w ′2 ′ ′2 ′ ′2 ′ + 4yw z w − ′ ′ ′ − 2xw x x − 2yzw x x − ′ ′2 w 2w y z w w w 2 2w x x ′ ′ ′ ′2 ′ 4yw x z x ′ ′ ′ 2xw y y + w − 2x y z x − w ′2 ′ ′ ′ ′ ′2 ′ − 2yzw y y − ′ ′ ′ + 4yw y z y − ′2 ′ − 2xw z z w 2 2w x y y w 2y z y w ′2 ′ ′ ′2 ′ ′ ′ ′ − 2yzw z z − ′ ′ ′ + 4yw z z − ′ ′2 + 2xw + 2yzw w 2 2w x z z w 2y z z w w 2 ′ ′ ′2 ′2 ′ ′ ′ ′ + ′ ′ − 2yz + 2xw w + 2yzw w + ′ ′ − 2yw z w + 2xw x x x w w w 2 w x w w w ′ ′ ′ ′ ′ ′ ′ ′ + 2yzw x x + ′2 − 2yx z x + 2xw y y + 2yzw y y + ′ ′ w 2 x x w w w 2 x y y ′ ′ ′ ′ ′ ′ ′2 − 2yy z y + 2xw z z + 2yzw z z + ′ ′ − 2yz z + ′ ′ + ′2 w w w 2 x z z w w w w ′ ′ ′2 ′ ′ ′ ′ ′ ′ ′ ′ 2yw & ′ ′ 2yw &w ′ ′ 2yw x &x +w x x + w y y + w z z − w + y & − w + w y &w − w ′ ′ ′ ′ ′ ′ 2yw y &y ′2 2yw z &z ′ ′ +x y &x − w + y &y − w + y z &z −zy−xw y + ′ ′ + ′ ′ − A H Kara ′2Symmetries+ − - ASD manifolds′ ′ (′ + ′ + ′ + ′ + ′ ) y z w x w 2 w 2 w w z w w x x y y z z ′ ′ ′ ′ ′ − (f + w fw + x fx + y fy + z fz ) = 0. (2.5) Introduction Lie & Noether symmetries and Killing vectors Summary
For the gauge f = 0, the separation by monomials and subsequent tedious calculations lead to the following vector fields, √ √ √ √ √ √ −2+ 2 −1+ 2 −2− 2 −1− 2 X1 = −( 2 − 1)yw ∂x + w ∂z , X2 = yw (1 + 2)∂x + w ∂z , X = ∂ , X = s∂ + w∂ + z∂ , 3 s √4 √s w √ z X = −(−1 + 2)zw 2∂ + w 1+ 2∂ , 5 √ √ x √ y − 2 1− 2 X6 = zw (1 + 2)∂x + w ∂y , X7 = −w∂w + x∂x , 1 X8 = s∂s + w∂w + y∂y , X9 = w ∂x . (2.6) Let Xp = X4 − X8 = −y∂y + z∂z .
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Noether first integrals
Each of these or a linear combination lead to a conservation law for the geodesic equations by Noether’s theorem above. Also, the non zero gauge may also be investigated. We list some for illustrative purposes. ′2 ′ ′ ′ 2 ′ ′ ′ ′ = yzw +ww (xw −2yz )+w (w x +y z ) i. X3: T3 w 2 ′ 2yzw ′ ′ ′ ii. X7: T7 = xw + + wx − 2yz √ w √ − 2 ′ ′ iii. X6: T6 = w − 1 + 2 zw − wz iv. X = 2s∂s + w∂w + x∂x + y∂y + z∂z : ′2 ′ = 2syzw + ′ − + 2sw − ′ + ′ ′ − ′ + T w 2 xw 3 w wx 2sw x zy ′ 4syw ′z′ ′ ′ yz − w + 2sy z
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary Killing vectors/Isometries
The algebra of Killing vectors based on the equivalent metric is a subalgebra of the algebra above, i.e., ℒX g = 0. These are generated by
X1 X2, Xp, X7, X9.
Whilst this is proved to be always the case (see Bokhari et al [14]-[17]), we note that X5 and X6 are independent of the arclength variable s and yet are not Killing vectors.
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary
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 which depends on the second heavenly equation. It was shown that the Killing vectors are contained in the Noether symmetries generated by the Lagrangian of the geodesic equations. Interestingly, a number of symmetries which are Noether and not Killing vectors are independent of the arclength variable ‘s’. One could further explore various equations on the manifold like the wave equation and also the nature of the ‘general’ heavenly equation.
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary
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A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary For Further ReadingI
M. F. Hogner, J. Math. Phys. 53, 103517 (2012) S. Chakravarty, L. Mason and E. T. Newman, 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, 404013 (2009) M. Dunajski, J. Phys. A: Math. Theor. 41, 315202 (2008) J. F. Plebanski, J. Math. Phys. 16, 2395 (1975) V. Husain, Class. Quantum Gmv. 11, 921 (1994)
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary For Further ReadingII
A. A. Malykh and M. B. Sheftel, J. Phys. A.: Math. Theor. 44 155201 (2011) B. Doubrov and E. V. Ferapontov, J. Geometry and Physics 60 1604 (2010) S. V. Manakov and P. M. Santini, J. Phys. A: Math. Theor. 44 345203 (2011) Springer-Verlag classics by G. Bluman & S. Kumei and P. Olver, on the Application of Lie groups to differential equations
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary 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-Dweik, A.H. Kara, M.Karim, and F.D. Zaman, Journal of Mathematical Physics,, 52 (6), 063511 (2011) S. Jamal, A.H. Kara and Ashfaque H. Bokhari, J. Phys., 90 667 (2012)
A H Kara Symmetries - ASD manifolds Introduction Lie & Noether symmetries and Killing vectors Summary For Further ReadingIV
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)
A H Kara Symmetries - ASD manifolds