HUEPHAH 0H3HKA, 2007, moM 70, № 3, c. 612-620
== ELEMENTARY PARTICLES AND FIELDS. THEORY
SYMMETRIES, INTEGRALS, AND THREE-DIMENSIONAL REDUCTIONS OF PLEBANSKI'S SECOND HEAVENLY EQUATION
© 2007 F. Neyzi1)*, M. B. Sheftel1)**, D. Yazici2)***
Received May 16, 2006
We study symmetries and conservation laws for Plebanski's second heavenly equation written as a firstorder nonlinear evolutionary system which admits a multi-Hamiltonian structure. We construct an optimal system of one-dimensional subalgebras and all inequivalent three-dimensional symmetry reductions of the original four-dimensional system. We consider these two-component evolutionary systems in three dimensions as natural candidates for integrable systems.
PACS numbers: 02.30.Ik
1. INTRODUCTION
Complex Einstein field equations that govern self-dual gravitational fields reduce to a single scalar-valued equation. This is either the first heavenly equation of Plebanski [1], or
UttUxx - Ux + uxz + Uty = 0, (1)
which is Pleban ski's second heavenly equation. In this paper we shall consider symmetries, recursions, and integrals of motion for the second heavenly equation presented as a two-component evolutionary system of two equations possessing a Hamiltonian structure. It can be formulated as a Hamiltonian system in three or more inequivalent ways [2] and, therefore, by the theorem of Magri [3], it is a completely integrable system. We consider all its inequivalent three-dimensional symmetry reductions in the hope that they will inherit multi-Hamiltonian structures and thus be integrable systems in three dimensions.
Earlier we constructed [4] a scalar recursion operator for one-component equation (1), which was determined by two different, though compatible, recursion relations, and now we cast it into 2 x 2 matrix form which naturally joins these two recursion relations into one matrix relation.
In Section 2 we introduce an evolutionary two-component form of the second heavenly equation (1) which we call the second heavenly system. To discover a Hamiltonian structure of this system of
'-'Department of Physics, Bogazici University, Istanbul, Turkey.
2)Department of Physics, Yildiz Technical University, Istanbul, Turkey.
E-mail: neyzif@boun.edu.tr E-mail: mikhail.sheftel@boun.edu.tr E-mail: yazici@yildiz.edu.tr
equations, we start with its symplectic structure. Then we invert the operator determining the symplectic two-form to find the Hamiltonian operator. We easily find the corresponding Hamiltonian density and thus present a Hamiltonian structure of the second heavenly equation in a two-component form. In Section 3 we construct explicitly a 2 x 2 matrix integral-differential recursion operator which incorporates naturally both recursion relations. This operator and the operator determining symmetries form a Lax pair for the two-component system. In Section 4 we find a complete Lie algebra of point symmetries of the two-component system and for all variational symmetries we construct the corresponding integrals of motion. Some simple symmetries of our equations are shown to be generated by certain integrals of motion via previously constructed Hamiltonian operator. In Section 5 we give examples of higher flows obtained with the aid of the Hermitian conjugate recursion operator. Those flows are nonlocal symmetries generated by local integrals. In Section 6 we construct an optimal system of one-dimensional Lie subalgebras using the adjoint representation of the total symmetry group on its Lie algebra. In Section 7 we construct the corresponding symmetry reductions of the second heavenly system. These are three-dimensional evolutionary two-component systems determining nonequivalent invariant solutions of the original four-dimensional system.
2. FIRST-ORDER FORM OF THE SECOND HEAVENLY EQUATION AND HAMILTONIAN STRUCTURE
The second heavenly equation is a second-order partial differential equation. In order to present its Hamiltonian structure we singled out an independent
variable t in (1) to play the role of "time" and express the second heavenly equation as a pair of first order nonlinear evolution equations [2]
ut = q, qt = —(q% - qy - uxz) = Q, (2)
uxx
where the auxiliary variable q together with u forms a set of two unknown functions. For the sake of brevity we shall refer to (2) as the second heavenly system. Now the vector field
X
d
qjr + qT~
du dq
(3)
defines the flow. We shall use matrix notation ui, i = = 1,2 for the variables u1 = u, u2 = q.
In [2] we have constructed the matrix differential operator
K =
qxDx + Dx qx Dy ux
ux
0
(4)
Q = / udV =
qxdu A dux —
1
— uxxdu A dq — - du A duy dV,
where dV = dxdydz and (5) is a closed two-form. The closure of the symplectic two-form (5) is equivalent to the satisfaction of Jacobi identities for the Hamiltonian operator.
The first Hamiltonian operator J for the second heavenly system was obtained as an inverse to K according to the definition f Kik(£, a)Jkj (a, n)da =
Sj S(£ — n) with the result 0
J
l
-xx
__^ D + D Jlx__
2 DDx DDx 2
-xx Uxx Uxx
(6)
V
—- Dy
-xx "-xx
/
It can be directly verified that 1
H = - ■q u^xx ¡rx Uxuz
(7)
is a conserved density for the flow (2) and, furthermore, H = f HdV is the Hamiltonian generating
the equations of motion (2) by the action of the operator (6):
J
S-H Sq H
determining the symplectic structure of the equations of motion (2), that is, the closed symplectic two-form u which satisfies the equation iXu = dH obtained by its contraction with the vector field X (3), where H is the Hamiltonian. The functional symplectic two-form [5] is obtained by integrating the density u = = 1 dui A Kijduj. Here K is given by (4), or explicitly
The direct proof of the Jacobi identity for the Hamiltonian operator (6) is straightforward but rather lengthy. A shorter proof follows from the fact that the inverse operator K in (4) determines a closed symplectic two-form (5).
3. RECURSION OPERATOR
Recently a recursion operator for second heavenly equation was obtained [4, 6]. We express it in a 2 x 2 matrix form suitable for the two-component system (2).
We introduce two components for symmetry characteristics [5] of the second heavenly system, so that the Lie equations with t as a group parameter have the matrix form
$
(8)
where 1 = Vt. From the Frechet derivative of the flow (2) we find
(5) A =
Dt
1
QDl + DxDz Dt — ^Dx + D
-xx
x
-xx
(9)
and the equation determining the symmetries of the second heavenly system is given by
A ($)=0. (10)
We note that the combination of the first and second determining equations (10) with the latter multiplied by an overall factor of uxx, coincides with the determining equation for symmetries of the original second heavenly equation (1). This determining equation has the divergence form (uxx1 - qxVx + Vy)t + (qtVx -— qx1 + Vz)x = 0 [4] rewritten in our new notation. This implies the local existence of the potential variable Vp such that
pt = qtVx - qx1 + Vz, (11)
px = -(uxx1 - qxVx + Vy)
which, as is proven in [4], satisfies the same equation determining symmetries of (1) and therefore is a "partner symmetry" for v [7]. In the two-component form we define the second component of this new symmetry similar to the definition of 1 as ip = Vpt.
t
Then the two-component vector $ =
satis-
fies the determining equation for symmetries in the form (10) and hence is a symmetry characteristic of the system (2) provided the vector (8) is also a symmetry characteristic. Thus the equations (11) become the recursion relation for symmetries in the two-component form $ = r($) with the recursion
operator given by
r
D-l(q, QD,
x Dx
- Dy) -D
+ Dz
Ux
-Qx
where D-1 is the inverse of Dx. The commutator of the recursion operator (12) and the operator determining symmetries (9) has the form
[R A] = I D-1 (Qt - Q)xx - (Qt - Q)x Dx1 (Ut - Q)xx I (13)
^ (ut - Q)xx + ut (Dy - 2QxDx)(Qt - Q) (Qt - Q)x
and as a consequence, the operators r and a commute, [r, a] = 0, by virtue of the second heavenly system (2). Moreover, r and a form a Lax pair for the second heavenly system. The recursion operator generates new Hamiltonian operators satisfying Magri's theorem [2].
4. SYMMETRIES AND INTEGRALS OF MOTION
Hamiltonian operators provide a natural link between commuting symmetries in evolutionary form [5] and conserved quantities, integrals of motion, in involution with respect to Poisson brackets.
The complete Lie algebra of point symmetries of the original one-component heavenly equation (1) was given in [4], but in the two-component representation the results look different. The basis generators of one-parameter subgroups of the complete Lie group of point symmetries for the second heavenly system (2) have the form
X1 = -2zdt + txdu + xdq, X2 = tdt + zdz + udu, X3 = tdt + xdx + 3udu + 2qdq,
x3
Zb = b(y)dz - b'(y)xdt - b"(y)—du,
6
Vd = dz(y, z)(tdu + dq) - dy(y, z)xc)u, Ya = ady + a'(xdx - tdt - zdz + qdq) +
(14)
tx2 x2 L>'JLj ^ 'JLJ
+ a ( xzdt - — du - — dq\ + a — du,
x2
Uc = Cydt + Czdx - Cyzx(tdu + dq) + Cyy — du +
+ Czz ^y du + tdq^J , Wf = f (y, z)du,
where a(y), b(y), C(y,z), d(y,z), and f (y,z) are arbitrary functions, primes denote derivatives of functions of one variable, and we used the shorthand notation dt = d/dt and so on. Since some of the generators contain arbitrary functions, the total symmetry group is an infinite Lie (pseudo)group. In the Table 1 the commutator adXi(Xj) = [Xi,Xj] of the generators (14) stands at the intersection of ith row and jth column, ad is the generator for the adjoint action of the symmetry group on its Lie algebra and we have used the shorthand notation
X2 = X2 - 1 X2 = 2X2 - 1
(15)
c = Cy(y, z)dz - Cz(y, z)dy, B(y) = j b(y)dy, 5a = a(y)dy + a'(y)1, Y'a = a(y)dy - a'(y)1, H(y) = J h(y)dy,
where I is the unit operator. Note the immediate consequences of definitions (15): X2f (y,z) = zfz(y,z), X2 f (y, z) = zfz - f, X'i f (y, z) = 2zfz - f, Ya(y)
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