Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates
aa r X i v : . [ n li n . S I] A ug Self-dual Einstein spaces and the general heavenly equation.Eigenfunctions as coordinates
B.G. Konopelchenko , W.K. Schief and A. Szereszewski Department of Mathematics and Physics,University of Salento and INFN, Sezione di Lecce,Lecce, 73100, Italy School of Mathematics and Statistics,The University of New South Wales,Sydney, NSW 2052, Australia Institute of Theoretical Physics,Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, PolandAugust 18, 2020
Abstract
Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaceswith the underlying governing equation being revealed as the general heavenly equation.The formalism developed here may be used to link algorithmically a variety of known heav-enly equations. In particular, the classical connection between Pleba´nski’s first and secondheavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, con-nections with travelling wave reductions of the recently introduced TED equation whichconstitutes a 4+4-dimensional integrable generalisation of the general heavenly equationare found. These are obtained by means of (partial) Legendre transformations. As a par-ticular application, we prove that a large class of self-dual Einstein spaces governed by acompatible system of dispersionless Hirota equations is genuinely four-dimensional in thatthe (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors.This generalises the known link between a particular class of self-dual Einstein spaces andthe dispersionless Hirota equation encoding three-dimensional Einstein-Weyl geometries.
A variety of important solution generation techniques applied to Einstein’s field equations havetheir origin in (or may be interpreted in terms of) integrable systems theory (see [1, 2] and ref-erences therein). Eigenfunctions play a key role in integrable systems theory since they encodeintegrable systems via the compatibility of the linear systems (Lax pairs) which they satisfy.Eigenfunctions, in turn, obey partial differential equations (eigenfunction equations) which areintimately related to the important Miura-type transformations. The latter provide a link be-tween integrable hierarchies and their associated modified versions such as the Korteweg-deVries (KdV) and modified KdV (mKdV) hierarchies. Eigenfunctions are the key ingredient inDarboux-type transformations which lie at the heart of solution generation techniques (B¨acklundtransformations) and their associated superposition principles (permutability theorems) for so-lutions of integrable systems. Eigenfunctions also encode conserved quantities associated withconservation laws hidden in integrable systems. Details on the above subjects and correspondingreferences may be found in, e.g., [2, 3, 4, 5, 6]. 1ierarchies of 1+1-dimensional modified integrable equations are known to admit integrablecounterparts which are related by reciprocal transformations. A geometrically important ex-ample is the integrable nonlinear Schr¨odinger (NLS) equation, the modified version of whichis given by the Heisenberg spin equation which, in turn, gives rise to the reciprocally relatedloop soliton equation (see, e.g., [2] and references therein). In the context of integrable systems,the analogue of reciprocal transformations for 2+1-dimensional hierarchies has been shown toinvolve eigenfunctions which, importantly, constitute some of the new independent variablesin the “reciprocally” related hierarchies. A prime example of the link between the associatedthree types of hierarchies of 2+1-dimensional integrable equations is provided by the connectionbetween the Kadomtsev-Petviashvili (KP) hierarchy, its modified (mKP) hierarchy and the re-ciprocally related 2+1-dimensional Dym hierarchy. In this connection, the reader may wish toconsult [7] for details and references.In this paper, we present a self-contained first application to general relativity of a generalscheme to be discussed elsewhere which, in any dimension, employs eigenfunctions of disper-sionless integrable systems as independent variables in the corresponding privileged equivalentsystems. Even though dispersionless integrable systems have been studied extensively (see, e.g.,[8, 9] and references therein) since the pioneering work of Zakharov and Shabat [10], the in-terpretation of eigenfunctions as independent variables has not been at the forefront of theseinvestigations. Here, we demonstrate that any four eigenfunctions of the self-dual Einstein equa-tions [11] corresponding to four distinct spectral parameters give rise to a unique representationof self-dual Einstein spaces in terms of a potential which depends on the eigenfunctions playingthe role of coordinates. Remarkably, the associated underlying dispersionless equation turns outto be the general heavenly equation introduced in [12, 13] as proven in Section 4. It is noted inpassing that particular coefficients of the Laurent expansions of eigenfunctions associated withthe self-dual Einstein equations have been identified in [12, 14] as the independent variablesin Pleba´nski’s important first and second heavenly equations [15] governing self-dual Einsteinspaces. An invariant definition of the above-mentioned key potential is presented in Section 5.It turns out that the assumption of one or two pairs of coinciding spectral parameters is alsoadmissible. This is discussed in Sections 7 and 8 respectively. In this manner, on the one hand,we retrieve the connection between the first heavenly equation and the Husain-Park equation [16]and, on the other hand, we demonstrate that the classical connection between Pleba´nski’s twoheavenly equations [15] may be viewed as a particular application of the algorithm developedin this paper. In Section 12, the classification in terms of (0, 1 or 2) pair(s) of coincidingspectral parameters is then shown to go hand in hand with travelling wave reductions of therecently introduced TED equation which constitutes a 4+4-dimensional integrable extensionof the general heavenly equation [17]. The connection between these two a priori unrelatedsubjects is provided by (partial) Legendre transformations. In particular, in Section 6, thelatter is shown to leave invariant the general heavenly equation. The investigation of Legendretransformations is motivated by the important observation that the gradient of the potentialsatisfying the general heavenly equation is, in fact, composed of another four eigenfunctionsassociated with the same spectral parameters.As a further application of the algorithm presented here, we show that the general heavenlyequation may be specialised to a system of four compatible dispersionless Hirota equations [18] bymatching the eigenfunction and scaling symmetries of the general heavenly equation [17, 19]. Itturns out that the self-dual Einstein metrics obtained in this manner generalise those associatedwith the three-dimensional Einstein-Weyl geometries known to be governed by the dispersionlessHirota equation [20] as discussed in Section 9. In general, in Section 10, the metrics generated bygeneric solutions of the dispersionless Hirota system are proven to be genuinely four-dimensionalin the sense that no conformal Killing vectors (including homothetic Killing vectors and Killingvectors) exist. The key to the proof of this property is the derivation of the action of theabove-mentioned eigenfunction symmetry constraint on the first heavenly equation, leading toa decomposition into three compatible differential equations. This is achieved in Section 11 byexploiting the connection between the general heavenly equation and Pleba´nski’s first heavenlyequation derived in Section 8. 2
The equations governing self-dual Einstein spaces
Self-dual Einstein spaces constitute four-dimensional manifolds equipped with a metric whichare characterised by a self-dual Riemann tensor. Self-duality implies that the Ricci tensor R ik vanishes and, hence, Einstein’s vacuum equations R ik = 0 are indeed satisfied [11]. Remarkably,the equations governing self-dual Einstein spaces have been shown to be equivalent to the self-dual Yang-Mills equations with four translational symmetries and the gauge group of volumepreserving diffeomorphisms [21]. The latter are encoded in the commutativity of two four-dimensional vector fields X and Y which are linear in a (spectral) parameter λ and divergencefree with respect to a volume form [4]. Specifically, consider two commuting vector fields X ( λ ) = A + λA , Y ( λ ) = A + λA (2.1)so that the commutativity condition [ X, Y ] = 0 is equivalent to[ A , A ] = 0 , [ A , A ] = 0 , [ A , A ] + [ A , A ] = 0 . (2.2)The latter constitute partial differential equations for the coefficients A iα of the vector fields A α = A iα ∂ x i , (2.3)where we have adopted Einstein’s summation convention over repeated indices. In addition, weassume that the vector fields X and Y are divergence free with respect to a volume formvol = f dx ∧ dx ∧ dx ∧ dx (2.4)encoded in a function f ( x i ) so that f div A α = ∂ x i ( f A iα ) = 0 . (2.5)The commutativity of X and Y implies the compatibility of the Lax pair [4] X Φ = 0 , Y
Φ = 0 . (2.6)Different choices of the coordinates x i lead to different but equivalent forms of the self-dualEinstein equations represented by the system (2.2), (2.5). For instance, Pleba´nski’s celebratedfirst heavenly equation [15] Ω x x Ω x x − Ω x x Ω x x = 1 , (2.7)which was, in fact, already recorded in 1936 in the context of “wave geometry” [22], correspondsto the choice (see, e.g., [23]) A = − ∂ x , A = Ω x x ∂ x − Ω x x ∂ x A = − ∂ x , A = Ω x x ∂ x − Ω x x ∂ x . (2.8)The latter vector fields are indeed seen to be divergence free with respect to the function f = 1and one may verify that the commutator relations (2.2) modulo the first heavenly equation (2.7)are satisfied. The corresponding metric reads [15] g = 2Ω x x dx dx + 2Ω x x dx dx + 2Ω x x dx dx + 2Ω x x dx dx (2.9)for which the Ricci tensor R ik may be shown to vanish. Within the general setting (2.1)-(2.6) of self-dual Einstein spaces, we now consider four (func-tionally independent) eigenfunctions φ i corresponding to four distinct parameters λ i , that is, X ( λ i ) φ i = 0 , Y ( λ i ) φ i = 0 , i = 1 , , , i as explained below). The important cases of one pair and twopairs of coinciding parameters are dealt with in Sections 7 and 8 respectively. In terms of thenew coordinates y i = φ i , (3.2)the vector fields X and Y are represented by X = ( Xφ i ) ∂ y i = ( λ − λ i )( A φ i ) ∂ y i Y = ( Y φ i ) ∂ y i = ( λ − λ i )( A φ i ) ∂ y i (3.3)and the volume form becomesvol = ˜ f dy ∧ dy ∧ dy ∧ dy , ˜ f = fJ , J = det (cid:18) ∂y i ∂x k (cid:19) . (3.4)Throughout this paper, indices on the parameters λ i are not taken into account when Einstein’ssummation convention is applied. Thus, expressions involving two repeated indices such as λ i a i in which one index is attached to λ do not denote a sum but expressions such as λ i a i b i = X i =1 λ i a i b i (3.5)encode summation over the index i . In particular, Einstein’s summation convention applies tothe vector field representation (3.3). The latter naturally leads to the introduction of the fourvector fields ˜ X ( i ) = ( A φ i ) X − ( A φ i ) Y. (3.6)Remarkably, even though the vector fields ˜ X ( i ) constitute linear combinations of the vector fields X and Y with non-constant coefficients, the following theorem obtains. Theorem 3.1.
The vector fields ˜ X ( i ) are divergence free.Proof. The proof of this theorem is based on the general (obvious) fact that if two commutingvector fields X and Y are divergence free then the vector field ( Y g ) X − ( Xg ) Y is likewisedivergence free for any function g . Hence, if we set g = φ i then0 = div[( Y φ i ) X − ( Xφ i ) Y ] = ( λ − λ i ) div[( A φ i ) X − ( A φ i ) Y ] (3.7)since ( A + λ i A ) φ i = 0 and ( A + λ i A ) φ i = 0.In terms of components, the vector fields ˜ X ( i ) adopt the form˜ X ( i ) = ( λ − λ k ) ˜ A ik ∂ y k , ˜ A ik = ( A φ i )( A φ k ) − ( A φ i )( A φ k ) . (3.8)It is observed that the coefficients of the skew-symmetric matrix ˜ A may be regarded as Pl¨uckercoordinates [24] of a line A φ φ φ φ ∧ A φ φ φ φ (3.9)in a three-dimensional projective space represented in terms of homogeneous coordinates. Ac-cordingly, the original Lax pair (2.6) may equivalently be formulated as the set of four equations˜ f ˜ X ( i ) Φ = A ik D k Φ = 0 , A ik = ˜ f ˜ A ik , D k = ( λ − λ k ) ∂ y k (3.10)since rank A = rank ˜ A = 2. The latter is reflected by the Pl¨ucker relationpf( A ) = A A + A A + A A = 0 , (3.11)where pf( A ) denotes the Pfaffian of A . The relevance of this observation is revealed below. Inparticular, the significance of the scaling of the matrix ˜ A is explained.4 The general heavenly equation
We are now in a position to present the key theorem of this paper.
Theorem 4.1.
Let φ i , i = 1 , , , be four eigenfunctions associated with any vector fieldrepresentation X ( x i ; λ ) , Y ( x i ; λ ) of self-dual Einstein spaces and distinct parameters λ i . Then,in terms of the independent variables y i = φ i , the self-dual Einstein equations are transformedinto the general heavenly equation ( λ − λ )( λ − λ )Θ y y Θ y y + ( λ − λ )( λ − λ )Θ y y Θ y y + ( λ − λ )( λ − λ )Θ y y Θ y y = 0 (4.1) for some potential Θ encoded in the divergence-free vector fields X and Y . Remark 4.1.
The differential equation (4.1), which may be formulated as c Θ y y Θ y y + c Θ y y Θ y y + c Θ y y Θ y y = 0 , c + c + c = 0 , (4.2)was originally derived as the continuum limit of the permutability theorem for both the classicalTzitz´eica equation of affine differential geometry and its integrable discretisation [12, 13, 25]. Inthis context, it was also observed that this equation constitutes yet another avatar of the self-dualEinstein equations. In connection with the classification of integrable symplectic Monge-Amp´ereequations, it was rediscovered and termed general heavenly equation [23]. The above theoremdemonstrates that the general heavenly equation is privileged in that the associated coordinatesplay the role of eigenfunctions regardless of the concrete realisation of the vector fields A α .In order to verify the above theorem, we begin by noting that the scaling of the matrix ˜ A ,leading to the matrix A as defined by (3.10) , has been introduced for convenience so that thevanishing divergence of the vector fields ˜ X ( i ) simplifies to the system of equations ∂ y k A ik = 0 , λ k ∂ y k A ik = 0 . (4.3)In terms of the skew-symmetric matrix ω defined by A lm = ǫ iklm ω ik , (4.4)where ǫ iklm denotes the totally anti-symmetric Levi-Civita symbol, these adopt the form ∂ y [ i ω kl ] = 0 , λ [ i ∂ y i ω kl ] = 0 (4.5)and may be regarded as the integrability conditions for the existence of a potential Θ definedby Θ y i y k = ω ik λ k − λ i . (4.6)Here, the square brackets indicate total anti-symmetrisation. Accordingly, the solution of (4.5)is parametrised in terms of Θ according to ω ik = ( λ k − λ i )Θ y i y k . (4.7)Now, the linear system (3.10) may be formulated as ω ω ω ω ω ω ω ω ω ω ω ω D Φ D Φ D Φ D Φ = 0 (4.8)with the associated rank 2 condition pf( A ) = 0 expressed aspf( ω ) = ω ω + ω ω + ω ω = 0 . (4.9)5he parametrisation (4.7) therefore leads to the general heavenly equation (4.1) with, for in-stance, ˜ X (3) Φ = 0 , ˜ X (4) Φ = 0 (4.10)constituting its standard Lax pair [12]. Here, it should be emphasised that the scaled vectorfields ˆ X (3) = ω ω D + ω ω D + D , ˆ X (4) = ω ω D + ω ω D + D (4.11)commute and are divergence free with respect to the volume formvol = Θ y y dy ∧ dy ∧ dy ∧ dy (4.12)as originally observed in [12]. Accordingly, the general heavenly equation may be regarded as an“invariant” form of the self-dual Einstein equations since neither the eigenfunctions φ i nor thepotential Θ are affected by coordinate transformations x i → f i ( x k ). In particular, any of theknown “heavenly” equations governing self-dual Einstein spaces (see, e.g., [23, 26] and referencestherein) may be mapped to the general heavenly equation by means of the algorithm presentedin this section. In order to interpret the potential Θ within the original self-dual Einstein setting associatedwith the vector fields A α ( x i ), we first make the following important observation. Remark 5.1.
The dependent variable Θ also encodes eigenfunctions in that the quantities ψ i = Θ y i (5.1)form another set of eigenfunctions corresponding to the parameters λ i . Indeed, since D k ψ i | λ = λ i = ( λ i − λ k )Θ y i y k = ω ki , i = k, (5.2)three equations of the linear system (4.8) for Φ = ψ i and λ = λ i are identically satisfied,while the remaining equation turns out to be pf( ω ) = 0. It is remarked that this implies thatthe general solution of the linear system at λ = λ i is an arbitrary function of the particulareigenfunctions φ i = y i and ψ i = Θ y i .The following theorem provides the basis of this section. Theorem 5.1.
Let φ i , i = 1 , . . . , be four eigenfunctions of the self-dual Einstein equations fordistinct parameters λ i . Then, the two-forms Ω k , k = 1 , . . . , defined by Ω k = 14 ǫ iklm A lm λ k − λ i dφ i ∧ dφ k , (5.3) where A lm = fJ ˜ A lm , ˜ A lm = ( A φ l )( A φ m ) − ( A φ l )( A φ m ) , J = det (cid:18) ∂φ i ∂x k (cid:19) , (5.4) have the following properties: d Ω k = 0 , Ω k ∧ Ω k = 0 , Ω k ∧ dφ k = 0 , X k =1 Ω k = 0 . (5.5) Here, the underbar indicates that there is no summation over the corresponding index. roof. Since the algebraic properties (5.5) , , are evident, it remains to show that d Ω k = 0. Interms of the coordinates y i , the latter is given by ǫ iklm ∂ y n A lm λ k − λ i dy n ∧ dy i ∧ dy k = 0 . (5.6)Hence, for any fixed k , the contribution of the terms proportional to dy m ∧ dy n ∧ dy k reads ∂ y m A l m λ k − λ n + ∂ y n A l n λ k − λ m = 0 , (5.7)wherein the indices k, l , m , n are fixed and distinct. On clearing the denominators of theabove, this may be formulated as ( λ k − λ m ) ∂ y m A l m = 0 (5.8)since the terms for m = k and m = l in the above sum vanish identically. Finally, the vanishingdivergence conditions (4.3) imply that the relations (5.8) are indeed satisfied and, hence, thetwo-forms Ω k are closed.The above theorem encodes the existence of a potential Θ which coincides with that derivedin Section 4 in connection with the parametrisation of the skew-symmetric matrix A in termsof the coordinates y i . Theorem 5.2.
There exist functions ψ k , k = 1 , . . . , such that Ω k = dψ k ∧ dφ k . (5.9) These constitute eigenfunctions for λ = λ k and give rise to the existence of a potential Θ via d Θ = ψ k dφ k . (5.10) Proof.
By virtue of Darboux’s theorem, the properties (5.5) , imply that the two-forms Ω k maybe written as exterior products of pairs of differentials. Property (5.5) then shows that Ω k isof the form (5.9) for some function ψ k . The compatibility condition dψ k ∧ dφ k = 0 associatedwith (5.10) guaranteeing the existence of the potential Θ is satisfied since dψ k ∧ dφ k = X k =1 Ω k = 0 (5.11)by virtue of (5.5) . Furthermore, since, in this case,Θ y k = ψ k , (5.12)comparison of (5.3) and (5.9) results inΘ y i y k = 14 ǫ iklm A lm λ k − λ i (5.13)which coincides with (4.6) by virtue of (4.4) so that ψ k = Θ y k indeed constitutes an eigenfunctionfor λ = λ k . It is emphasised that, on use of the pair (5.3), (5.9) regarded as a definition of thefunctions ψ k , one may also directly show that X ( λ k ) ψ k = Y ( λ k ) ψ k = 0. It is evident that Theorem 4.1 applied to the general heavenly equation( µ − µ )( µ − µ ) H x x H x x + ( µ − µ )( µ − µ ) H x x H x x + ( µ − µ )( µ − µ ) H x x H x x = 0 (6.1)7rovides an invariance of the general heavenly equation since it maps (6.1) to the general heavenlyequation (4.1). In general, the associated spectral parameters λ i do not have to be the fourparameters µ i in the general heavenly equation (6.1) but if we do make this special choice thenthe corresponding eigenfunctions are given by φ i = f i ( x i , H x i ) (6.2)as pointed out in the previous section. In particular, the choice φ i = H x i (6.3)is admissible. This raises the question as to whether the quantities x i and x i := H x i playsymmetric roles in the general heavenly equation (6.1). In order to demonstrate that this is thecase, we observe that the general heavenly equation may be formulated as( µ − µ )( µ − µ )( H x x H x x − H x x H x x )+ ( µ − µ )( µ − µ )( H x x H x x − H x x H x x ) = 0 (6.4)and state the following theorem. Theorem 6.1.
The general heavenly equation in the form ( µ − µ )( µ − µ ) dx ∧ dx ∧ dx ∧ dx + ( µ − µ )( µ − µ ) dx ∧ dx ∧ dx ∧ dx = 0 , d H = x i dx i (6.5) is invariant under the Legendre transformation H ( x k ) → ˜ H ( x k ) = x i x i − H . (6.6) Proof.
In terms of the new variables ˜ H , x i , relation (6.5) becomes d ˜ H = x i dx i (6.7)which proves the invariance since the remaining relation (6.5) is symmetric in the upper andlower indices.It is remarked that the general heavenly equation is also invariant under a “partial” Legen-dre transformation which interchanges any chosen number of corresponding variables carryingupper and lower indices, that is, the summation over i in (6.6) may be restricted to any sub-set of { , . . . , } with ˜ H depending on the associated appropriate variables. Partial Legendretransformations are further discussed in Section 12. For any fixed spectral parameter λ , the general eigenfunction is a function of two functionallyindependent particular eigenfunctions. Hence, instead of demanding that all parameters λ i bedistinct, it is also admissible to choose up to two pairs of coinciding spectral parameters. In thissection, we consider the case of one pair of coinciding parameters, that is, λ = λ without lossof generality. Thus, the task is to find the analogue of the parametrisation (4.7) which resolvesthe vanishing divergence conditions (4.5). To this end, we observe that the defining equations(4.6) for the potential Θ are still meaningful and compatible as long as (1 , = ( i, k ) = (2 , ω ik = ( λ k − λ i )Θ y i y k (7.1)with the coefficients ω and ω = − ω being excluded. The latter are determined by theremaining vanishing divergence conditions which read ∂ y ω = 0 , ∂ y ω = 0 (7.2)8o that the vanishing Pfaffian condition (4.9) becomes f Θ y y + Θ y y Θ y y − Θ y y Θ y y = 0 , (7.3)where f = λ − λ ( λ − λ )( λ − λ ) ω ( y , y ) . (7.4)Hence, on application of an appropriate member of the class of coordinate transformations( y , y ) → f ( y , y ), the Husain-Park equation corresponding to f = 1 is obtained (see [16]and references therein). It is noted that this is consistent with the fact that any function of twoeigenfunctions corresponding to the same spectral parameter constitutes another eigenfunctionso that it is a priori known that the differential equation (7.3) must be invariant under thisclass of coordinate transformations. Furthermore, it should be emphasised that rather thanconsidering the coinciding pair λ = λ and applying the above algorithm, one may apply aconfluence limit of the typeΘ → Θ + g ( y , y ) ǫ , λ = λ + ǫ, ǫ → X ( λ ) and Y ( λ ) are tangent to the coordinate surfaces( y , y ) = const since X ( λ ) y i = 0 , Y ( λ ) y i = 0 , i = 1 , . (7.6)Hence, the commutativity of X ( λ ) and Y ( λ ) guarantees the existence of a natural coordinatesystem ( z , z , y , y ) defined by the additional relations X ( λ ) z = 1 , X ( λ ) z = 0 , Y ( λ ) z = 0 , Y ( λ ) z = 1 (7.7)in terms of which these two vector fields are “straight”, that is, X ( λ ) = ∂ z , Y ( λ ) = ∂ z . (7.8)It is observed that even though, for any fixed spectral parameter, say, λ , the correspondingeigenfunction is an arbitrary function of two functionally independent particular eigenfunctions,the kernel of each of the vector fields X ( λ ) and Y ( λ ) is encoded in three functionally indepen-dent solutions of X ( λ ) z ( X ) = 0 and Y ( λ ) z ( Y ) = 0 respectively. In fact, the relations (7.6) and(7.7) show that, in the above situation, z ( X ) = F ( z , y , y ) and z ( Y ) = G ( z , y , y ). Hence,the introduction of the coordinate system ( z , z , y , y ) constitutes a natural way of gettingaround the fact that it is impossible to have three or four coinciding spectral parameters in theformalism presented in this paper. The general relationship between the latter and the classi-cal theory of “straightening” commuting vector fields using “eigenfunctions” (integrals) [27] iscurrently under investigation. As a first illustration, we now show explicitly how the general heavenly equation (6.1) may bemapped to the Husain-Park equation (7.3). Since it is natural to select the parameters λ = λ = µ , λ = µ , λ = µ , (7.9)we may make the choice φ = H x , φ = x , φ = x , φ = x . (7.10)If we consider the case µ → ∞ and µ = 0 without loss of generality then the Lax pair for thegeneral heavenly equation in the form( µ − µ ) H x x H x x = µ H x x H x x − µ H x x H x x (7.11)9s given by [12] Φ x = 1( λ − µ ) H x x ( λ H x x Φ x − µ H x x Φ x )Φ x = 1( λ − µ ) H x x ( λ H x x Φ x − µ H x x Φ x ) (7.12)so that A = ∂ x − H x x H x x ∂ x , A = ∂ x − H x x H x x ∂ x . (7.13)The latter vector fields are divergence free with respect to f = H x x as pointed out at the endof Section 4. Accordingly, the entries (5.4) of the skew-symmetric matrix A = ˜ A (by virtue of J = H x x ) read A = H x x H x x − H x x H x x H x x , A = H x x − H x x H x x H x x A = H x x − H x x H x x H x x , A = H x x H x x , A = H x x H x x , A = 1 . (7.14)On use of the identity ∂ x = H x x ∂ y , (7.15)we therefore conclude that the connection between the potentials H and Θ encoded in (5.13)may be formulated as Θ y y = − ∂ y H x µ , Θ y y = − ∂ y H x µ (7.16)and similar expressions for the remaining mixed derivatives of Θ except for Θ y y . Hence,integration leads, without loss of generality, to the first-order relationsΘ y = − H x µ , Θ y = − H x µ . (7.17)One may now directly verify that the above pair is compatible modulo the general heavenlyequation (7.11) and Θ indeed satisfies the Husain-Park equation (7.3) with f = ( µ − − µ − ) / The connection between Pleba´nski’s first heavenly equation and the (elliptic) Husain-Park equa-tion has been established explicitly in [16]. Here, we demonstrate how this connection may befound algorithmically using our formalism. It is recalled (see Section 2) that the standard Laxpair for the first Pleba´nski equationΩ x x Ω x x − Ω x x Ω x x = 1 (7.18)reads Φ x = λ (Ω x x Φ x − Ω x x Φ x ) = λA ΦΦ x = λ (Ω x x Φ x − Ω x x Φ x ) = λA Φ . (7.19)For a non-vanishing spectral parameter λ , this is equivalent to the pairΦ x = λ − (Ω x x Φ x − Ω x x Φ x )Φ x = λ − (Ω x x Φ x − Ω x x Φ x ) . (7.20)In the following, the most convenient specialisation of the parameters λ i and associatedeigenfunctions φ i for i = 1 , λ = λ = 0 , φ = x , φ = x (7.21)10ith the remaining eigenfunctions φ and φ corresponding to the parameters λ and λ beingarbitrary. Accordingly, we obtain˜ A = φ x , ˜ A = φ x , ˜ A = φ x , ˜ A = φ x ˜ A = 1 , ˜ A = φ x φ x − φ x φ x . (7.22)Moreover, inversion of the identities ∂ x = φ x ∂ y + φ x ∂ y , ∂ x = φ x ∂ y + φ x ∂ y (7.23)yields ∂ y = φ x ∂ x − φ x ∂ x J , ∂ y = φ x ∂ x − φ x ∂ x J (7.24)with J = φ x φ x − φ x φ x so that the relations (7.22) , , , become˜ A = Jλ ∂ y Ω x , ˜ A = Jλ ∂ y Ω x , ˜ A = Jλ ∂ y Ω x , ˜ A = Jλ ∂ y Ω x (7.25)by virtue of the Lax pair (7.20). Now, since f = 1 so that A lm = ˜ A lm /J , four of the relations(5.13) may be integrated to obtainΘ y = 12 λ λ [Ω x + p ( x , x )] , Θ y = 12 λ λ [Ω x + q ( x , x )] , (7.26)where p ( y , y ) = p ( x , x ) and q ( y , y ) = q ( x , x ) are functions of integration. By construc-tion, the above pair must be compatible modulo the first Pleba´nski equation (7.18) and the Laxpair (7.19) for φ and φ corresponding to the parameters λ and λ . Indeed, cross-differentiationproduces the relation λ λ ( p x − q x ) = λ + λ . (7.27)Hence, without loss of generality, we may choose p = λ + λ λ λ x , q = − λ + λ λ λ x . (7.28)Finally, the remaining compatible relation (5.13) i =3 ,k =4 , namelyΘ y y = 12 J ( λ − λ ) , (7.29)guarantees that Θ is a solution of the Husain-Park equation (7.3) for f = ( λ − λ ) / λ λ .It is evident that the choice λ = − λ is privileged since, in this case, the pair (7.26) simplifiesto Θ y = − λ Ω x , Θ y = − λ Ω x • (7.30)The latter represents the analogue of the relations derived in [16] for the elliptic Husain-Parkequation. Indeed, if one sets aside the “normalisation” (7.29) then Θ defined by the compatiblepair (7.30) constitutes a solution of the Husain-Park equation (7.3) modulo a suitable gaugetransformation of the form Θ → Θ + f ( y , y ). Here, we consider the case of two pairs of coinciding parameters, say, λ = λ and λ = λ .This case can be dealt with in the same manner as before, leading to the parametrisation ω ik = ( λ k − λ i )Θ y i y k , ω = ω ( y , y ) , ω = ω ( y , y ) , (8.1)11herein ( i, k )
6∈ { (1 , , (2 , , (3 , , (4 , } . The Pfaffian condition (4.9) then becomes f f + Θ y y Θ y y − Θ y y Θ y y = 0 (8.2)with f = ω ( y , y ) λ − λ , f = ω ( y , y ) λ − λ , (8.3)which, on application of a suitable coordinate transformation of the form ( y , y ) → f ( y , y )and ( y , y ) → f ( y , y ), constitutes Pleba´nski’s first heavenly equation corresponding to f = f = 1. Once again, it is remarked in passing that a confluence limit of the typeΘ → Θ + g ( y , y ) ǫ + g ( y , y ) ǫ , λ = λ + ǫ, λ = λ + ǫ, ǫ → The classical link [15, 23] between Pleba´nski’s second heavenly equationΛ x x + Λ x x + Λ x x Λ x x − Λ x x = 0 (8.5)and the first heavenly equation may be formulated in terms of the two-formˆΩ = ( dx − Λ x x dx + Λ x x dx ) ∧ ( dx + Λ x x dx − Λ x x dx ) (8.6)which has the properties d ˆΩ = 0 , ˆΩ ∧ ˆΩ = 0 . (8.7)Accordingly, Darboux’s theorem guarantees the existence of functions y and y such thatˆΩ = dy ∧ dy . (8.8)Comparison with (8.6) shows that there exist expansions of the form dx − Λ x x dx + Λ x x dx = u dy + u dy dx + Λ x x dx − Λ x x dx = u dy + u dy (8.9)for some functions u , u , u and u subject to u u − u u = 1 . (8.10)If we now regard y , y and y = x , y = x as independent variables then the expansions (8.9)imply that x y = x y so that there exists a potential Θ defined according to x = Θ y , x = Θ y . (8.11)Hence, (8.9) gives rise to the parametrisation u ik = Θ y i y k (8.12)which, in turn, reveals that the algebraic relation (8.10) encodes Pleba´nski’s first heavenlyequation Θ y y Θ y y − Θ y y Θ y y = 1 . (8.13)12he connection with the present formalism is now uncovered by investigating the nature ofthe coordinates y i . Thus, if we solve (8.9) for dy and dy then we obtain dy = u ( dx − Λ x x dx + Λ x x dx ) − u ( dx + Λ x x dx − Λ x x dx ) dy = u ( dx + Λ x x dx − Λ x x dx ) − u ( dx − Λ x x dx + Λ x x dx ) . (8.14)The latter implies that u = y x , u = y x , u = − y x , u = − y x (8.15)so that (8.14) reduces to y ix = Λ x x y ix − Λ x x y ix ,y ix = Λ x x y ix − Λ x x y ix , i = 1 , . (8.16)Hence, comparison with the Lax pair [23] X ( λ )Φ = 0 , X ( λ ) = ∂ x − Λ x x ∂ x + Λ x x ∂ x − λ∂ x Y ( λ )Φ = 0 , Y ( λ ) = ∂ x − Λ x x ∂ x + Λ x x ∂ x + λ∂ x (8.17)for the second heavenly equation (8.5) shows that the pairs y , y and y , y constitute eigen-functions for λ = 0 and λ → ∞ respectively.It is evident that our formalism is directly applicable even if one or two parameters λ i vanishor, due to the symmetry λ → λ − , one or two parameters tend to infinity. If vanishing andinfinite parameters are simultaneously present then the situation is more subtle but it is easyto verify that the formalism also applies mutatis mutandis in this case. In the current situa-tion, the second-order relations between the potential Θ and the original quantities associatedwith the second heavenly equation are obtained by eliminating u ik between (8.12) and (8.15).Accordingly, the link between the two heavenly equations presented in Pleba´nski’s pioneeringwork [15] may be regarded as a particular application of the scheme presented here. Since the connection between the general heavenly equation and the first Pleba´nski equationconstitutes the basis of Section 11 which, in turn, justifies the reasoning employed in Section 10,we now derive the differential relations between the potentials satisfying those two equations.Thus, in order to map the general heavenly equation (7.11) to the first heavenly equation, it isconvenient to choose the eigenfunctions φ = H x , φ = x , φ = H x , φ = x (8.18)corresponding to the parameters λ = λ = µ , λ = λ = µ . (8.19)Then, evaluation of (5.4) for the vector fields A and A given by (7.13) results in˜ A = (cid:16) H x x − H x x H x x H x x (cid:17)(cid:16) H x x − H x x H x x H x x (cid:17) − (cid:16) H x x − H x x H x x H x x (cid:17)(cid:16) H x x − H x x H x x H x x (cid:17) ˜ A = (cid:16) H x x − H x x H x x H x x (cid:17) , ˜ A = 1˜ A = (cid:16) H x x − H x x H x x H x x (cid:17) . (8.20)13y construction, the mixed derivativesΘ y y = 12 A µ − µ , Θ y y = 12 A µ − µ Θ y y = 12 A µ − µ , Θ y y = 12 A µ − µ , (8.21)where A ik = fJ ˜ A ik , f = H x x , J = H x x H x x − H x x H x x , (8.22)are compatible modulo the general heavenly equation (7.11) and the potential Θ obeys Pleba´nski’sfirst heavenly equation in the formΘ y y Θ y y − Θ y y Θ y y = σ, σ = − µ µ ( µ − µ ) (8.23)as required.As indicated above, the relations (8.21) provide a key link in the remaining discussion. Inorder to motivate the content of Sections 10 and 11, we first review an important known factin the context of the general heavenly equation. Before we do so, we may summarise the keyresults of Sections 4, 7 and 8 in the following table. λ i Canonical form of the self-dual Einstein equations0 General heavenly equation1 Husain-Park equation2 First heavenly equation
The Jones-Tod procedure [28] provides a connection between four-dimensional spacetimes withanti-self-dual Weyl tensor and a conformal Killing vector and three-dimensional Einstein-Weylgeometries. In particular, in [20], it has been shown how the dispersionless Hirota equation(see, e.g., [18]) simultaneously gives rise to Einstein-Weyl geometries and a particular class of(anti-)self-dual Einstein spaces. Here, we discuss this observation in connection with the generalheavenly equation with a view to the generalisation presented in Section 10.The metric of self-dual Einstein spaces governed by the general heavenly equation( µ − µ ) H x x H x x = µ H x x H x x − µ H x x H x x (9.1)with associated Lax representationΦ x = 1( λ − µ ) H x x ( λ H x x Φ x − µ H x x Φ x )Φ x = 1( λ − µ ) H x x ( λ H x x Φ x − µ H x x Φ x ) (9.2)is known to be given by [12] g = q − [ H x x H x x H x x ( dx ) + H x x ( H x x H x x + H x x H x x ) dx dx + · · · ] , (9.3)where q = f and f iklm = H x i x k H x l x m − H x i x l H x k x m (9.4)for distinct indices i, k, l, m . Indeed, one may directly verify that R ik = 0 modulo the generalheavenly equation (9.1). It is noted that there exist only three essentially different quantities14 iklm and their ratios are constant due to the structure of the general heavenly equation. Hence,up to an irrelevant constant factor, the metric (9.3) is completely symmetric in the indices.We now single out a coordinate, say, x and split the metric (9.3) into a “three-dimensional”metric and a “perfect square” according to g = − f h + H x x H x x H x x f ( dx + η ) , (9.5)where h = H x x H x x H x x ( dx ) + α H x x H x x H x x ( dx ) + β H x x H x x H x x ( dx ) + 2 α H x x dx dx + 2 β H x x dx dx − αβ H x x dx dx (9.6)with the constants α = f f , β = f f (9.7)and η = 12 (cid:18) H x x H x x + H x x H x x (cid:19) dx + 12 (cid:18) H x x H x x + H x x H x x (cid:19) dx + 12 (cid:18) H x x H x x + H x x H x x (cid:19) dx . (9.8)Furthermore, we consider the admissible reduction H = f ( x ) ω ( x , x , x ) (9.9)which specialises the general heavenly equation (9.1) to the dispersionless Hirota equation( µ − µ ) ω x x ω x − µ ω x x ω x + µ ω x x ω x = 0 . (9.10)Since, the dependence of the metric (9.3) on x is now merely encoded in the overall factor f ( x ),it is evident that the reduction (9.9) leads to self-dual Einstein spaces admitting a homotheticKilling vector. Moreover, we obtain h ∼ ω x ω x ω x ( dx ) + α ω x ω x ω x ( dx ) + β ω x ω x ω x ( dx ) + 2 αω x dx dx + 2 βω x dx dx − αβω x dx dx , (9.11)up to an irrelevant factor depending on x , which is precisely the metric governing the Einstein-Weyl geometry associated with the dispersionless Hirota equation [20].
10 A dispersionless Hirota system. Self-dual Einsteinspaces not admitting conformal Killing vectors
A non-trivial reduction of the general heavenly equation is obtained by matching its scalingsymmetry H ∂ H with the symmetry Φ ∂ H . The fact that any eigenfunction of the general heavenlyequation constitutes a symmetry of the general heavenly equation was first observed in [19] andextends to its 4+4-dimensional version (TED equation) [17] as further discussed in Section 12.Thus, if we set Φ = H then the Lax pair (9.2) gives rise to the pair of dispersionless Hirotaequations ( λ − µ ) H x H x x − λ H x H x x + µ H x H x x = 0( λ − µ ) H x H x x − λ H x H x x + µ H x H x x = 0 (10.1)which is, by construction, compatible with the general heavenly equation (9.1). Moreover, thepair of dispersionless Hirota equations( µ − µ ) H x H x x + ( λ − µ ) H x H x x − ( λ − µ ) H x H x x = 0 λ ( µ − µ ) H x H x x + µ ( λ − µ ) H x H x x − µ ( λ − µ ) H x H x x = 0 (10.2)15s an algebraic consequence of the dispersionless Hirota equations (10.1) and the general heav-enly equation. In fact, any three of the four dispersionless Hirota equations imply the fourth andthe general heavenly equation. It is noted that the dispersionless Hirota system is completelysymmetric in its indices if the symmetry in the parameters µ = ∞ , µ = 0, µ , µ is restored bymeans of a suitable fractional linear transformation of the parameters, leading to the fully sym-metric form (6.1) of the general heavenly equation. The compatibility of copies of dispersionlessHirota equations was first observed in [18] and is a direct consequence of the multi-dimensionalconsistency of the general heavenly equation [29] or, more generally, the 4+4-dimensional TEDequation [17]. It is also emphasised that the above dispersionless Hirota system is invariantunder H → F ( H ). Even though this invariance may be proven directly, it is a consequence of thefact that any function of an eigenfunction of the general heavenly equation constitutes anothereigenfunction. Remark 10.1.
If we impose the constraint (9.9) and make the choice λ = µ then the disper-sionless Hirota system (10.1), (10.2) reduces to the dispersionless Hirota equation (9.10). Hence,remarkably, the Einstein-Weyl geometry associated with the dispersionless Hirota equation iscaptured as a special case by the class of self-dual Einstein spaces governed by the eigenfunc-tion symmetry reduction leading to the dispersionless Hirota system. In fact, the dispersionlessHirota system specialises to the dispersionless Hirota equation if any of the four constraints H = f ( x i ) ω ( x k , x l , x m ) , λ = µ i , (10.3)where the indices i, k, l, m are distinct, is imposed. Accordingly, the Einstein-Weyl geometrydiscussed in Section 9 is encoded in four different ways in the self-dual Einstein spaces examinedin this section so that, in this sense, the algebraic multi-dimensional consistency of the disper-sionless Hirota equation has its geometric counterpart in the “multi-dimensional consistency”of its associated Einstein-Weyl geometry. The exact nature of this geometric property is thesubject of ongoing research. H → F ( H ) In the preceding, it has been demonstrated that the four-dimensional general heavenly equationadmits a decomposition into four compatible three-dimensional dispersionless Hirota equations(of which only three are independent). It is therefore natural to inquire as to whether the solu-tions of the general heavenly equation obtained in this manner are genuinely four-dimensionalin the sense that the corresponding self-dual Einstein spaces do not admit conformal Killingsymmetries. Furthermore, it is desirable to show that the symmetry H → F ( H ) really actsnon-trivially on the metric. At first glance, this appears to be likely due to the appearance ofthe function F in the transformed metric. We begin by addressing the latter problem.We first observe that H = z + f ( z ) , z = X i =1 x i , z = X i =1 α i x i (10.4)constitutes a trivial solution of the general heavenly equation (9.1) and, in order for it to satisfythe pair (10.1) of dispersionless Hirota equations, the constants α i and µ k must be related by µ k = λ α ( α − α k ) α ( α − α k ) , k = 3 , . (10.5)However, this solution does not correspond to a viable metric since f = 0. This may berectified by boosting the solution using the symmetry H → F ( H ) to obtain H = F ( z + f ( z )) . (10.6)Indeed, the condition for non-vanishing f is given by ( α − α )( α − α ) F ′′ ( u ) f ′′ ( v ) = 0.For example, the solution H = e z cosh z , (10.7)16orresponding to F ( u ) = e u , f ( v ) = ln cosh v , falls into this category, provided that α = α and α = α being excluded due to (10.5). This solution generates a flat metric if, for instance, α = − α = 1. More generally, for this particular choice of the function f and the constants α i but arbitrary function F so that H = F ( z + ln cosh z ), the components of the Riemann tensorwhich are not identically zero turn out to be proportional to R iklm ∼ α α (cid:16) [ F ′ ( u )] F ′′ ( u ) F ′′′′ ( u ) − F ′ ( u )] [ F ′′′ ( u )] + 2[ F ′′ ( u )] (cid:17) , (10.8)where u = z + ln cosh z . Thus, for a four-parameter family of functions F , which includes F ( u ) = e u , the metric is flat but, generically, the invariance H → F ( H ) maps the flat metricassociated with the solution (10.7) to a non-flat metric. Here, we exclude the case α α = 0.This proves that this invariance of the dispersionless Hirota system is non-trivial at the level ofthe metric. In this section, we prove the following theorem.
Theorem 10.1.
The generic metric of self-dual Einstein spaces governed by the dispersionlessHirota system (10.1) , (10.2) does not admit conformal Killing vectors (including homotheticKilling vectors and Killing vectors). It turns out convenient to examine the above problem in the setting of Pleba´nski’s firstheavenly equation Θ y y Θ y y − Θ y y Θ y y = 1 (10.9)with associated Lax pair Φ y = ¯ λ (Θ y y Φ y − Θ y y Φ y )Φ y = ¯ λ (Θ y y Φ y − Θ y y Φ y ) . (10.10)In Section 8, it has been demonstrated how the current formalism may be used to establishthe link between the general and first heavenly equations. Extension of this link to the Laxpairs is readily shown to lead to the above standard Lax pair with ¯ λ being related to λ by afractional linear transformation (cf. (11.9)). In Section 11, this link is exploited to reveal howthe decomposition of the general heavenly equation into the dispersionless Hirota system actson the first heavenly equation. Remarkably, this decomposition corresponds to the assumptionthat Φ = Θ − y Θ y − y Θ y (10.11)constitutes an eigenfunction of the first heavenly equation. Insertion of Φ as given by (10.11)into the Lax pair (10.10) leads to two constraints on Θ which are compatible with the Pleba´nskiequation (10.9). This is discussed in Section 11 in more detail. In order to understand the natureof these constraints, we first briefly examine the symmetries of the first heavenly equation. Here, we consider symmetries of the first Pleba´nski equation, that is, flowsΘ s = ∆ (10.12)which leave invariant the first heavenly equation (10.9). Thus, differentiation of the latter withrespect to the symmetry parameter s produces the linear partial differential equationΘ y y ∆ y y + Θ y y ∆ y y − Θ y y ∆ y y − Θ y y ∆ y y = 0 (10.13)for the function ∆. Any solution of this differential equation gives rise to a symmetry of thefirst heavenly equation. For instance, it is evident that the latter is invariant under the scalingΘ → ǫ Θ, y i → ǫ y i for any fixed i . This corresponds to∆ = Θ − y i Θ y i . (10.14)17ne may directly verify that this is indeed a solution of the differential equation (10.13). Anothersymmetry is given by ∆ = Φ (10.15)which may be seen immediately by cross-differentiating the Lax pair (10.10), leading toΘ y y Φ y y + Θ y y Φ y y − Θ y y Φ y y − Θ y y Φ y y = 0 , (10.16)provided that ¯ λ = 0. This symmetry has been recorded in [30] in connection with the notion ofpartner symmetries and is also a direct consequence of the eigenfunction symmetry of the 4+4-dimensional TED equation. We conclude that the admissible constraint (10.11) correspondsto a symmetry reduction of the first heavenly equation associated with an appropriate linearcombination of the symmetries (10.14) and (10.15).Less obvious symmetries are provided by conservations laws associated with the first heavenlyequation. Thus, the compatibility condition associated with the pair φ ay = Θ y Θ y y − Θ y Θ y y + 2 y φ ay = Θ y Θ y y − Θ y Θ y y (10.17)is precisely the first heavenly equation (10.9) so that the existence of the potential φ a is guar-anteed. For reasons of symmetry, the existence of a potential φ b defined according to φ by = Θ y Θ y y − Θ y Θ y y − y φ by = Θ y Θ y y − Θ y Θ y y (10.18)is likewise guaranteed. Differentiation of (10.17) and (10.18) with respect to y , y and y , y respectively then shows that the quantities∆ a = φ a , ∆ b = φ b (10.19)satisfy the differential equation (10.13) and therefore constitute symmetries of the first heavenlyequation. We will now investigate the existence of homothetic Killing vectors of the class of metrics asso-ciated with solutions of the dispersionless Hirota system and then address the (non-)admittanceof more general proper conformal Killing vectors. It is recalled (Section 2) that, in terms ofsolutions of the first heavenly equation (10.9), the metric of self-dual Einstein spaces adopts theform g = 2Θ y y dy dy + 2Θ y y dy dy + 2Θ y y dy dy + 2Θ y y dy dy . (10.20)A vector field V k constitutes a homothetic Killing vector field if its covariant representation V k = g kl V l satisfies Killing’s equations [1] given by ∇ ( i V k ) = χg ik , (10.21)wherein ∇ i denotes the covariant derivative, the round brackets indicate the standard sym-metrised sum and χ is a constant. If the latter vanishes then V k is a Killing vector giving riseto an isometry, while χ = 0 corresponds to a proper homothetic Killing vector. Based on aspinor formulation, it has been demonstrated in [31] that the above Killing equations may becompletely resolved. In fact, in the present formulation, one may directly integrate this set oflinear partial differential equations to obtain V = c a (Θ y Θ y y − Θ y Θ y y ) − a ( y , y )Θ y y − b ( y , y )Θ y y V = c a (Θ y Θ y y − Θ y Θ y y ) − a ( y , y )Θ y y − b ( y , y )Θ y y V = c b (Θ y Θ y y − Θ y Θ y y ) + c ( y , y )Θ y y + d ( y , y )Θ y y V = c b (Θ y Θ y y − Θ y Θ y y ) + c ( y , y )Θ y y + d ( y , y )Θ y y , (10.22)18here c a , c b are constants, so that the components of the homothetic Killing vector are given by V = c b Θ y + c ( y , y ) , V = − c b Θ y + d ( y , y ) V = c a Θ y − a ( y , y ) , V = − c a Θ y − b ( y , y ) . (10.23)The Killing equations reduce to the “master equation” (in the terminology of [31])∆ h = 0 , (10.24)where∆ h = c a φ a + a ( y , y )Θ y + b ( y , y )Θ y + 2 χ Θ − c ( y , y )Θ y − d ( y , y )Θ y − c b φ b (10.25)and φ a and φ b are the potentials defined by (10.17) and (10.18) respectively. Moreover, thefunctions a, b, c, d are constrained by the linear equation a y + b y + 4 χ − c y − d y = 0 . (10.26)In fact, the latter constraint constitutes the trace ∇ i V i = 4 χ of the Killing equations (10.21).It may be solved explicitly since the dependence of the pairs a, b and c, d on different variablesimplies the separation a y + b y + 2 χ − µ = 0 , c y + d y − χ − µ = 0 , (10.27)where µ is an arbitrary constant. In summary, the first heavenly equation (10.9) admits ahomothetic Killing vector if and only if it is constrained by the (non-local) condition (10.24).It turns out that the constraint (10.24) is compatible with the first heavenly equation (10.9).For instance, if c a = c b = 0 then ∆ h = 0 constitutes a first-order constraint which may be shownto lead to a three-dimensional reduction of the first heavenly equation. If c a c b = 0 then thenecessary conditions ∆ hy y = ∆ hy y = ∆ hy y = ∆ hy y = 0 lead to four third-order differentialconstraints which are compatible with the first heavenly equation. Conversely, if those fourconstraints are satisfied then the functions of integration in the definitions (10.17) and (10.18)of the potentials φ a and φ b respectively may be chosen in such a manner that the non-localcondition ∆ h = 0 is satisfied. The reason for the compatibility of the master equation ∆ h = 0 isreadily revealed by examining the structure of ∆ h as given by (10.25). Indeed, the discussion inthe previous subsection has revealed that φ a and φ b are symmetries of the first heavenly equationand if the functions a, b, c, d are suitable multiples of y , y , y , y respectively then ∆ h | c a = c b =0 also constitutes a symmetry. In fact, in general, the condition (10.26) is exactly the conditionwhich guarantees that ∆ h represents a symmetry of the first heavenly equation, that is, ∆ h satisfies the symmetry condition (10.13), thereby justifying the notation ∆ h . Accordingly, themaster equation ∆ h = 0 is nothing but a symmetry reduction of the first heavenly equation.In order to address the question as to whether the decomposition of the general heavenlyequation into the dispersionless Hirota system corresponds to the assumption of a homotheticKilling vector, it is now required to determine whether the symmetry constraint (10.11) on thefirst heavenly equation constitutes a special case of the symmetry constraint ∆ h = 0. To thisend, we first observe (as mentioned earlier, cf. Section 11.2) that insertion of Φ as given by(10.11) into the Lax pair (10.10) leads to two second-order constraints on the first heavenlyequation. These constraints together with the first heavenly equation may then be formulatedas a system of the typeΘ y y = F, Θ y y = G, Θ y y = H, F, G, H ∈ S . (10.28)Here, the exact form of the functions F, G and H is not important. The key property of the abovesystem is that these functions are contained in S which denotes the set of functions dependingon Θ , Θ y , Θ y and their derivatives of any order with respect to y and y . The functions of thisset may also depend explicitly on the independent variables y i . By construction, the associated19ompatibility conditions are satisfied which, in turn, implies that, generically, the solution ofthe triple (10.28) is determined by the Cauchy dataΘ = f ( y , y ) , Θ y = f ( y , y ) , Θ y = f ( y , y ) at ( y , y ) = ( y , y ) . (10.29)On the other hand, differentiation of ∆ h and use of (10.17) and (10.18) evidently yields ∆ hy y ∈ S .Hence, ∆ hy y = 0 evaluated at ( y , y ) = ( y , y ) constitutes a differential constraint on theCauchy data f k ( y , y ). Accordingly, generically, the constrained Pleba´nski system (10.28),which, as stated earlier, is equivalent to the dispersionless Hirota system, does not give rise to ahomothetic Killing vector. Finally, it is well known [32] that any metric satisfying Einstein’s vac-uum equations R ik = 0 which admits a proper conformal Killing vector also possesses a Killingvector. It is recalled that V k constitutes a proper conformal Killing vector if it satisfies Killing’sequations (10.21) for a non-constant function χ . This completes the proof of Theorem 10.1.
11 Decomposition of the first heavenly equation
In the previous section, it has been demonstrated that the dispersionless Hirota system (10.1),(10.2) gives rise to self-dual Einstein spaces which, generically, do not admit any conformalKilling vectors. Since the dispersionless Hirota system has been obtained as a symmetry re-duction of the general heavenly equation, it should be investigated whether the correspondingsymmetry constraint may be formulated in an invariant manner so that it may be applied to anyof the known heavenly equations. The results of this investigation will be presented elsewhere.Here, we briefly present the action of this symmetry constraint on the first heavenly equationsince it has been exploited in the previous section.
In Section 8.2, it has been shown that the potentials H and Θ obeying the general heavenlyequation (7.11) and the first heavenly equation (8.23) respectively are linked by the second-orderrelations (8.20)-(8.22). It turns out that an appropriate extension of these relations admits a firstintegral if the solutions of the general heavenly equation are restricted to the class of solutionssatisfying the dispersionless Hirota system( λ − µ ) H x H x x − λ H x H x x + µ H x H x x = 0( λ − µ ) H x H x x − λ H x H x x + µ H x H x x = 0( µ − µ ) H x H x x + ( λ − µ ) H x H x x − ( λ − µ ) H x H x x = 0 λ ( µ − µ ) H x H x x + µ ( λ − µ ) H x H x x − µ ( λ − µ ) H x H x x = 0 . (11.1)Specifically, if we differentiate the ansatz κ H = Θ − y Θ y − y Θ y (11.2)with respect to y i , where κ is a constant, then we obtain the additional second derivativesΘ y y = − κ H y + y Θ y y y , Θ y y = Θ y − κ H y − y Θ y y y Θ y y = − κ H y + y Θ y y y , Θ y y = Θ y − κ H y − y Θ y y y . (11.3)It is evident that all terms of the right-hand sides of these relations except for Θ y and Θ y maybe expressed entirely in terms of the potential H and the associated independent variables x i .One may now directly verify (using computer algebra) that these relations are compatible with(8.21) modulo the dispersionless Hirota system (11.1) provided that κ = − λ ( λ − µ )( λ − µ ) . (11.4)20ence, taking into account the invariance H → H + const of the general heavenly equation, theansatz (11.2) is equivalent to the system (11.3) and may be interpreted as a first integral of theextended system (8.21), (11.3). In order to motivate the first-order link (11.2) between the potentials H and Θ in the case ofthe restricted class of solutions of the general heavenly equation governed by the dispersionlessHirota system, we note that since the latter is a result of the imposition of a symmetry constraintinvolving the eigenfunction and a scaling symmetry of the general heavenly equation, one expectsto find a similar result if one formulates this symmetry constraint in terms of Pleba´nski’s firstheavenly equation. In the previous section, we have demonstrated that the symmetry constraintΦ = Θ − y Θ y − y Θ y (11.5)on the first heavenly equation is admissible, where Φ obeys the Lax pairΦ y = ¯ λ (Θ y y Φ y − Θ y y Φ y )Φ y = ¯ λ (Θ y y Φ y − Θ y y Φ y ) (11.6)for the first heavenly equation which, in the current context, is of the form (8.23). Now, takinginto account that H constitutes an eigenfunction if the general heavenly equation is specialisedto the dispersionless Hirota system, the identification of the two eigenfunctions Φ and κ H leadsto the ansatz (11.2).Insertion of Φ as given by (11.5) into the Lax pair (11.6) leads toΘ y y = ( y Θ y y − Θ y )Θ y y − ( y Θ y y + y Θ y y − Θ y )¯ λ − − σy y Θ y y Θ y y = ( y Θ y y − Θ y )Θ y y + ( y Θ y y + y Θ y y − Θ y ) σ ¯ λ − σy y Θ y y , (11.7)where we have exploited the Pleba´nski equation (8.23) formulated asΘ y y = Θ y y Θ y y + σ Θ y y . (11.8)By construction, the system (11.7), (11.8) is compatible. Finally, it may be verified that thissystem is satisfied if Θ is related to the potential H via the system (8.21), (11.3) provided that¯ λ = 2 ( µ − µ ) ( λ − µ ) µ ( λ − µ ) . (11.9)Hence, the decomposition (11.7), (11.8) of Pleba´nski’s first heavenly equation (8.23) constitutesan incarnation of the decomposition of the general heavenly equation into the dispersionlessHirota system (11.1). It is noted that (11.9) may also be formulated as − ( σ ¯ λ ) − = 2 ( µ − µ ) ( λ − µ ) µ ( λ − µ ) (11.10)which highlights the symmetry of the pair (11.7).
12 Partial Legendre transformations. The TED equation
In Section 6, it has been demonstrated that the general heavenly equation is invariant under aLegendre transformation. The analysis in the previous section naturally leads to the considera-tion of partial Legendre transformations applied to Pleba´nski’s first heavenly equation and, byextension, to the Husain-Park equation. 21
The constraint (11.5) suggests considering the partial Legendre transformation(Θ; y , y , y , y ) → ( ¯Θ; y , y , y , y ) , (12.1)where ¯Θ = Θ − y Θ y − y Θ y , y i = Θ y i , (12.2)so that it is natural to examine the first heavenly equation (8.23) in the form dy ∧ dy ∧ dy ∧ dy = σdy ∧ dy ∧ dy ∧ dy . (12.3)In terms of the new potential ¯Θ, the definition of the variables y i formulated as d Θ = y i dy i (12.4)implies that y = − ¯Θ y , y = ¯Θ y , y = − ¯Θ y , y = ¯Θ y , (12.5)leading to the partial Legendre transform¯Θ y y ¯Θ y y − ¯Θ y y ¯Θ y y = σ (cid:0) ¯Θ y y ¯Θ y y − ¯Θ y y (cid:1) (12.6)of the first heavenly equation. This form of the self-dual Einstein equations together with itsLegendre-type connection with Pleba´nski’s first heavenly equation has been recorded in [30]. Inthe current context, the constraint (11.5) shows that the counterpart of the decomposition ofthe general heavenly equation into the dispersionless Hirota system is the decomposition of theheavenly equation (12.6) into a system analogous to the system (11.7), (11.8) which is generatedby matching the eigenfunction and the potential ¯Θ, that is,Φ = ¯Θ (12.7)as in the case of the general heavenly equation. It turns out that the above observation is not a coincidence. As indicated in the preceding, theTED equation constitutes a 4+4-dimensional integrable generalisation of the general heavenlyequation and, in fact, exists in 2 n + 2 n dimensions [17]. The TED equation(Θ y z − Θ y z )(Θ y z − Θ y z )+ (Θ y z − Θ y z )(Θ y z − Θ y z )+ (Θ y z − Θ y z )(Θ y z − Θ y z ) = 0 (12.8)is multi-dimensionally consistent [17] and encodes many (if not all) known heavenly equationsand also, for instance, the six-dimensional second heavenly equation (see, e.g., [23] and referencestherein). In particular, the travelling wave reductionΘ z i = λ i Θ y i (12.9)leads to the general heavenly equation( λ − λ )( λ − λ )Θ y y Θ y y + ( λ − λ )( λ − λ )Θ y y Θ y y + ( λ − λ )( λ − λ )Θ y y Θ y y = 0 . (12.10)Moreover, it has been shown that any eigenfunction Φ obeying the associated Lax pair con-stitutes a symmetry of the TED equation so that matching this eigenfunction symmetry withthe scaling symmetry of the TED equation encapsulated in the constraint Φ = Θ leads to a22igher-dimensional integrable extension of the dispersionless Hirota system consisting of fourcompatible 3+3-dimensional generalised dispersionless Hirota equations, namely(Θ y i z k − Θ y k z i )(Θ z l − λ Θ y l )+ (Θ y k z l − Θ y l z k )(Θ z i − λ Θ y i )+ (Θ y l z i − Θ y i z l )(Θ z k − λ Θ y k ) = 0 . (12.11)Here, the indices i, k, l ∈ { , , , } are distinct. Indeed, in the travelling wave reduction (12.9),the fully symmetric avatar of the dispersionless Hirota system is obtained. One may also directlyverify that the TED equation (12.8) is an algebraic consequence of the generalised dispersionlessHirota system (12.11).Another travelling wave reduction of the TED equation generated byΘ z = λ Θ y , Θ z = λ Θ y + ν Θ y Θ z = λ Θ y , Θ z = λ Θ y + ν Θ y (12.12)reads ( λ − λ ) (Θ y y Θ y y − Θ y y Θ y y ) = ν ν (Θ y y Θ y y − Θ y y ) (12.13)which is exactly of the form (12.6) (with a slightly different labelling of the dependent andindependent variables). Its decomposition into a system of compatible equations via the sym-metry constraint Φ = Θ is obtained by imposing the travelling wave constraints (12.12) onthe generalised dispersionless Hirota system (12.11). This explains why the heavenly equation(12.6) admits the (symmetry) constraint (12.7). It is important to note that the travelling waveconstraints (12.12) constitute an extension of the travelling wave constraints (12.9) subject tothe choice λ = λ and λ = λ . The latter is precisely the specialisation employed in Section 8which has led to Pleba´nski’s first heavenly equation. In fact, this suggests that one should alsoconsider the “intermediate” case given byΘ z = λ Θ y , Θ z = λ Θ y + ν Θ y Θ z = λ Θ y , Θ z = λ Θ y (12.14)and corresponding to the choice λ = λ in the above-mentioned sense, leading to the reductionΘ y y Θ y y − Θ y y Θ y y = ˜ σ (Θ y y Θ y y − Θ y y Θ y y )˜ σ = ν ( λ − λ )( λ − λ )( λ − λ ) (12.15)of the TED equation. Indeed, application of the partial Legendre transformation(Θ; y , y , y , y ) → ( ˜Θ; y , y , y , y ) , (12.16)with ˜Θ = Θ − y Θ y , d Θ = y i dy i (12.17)is readily shown to produce the Husain-Park equation˜ σ ˜Θ y y = ˜Θ y y ˜Θ y y − ˜Θ y y ˜Θ y y . (12.18)Hence, we conclude that the three sets of travelling wave constraints (12.9), (12.12) and (12.14)of the TED equation correspond to the three canonical choices of the parameters λ i in the currentformalism with the associated reductions (12.10), (12.13) and (12.15) being linked to the generalheavenly equation, Husain-Park equation and first heavenly equation by the respective (partial)Legendre transformation.In summary, a link between travelling wave reductions of the TED equation and the classi-fication within the present formalism with respect to the number of pairs of coinciding spectralparameters has been established. This highlights, once again, the significance of the TED equa-tion. In this connection, it is interesting to recall [17] that, as mentioned at the beginning ofthis section, the Husain-Park and first heavenly equations may also be obtained directly fromthe TED equation by imposing appropriate constraints.23 cknowledgements W.K.S. wishes to express his gratitude to his colleague John Steele for sharing his expertise inthe area of conformal Killing symmetries.
References [1] H. Stephani, D. Kramer, M.A.H. MacCallum, C.A. Hoenselaers and E. Herlt,
Exact Solu-tions of Einstein’s Field Equations , 2nd edition, Cambridge University Press (2003).[2] C. Rogers and W.K. Schief,
B¨acklund and Darboux Transformations. Geometry and Mod-ern Applications in Soliton Theory , Cambridge Texts in Applied Mathematics, CambridgeUniversity Press (2002).[3] M.J. Ablowitz and H. Segur,
Solitons and the Inverse Scattering Transform , SIAM, Philadel-phia (1981).[4] M.J. Ablowitz and P. Clarkson,
Solitons, Nonlinear Evolution Equations and Inverse Scat-tering , Cambridge University Press (1991).[5] A.P. Fordy, ed,
Soliton Theory: A Survey of Results , Manchester University Press, Manch-ester and New York (1990).[6] B.G. Konopelchenko, Soliton eigenfunction equations: the IST integrability and some prop-erties,
Rev. Math. Phys. (1990) 399–440.[7] W. Oevel and C. Rogers, Gauge transformations and reciprocal links in 2+1 dimensions, Rev. Math. Phys. (1993) 299–330.[8] L.V. Bogdanov and B.G. Konopelchenko, Grassmannians Gr( N -1, N +1), closed differential N -1-forms and N -dimensional integrable systems, J. Phys. A: Math. Theor. (2013)085201 (17pp).[9] S.V. Manakov and P.M. Santini, Inverse scattering problem for vector fields and the Cauchyproblem for the heavenly equation, Phys. Lett. A (2006) 613–619.[10] V.E. Zakharov and A.B. Shabat, Integration of nonlinear equations of mathematical physicsby the method of inverse scattering, II.
Func. Anal. Appl. (1979) 166–174.[11] C.P. Boyer, The geometry of complex self-dual Einstein spaces, in K.B. Wolf, ed, NonlinearPhenomena , Lecture Notes in Physics (1983) 25–546.[12] W.K. Schief, Self-dual Einstein spaces via a permutability theorem for the Tzitzeica equa-tion,
Phys. Lett. A (1996) 55–62.[13] W.K. Schief , Self-dual Einstein spaces and a discrete Tzitzeica equation. A permutabilitytheorem link, in P. Clarkson and F. Nijhoff, eds,
Symmetries and Integrability of DifferenceEquations , London Mathematical Society, Lecture Note Series , Cambridge UniversityPress (1999) 137–148.[14] K. Takasaki, Aspects of integrability in self-dual Einstein metrics and related equations,
Publ. RIMS, Kyoto Univ. (1986) 949–990.[15] J.F. Pleba´nski, Some solutions of complex Einstein equations, J. Math. Phys. (1975)2395–2402.[16] M. Jakimowicz and J. Tafel, Self-dual metrics in Husains approach, Class. Quantum Grav. (2006) 4907–4914. 2417] B.G. Konopelchenko and W.K. Schief, On an integrable multi-dimensionally consistent2 n + 2 n -dimensional heavenly-type equation, Proc. R. Soc. London A (2019) 20190091(21pp).[18] W. Kry´nski, On deformations of the dispersionless Hirota equation,
J. Geom. Phys. (2018) 46–54.[19] A. Sergyeyev, A simple construction of recursion operators for multidimensional dispersion-less integrable systems,
J. Math. Anal. Appl. (2017) 468–480.[20] M. Dunajski and W. Kry´nski, EinsteinWeyl geometry, dispersionless Hirota equation andVeronese webs,
Math. Proc. Camb. Phil. Soc. (2014) 139–150.[21] L.J. Mason and E.T. Newman, A connection between the Einstein and Yang-Mills equa-tions,
Commun. Math. Phys. (1989) 659–668.[22] T. Sibata and K. Morinaga, Complete and simpler treatment of wave geometry,
J. Sci.Hiroshima Univ. Ser. A (1936) 173–189.[23] B. Doubrov and E.V. Ferapontov, On the integrability of symplectic Monge-Amp`ere equa-tions. J. Geom. Phys. (2010) 1604–1616.[24] W.V.D. Hodge and D. Pedoe, Methods of Algebraic Geometry. Vol. 1 , Cambridge Mathe-matical Library, Cambridge University Press (1994).[25] A.I. Bobenko and W.K. Schief, Discrete indefinite affine spheres, in A. Bobenko and R.Seiler, eds,
Discrete Integrable Geometry and Physics , Oxford University Press (1999) 113–138.[26] J.F. Pleba´nski and M. Przanowski, The Lagrangian of a self-dual gravitational field as alimit of the SDYM Lagrangian,
Phys. Lett. A (1996) 22-28.[27] S. Sternberg,
Lectures on Differential Geometry , AMS Chelsea Publishing, Providence,Rhode Island (1999).[28] P.E. Jones and K.P. Tod, Minitwistor spaces and Einstein-Weyl spaces,