d. It is, and the

Physics Letters A 320 (2004) 383–388

www.elsevier.com/locate/pla

A hodograph transformation which appliesto the Boyer–Finley equation✩

Manuel Mañas, Luis Martínez Alonso

Departamento de Física Teórica II, Universidad Complutense, E28040 Madrid, Spain

Received 27 January 2003; accepted 4 July 2003

Communicated by A.P. Fordy

Abstract

A hodograph transformation for a wide family of multidimensional nonlinear partial differential equations is presenteused to derive solutions of the Boyer–Finley equation (dispersionless Toda equation), which are not group invariantcorresponding family of explicit ultra-hyperbolic selfdual vacuum spaces. 2003 Elsevier B.V. All rights reserved.

MSC: 58B20

Keywords: Boyer–Finley equation; Hodograph transformations; Einstein–Weyl spaces

on-ialtheua-heac-ma-

byro-tion

xting

tolyleytheolicto

tion

it

1. Introduction

This work introduces a hodograph method to cstruct solutions of a ample family of nonlinear partdifferential equations (PDE) among which we havedispersionless Kadomtsev–Petviashvili (dKP) eqtion and the Boyer–Finley equation, relevant in tfinding of Einstein–Weyl 3D spaces and selfdual vuum Einstein spaces [1,4,16]. Hodograph transfortions goes back the XIX century and as was shownRiemann they are relevant in the discussion of hyddynamic type systems, this hodograph transformawas generalized recently by Tsarev [15].

The layout of this Letter is as follows. The nesection is devoted to describe our scheme. Us

✩ Partially supported by CICYT proyecto PB98-0821.E-mail addresses: manuel@darboux.fis.ucm.es (M. Mañas),

luism@fis.ucm.es (L.M. Alonso).

0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2003.10.023

solutions to a implicit relation we find solutionsnonlinear PDEs. Finally, in the Section 3 we appthese results to find new solutions of the Boyer–Finequation—which are non-group invariant—andassociated Einstein–Weyl spaces and ultra-hyperbself-dual vacuum Einstein spaces. We would likestress that the solutions of the Boyer–Finley equagiven in [16] are group invariant.

2. The hodograph transformation

Our method begins with the following implicequation for determining a scalar functionu = u(x)depending onn variablesx = (x1, . . . , xn),

(1)X0(u)+n∑

i=1

xi Xi(u)= 0,

.

384 M. Mañas, L.M. Alonso / Physics Letters A 320 (2004) 383–388

hes

ns

-

ehe

e

phe

weg

m-

he

as

whereXi , i = 0,1, . . . , n, are given functions ofu.By denotingx = x1, ti = xi+1 (i = 1, . . . , n − 1), itfollows that (1) is a hodograph transformation for tfamily of one-dimensional hydrodynamical system

(2)uti = Ci(u)ux, i = 1, . . . , n− 1,

where

(3)Ci(u) := Xi+1(u)

X1(u).

Our main observation is that (1) provides solutiofor the family of nonlinear PDEs:

(4)∑

|α|=m

cαDαφ =DβF(φ), |β| =m,

whereDα andDβ denote partial differentiation operations of a given orderm corresponding ton-component multi-indicesα,β ∈ N

n, F = F(φ) is anarbitrary function andcα are arbitrary constants. Ware going to prove that a solution of (4) is given by tfunction

φ(x) :=G(Q(u)),

(5)Q(u) :=∑

|α|=m cαXα(u)

Xβ(u), Xσ :=Xσ1 · · ·Xσn,

where G := (Fu)−1 is the inverse function of th

derivativeFu of F with respect tou.From (5) we deduce that

φxi =G′(Q(u))Q′(u)uxi =G′(Q(u))Q′(u)Xi(u)

Xj (u)uxj

= ∂

∂xj

u∫G′(Q(u))Q′(u) Xi(u)

Xj (u)du

and therefore

Dαφ =Dβ

u∫G′(Q(u))Q′(u)X

α(u)

Xβ(u)du.

From this relation we conclude∑|α|=m

cαDαφ

=Dβ

u∫G′(Q(u))Q′(u)

∑|α|=m cαX

α(u)

Xβ(u)du

=Dβ

u∫G′(Q(u))Q′(u)Q(u)du.

Now, if H := F ◦G then

(H)′(Q) = (F ′ ◦G)(Q)G′(Q) =QG′(Q)

and hence

∑|α|=m

cαDαφ =Dβ

u∫H ′(Q(u))Q′(u)du

=DβH(Q)=DβF(φ).

2.1. Observations

1. In spite of the implicit nature of the hodograrelation (1) we can easily find explicit examples. Wshall point out two of such cases:

• Assuming that

Xi(u)=N∑j=0

aikuk, i = 0,1, . . . , n,

(1) reads as

N∑k=0

Akuk = 0, Ak = a0k +

n∑i=1

aikxi,

and we will haveN complex roots

ul = ul(A0, . . . ,AN), l = 1, . . . ,N.

For each of these roots we can evaluateXi(ul) andget a family of solutions. As we know ifN � 4the roots can be gotten explicitly and thereforewill have an explicit algebraic function dependinon the parameters{aik}k=0,...,4,i=1,...,n.

• Another example appears by considering the Labert functionW(z) which solves

W exp(W) = z

and has been studied with certain detail [5]. Tpoint here is that the implicitfundamental relation

a + bu+ cexp(u)= 0

is solved in terms of the Lambert functionfollows:

u= −W

(c

bexp

(−a

b

))− a

b.

M. Mañas, L.M. Alonso / Physics Letters A 320 (2004) 383–388 385

o-

DE

foreence

han

ourKo-ruc-[4].f

eyoda

vac-

],to

Thus, taking

Xi(u)= ai + biu+ ci exp(u),

with ai, bi and ci arbitrary constants, the hodgraph relation is

A+Bu+C exp(u)= 0,

with

A := a0 +n∑

i=1

aixi, B := b0 +n∑

i=1

bixi,

C := c0 +n∑

i=1

cixi,

and the solution is

u= −W

(C

Bexp

(−A

B

))− A

B.

Hence, introducing the rational functions

r(x) := a0 + ∑ni=1 aixi

b0 + ∑ni=1 bixi

,

s(x) := b0 + ∑ni=1bixi

c0 + ∑ni=1 cixi

we can evaluate

Xi(x)= ai − bir(x)

− (biC − cis(x)

)W

(1

s(x)expr(x)

)and using (5) get a solution to the nonlinear P(4) in terms of the Lambert function.

2. We can employ the freedom in the choicethe functions{Xi(u)}ni=1 to generate solutions of morgeneral equations. Suppose a functional dependof the form

φ = (F ′α

)−1(Qα)= (

F ′β

)−1(Qβ),

for all α,β ∈ I, beingI a set ofr = cardI multi-indices of orderm, and

QγXγ =

∑|δ|=m

aδXδ.

Then,φ satisfies∑|δ|=m

aδDδφ = 1

r

∑γ∈I

DγFγ (φ).

For example, the hodograph relation

tT (u)+ xX(u)+ yY (u)=H(u)

provides solutions to

1

2(φxx + φyy) = (

exp(φ))t t, φ = log

X2 + Y 2

2T 2

as well to

1

2(φxx + φyy) = (

exp(2φ))xt, φ = log

√X2 + Y 2

4XT.

Thus, we need to fulfill

X3 + Y 2X = T 3.

So that, the solutions of

t3√X(u)+ Y 2(u)/X(u)+ xX(u)+ yY (u)=H(u)

gives

φxx + φyy = (exp(φ)

)t t

+ (exp(2φ)

)xt,

φ = 1

3log

(1+ (Y/X)2

) − log(2).

3. Applications in general relativity

Among the nonlinear PDEs of the form for whicour hodograph technique is applicable one findsintegrable equation: the dKP equation

φtx + φyy = (φ2)

xx.

This equation is relevant in hydrodynamics andhodograph solutions were already discussed bydama in [7], the dKP equation appears in the consttion of three-dimensional Einstein–Weyl spacesAnother integrable equation within our family oPDEs is known with different names: Boyer–Finlequation, Boyer–Finley equation, dispersionless Tand SU(∞)-Toda equation:

(6)φzz̄ + κ(eφ

)t t

= 0, κ = ±1,

wherez = x + iy andz̄ = x − iy, x, y, t, φ ∈ R. Thisequation has been found to characterize self-dualuum Einstein spaces—of signature(+ + −−) (ultra-hyperbolic) forκ = −1 and(+ + ++) (Euclidean)whenκ = 1—having a non-selfdual Killing vector [1while those having a selfdual Killing vector appear

386 M. Mañas, L.M. Alonso / Physics Letters A 320 (2004) 383–388

the

nbles

s

hicitn[3]

[10,ith

2].is

thexamennd

er-dor-

ingee-ns,ym-gual[16]rate

on-all

t

ot

ifs.

be related to the wave (or Laplace) equation andmetrics are of Gibbons–Hawking type [6].

Very few solutions of the Boyer–Finley equatiohave been found. In first place a separation of variaφ(z, z̄, t) = log(f (t))+Φ(z, z̄) leads to the Liouvilleequation [9]Φzz̄ = eΦ , whose general solution iwell known. If one imposes a symmetry, sayz =z̄, then the equation linearize, after a hodograpchange of variable [16] and in this form implicsolutions are gotten. Also in [14] an implicit solutiobased on the Painlevé III equations was given. Ina new explicit solution was presented, see also11]. Further studies of the geometry associated wthe equation can be found in, for example, Ref. [See also [12,13] for further information regarding thequation.

The Boyer–Finley equation is also known asdispersionless Toda equation and appears as an eple of the so-called Whitham hierarchies. It has beapplied to the study of conformal transformations atopological field theory [8].

Our scheme provides solutions to the ultra-hypbolic Boyer–Finley equation. The problem is to finsolutions of the Boyer–Finley equation so that the cresponding metric does not have an additional Killvector. Hence, following [11] the solutions of thBoyer–Finley equation must be non-invariant [10] (bing the symmetry group composed of translatioscaling and conformal transformations), as these smetries will carry to corresponding additional Killinvectors. This construction is equivalent to self-dhyper-Kähler spaces and, as was shown by Wardthe Boyer–Finley equation can be used to geneEinstein–Weyl spaces in 3D.

To check that our scheme gives solutions of ninvariant type, for the ultra-hyperbolic case, we shuse the hodograph equation in the following form:

t + ρe−iα(ρ)z+ ρeiα(ρ)z̄ = h(ρ),

whereα andh are arbitrary functions ofu= ρ and thesolution of the Boyer–Finley equation is given by

φ = log(ρ2),

this form of the hodograph equation ensures thaφ

takes real values. Using polar coordinatesz = reiθ weget the following hodograph relation

(7)t + 2ρr cos(α(ρ)− θ)= h(ρ).

-

Now, following [10] we must check whether or nis possible to find constantsα andβ and functionsa(z)andb(z̄) such that the following equation holds

(8)(α + βt)φt + a(z)φz + b(z̄)φz̄ = 2β − a′(z)− b′(z̄).

Now, recalling that the hodograph relation implies

ρz = ρe−iα

D, ρz̄ = ρeiα

D, ρt = 1

D,

with

D := h′ − (1− iρα′)e−iαz− (1+ iρα′)eiαz̄,

and introducing the notation

A(z) := a(z)− βz, B(z̄) := b(z̄)− βz̄,

we can write (8) in the following form:

(9)α + βh+Aρe−iα +Bρeiα = −(A′ +B ′)F,

with

F := ρD

2.

Now, if the functions{1, ρe−iα,ρeiα,h} are linearlydependent,

(10)λ11+ λ2ρe−iα + λ3ρeiα + λ4h= 0,

for some constantsλi , i = 1,2,3,4, then (9) will beidentically satisfied ifα = λ1, β = λ2, A = λ3 andB = λ4. The invariant solutions should appear also(9) holds takingx, y, t andu as independent variableIn doing so must imposeA = A1z + A0 and B =B1z̄+B0 together with the equations

A1 − 1

2(1− iρα′)(A1 +B1)= 0,

B1 − 1

2(1+ iρα′)(A1 +B1)= 0,

α + βh+A0ρe−iα +B0ρeiα + 1

2(A1 +B1)ρh

′ = 0.

The two first are equivalent to the ODE

α′ = iA1 −B1

A1 +B1

1

ρ

that implies

α = iγ logρ +C �⇒ e−iα = cργ ,

(11)γ := A1 −B1,

A1 +B1

M. Mañas, L.M. Alonso / Physics Letters A 320 (2004) 383–388 387

ld-

ust

be

→

ess

ilyr-

do-of

in-

nsei

1)

01)

.E.

ys.

while the third determinesh as a solution of thefollowing ODE

α + βh+A0cρ1+γ +B0c

−1ρ1−γ

+ 1

2(A1 +B1)ρh

′ = 0,

whose solution is

h(ρ) = Cρ2 βA1+B1 − α

β− cA0

A1 +B1ρ

2 A1A1+B1

(12)− B0

c(A1 +B1)ρ

2 B1A1+B1 .

Generically, if neither (10) nor (11) and (12) hoit would be difficult to have an invariant solution. Introducing the notationf±(ρ) := ρe∓iα , F± := f ′±/F ′andH := h′/F ′ takingt-derivatives of (9) we get

(13)βH +AF+ +BF− = −(A′ +B ′),(14)βH(n) +AF

(n)+ +BF

(n)− = 0, n � 1.

Thus, in order to have invariant solutions we mimpose∣∣∣∣∣∣∣H(n1) F

(n1)+ F(n1)−

H(n2) F(n2)+ F

(n2)−H(n3) F

(n3)+ F(n3)−

∣∣∣∣∣∣∣ = 0,

with 0< n1 < n2 < n3, nj ∈ N,

and therefore an infinite set of equations need tosatisfied by the solutionρ(z, z̄, t) of the hodographrelation.

Following [16] we know that any solution ofφ of(6) defines an Einstein–Weyl space given by

(15)dl2 = dt2 − 4ρ2(dr2 + r2 dθ2), ω = 2φt dt .

Thus, by introducing the change of variables(t, r, θ)

(ρ, r,ψ)

t = h(ρ)− 2rρ cosψ, r = r,

(16)θ =ψ + α(ρ),

the corresponding Einstein–Weyl structure becomexplicitly given in terms of two arbitrary functionα(ρ) andh(ρ) by

dl2 = [(h′ − 2r cosψ)2 − 4r2ρ2(α′)2

]dρ2

− 4ρ2 sin2ψ dr2 − 4r2ρ2 cos2ψ dψ2

− 2ρ cos(ψ)(h′ − 2r cosψ)dρ dr

− 4rρ[(h′ − 2r cosψ)sin(ψ)− 2rρα′]dρ dψ

− 2rρ2 sin2ψ dr dψ,

and

ω = 4

ρ

[(h′ − 2r cosψ)dρ − 2ρ cosψ dr

+ 2rρ sinψ dψ]

(17)× [h′ − 2r cosψ − 2rρα′ sinψ]−1 dt .

It should be noticed that (16) and (17) define a famof Einstein–Weyl structures different from that chaacterized by Ward in [16]. Indeed, Ward uses an hograph transformation for determining all solutions(6) independent on one of the spatial variablesx or y.

The corresponding ultra-hyperbolic vacuum Estein metric in 4D is given by [11]

ds2 = φt dl2 − 1

φt

[dt̃ + i(φz dz− φz̄ dz̄)

]2

= 2

ρDdl2 − ρD

2

(dt̃ − 4

D

(sinψ dr + r cosψ dψ

+ α′r cosψ dρ))2

,

with D = h′ − 2r cosψ − 2rρα′ sinψ .

Acknowledgements

The authors would like to acknowledge discussiowith Alexander Mikhailov, Ian Strachan and SergTsarev.

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