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Constructing multiple prolongation structures from homotopic maps
REPORTS ON MATHEMATICAL PHYSICS
Vol. 67 (2011)
No. 1
CONSTRUCTING MULTIPLE PROLONGATION STRUCTURES FROM HOMOTOPIC MAPS E. O. I FIDON Department of Mathematics, University of Benin, Benin-City, Nigeria (e-mail: ifi[email protected]) (Received May 4, 2010) In this paper, we show how multiple prolongation structures developed out of homotopy theory, can be constructed from a differential ideal corresponding to an exterior differential system. We use this method to construct multiple prolongation structures for the Robinson–Trautman equations of Petrov type III. It is found that the introduction of two arbitrary pseudo-potentials in the carrier space of the vector fields of this equation imposes nontrivial constraints on the prolongation structures which prevents the algebra from growing rapidly. Specific choices of the newly introduced pseudo-potentials result a coupled Kac–Moody A1 ⊕ A1 and Virasoro algebra as prolongation structure. Other choices of the potentials reproduce previously established results, namely the contragradient algebra K2 of infinite growth. The Lax pair and Riccati equations for pseudo-potentials can be formulated respectively from linear and nonlinear realizations of the prolongation structure. 1991 Mathematics Subject Classification: 17B80, 37K10, 17B66, 83C35. Keywords: B¨acklund transformations, Einstein equations, gravitational waves.
1.
Introduction
B¨acklund transformations have in the past two decades emerged as an important tool in the study of a wide range of nonlinear partial differential equations arising in mathematical physics. A collection of several of its applications is given in [1]. Perhaps till date, the most widely used method of deriving B¨acklund transformations is the prolongation procedure of Wahlquist and Estabrook which is based on both the notion of pseudo potentials and on Cartan’s method of describing differential equations via the construction of a ‘contact module’. The algebras obtained from these potentials are called ‘prolongation structures’. Appropriate modifications of the prolongation procedure proved to be sufficient in finding general (even nonlinear) pseudo-potentials and sometimes B¨acklund transformations (see for example [2–4] for various variants of the method). It has however been shown that the existence of various distinct subideals of the contact module causes different prolongation structures to be generated. A good example of this can be seen in [1] (p. 150), where two distinct (linearly independent) sub-ideals 1 , 3 are considered as effective for the sine-Gordon equation. However, the B¨acklund map which defines the classical B¨acklund transformations [53]
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E. O. IFIDON
obtained from 3 , cannot be obtained by applying the same procedure to 1 . Another example can be found in [5] where all the nonisomorphic ideals of 2-forms that can be used to generate the sine-Gordon equation were considered. In that paper, it was demonstrated that these differences lead to major differences in the prolongation algebra generated. There are however situations in which the prolongation structures generated are independent of the choice of the generating ideal as each in-equivalent sub-ideals give rise to prolongation structures which are the same or are related by gauge transformations. Typical examples of this class of generating ideals can be found in [6]. In such situations, one may be lead to an algebra of infinite dimension as well as infinite growth with no known realizations, thus impeding the ability of the method to generate new (nontrivial) solutions or B¨acklund transformations. The prolongation technique is obviously not sufficient to guarantee manageable results that could eventually lead to the construction of B¨acklund maps. This suggest to us that we need to use prolongation in conjunction with other methods. It seems desirable therefore to seek methods that can generate more than one distinct prolongation structure from a given differential ideal with the hope that at least one of these prolongation structures can be represented by finite matrices in order to obtain the corresponding B¨acklund transformation which can then be used to generate new nontrivial solutions from a known trivial solution. Such a method can be regarded as a generalization of the Wahlquist–Estabrook procedure for which a means is devised whereby additional ‘pseudo-potentials’ is introduced into the usual Wahlquist–Estabrook prolongation space. The commutator of the newly introduced potentials with the Wahlquist– Estabrook pseudo-potentials are expected to impose nontrivial conditions on the prolongation algebra. A brief outline of the method is given in the next section. 2.
Outline of the method Our theory begins with the introduction of a parameter μ ∈ R in the defining equations of the system under consideration. Of particular interest would be equations of the form A a α μ ∈ [0, 1] , A = 1, 2, 3, . . . , (2.1) uA τ = F (x , u , μ), which are nonevolution equations with coordinates x a , a = 1, . . . , m, uα , α = 1, . . . , n, and the multi-index τ ∈ {i1 , i2 , . . . , ik−1 , ik }. We shall adopt Cartan’s approach (systematized by Wahlquist and Estabrook) for studying differential equations. We would assume in the first instance that μ is a fixed parameter, then define our ‘contact module’ and exterior differential system in terms of μ. (2.1) may be realized as a submanifold of the k-jet bundle Y k ⊂ J k (M, N ) where M ∈ Rm and N ∈ Rn . Cartan’s approach involves the construction of a ‘contact module’ generated by a set of 1-forms: ⎧ ⎪ θ α = dzα − ziα dx i , ⎪ ⎪ ⎨ α α α j k (M, N ) := θi = dzi − zij dx , (2.2) ⎪ . . . ⎪ ⎪ ⎩θα α α ik i1 i2 ...ik−1 = dzi1 i2 ...ik−1 − zi1 i2 ...ik−1 ik dx ,
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MULTIPLE PROLONGATION STRUCTURES
where the summation convention has been used with respect to the repeated (one upper, one lower) indices. The ziα1 ...iq are symmetric in their subscripts. We let the contact module ‘know’ about the pde we want to study by restricting it to Y k . k The restricted contact module would be denoted by (M, N ). We prolong this space by ‘adjoining’ a space, N ∈ Rn , for the desired ‘potentials’ to live; these are additional functions of the underlying coordinates x a ∈ M . The exterior differential system of m-forms on J k−1 (M, N ) associated with (2.1) is then generated by η1 = dzα ∧ dx i2 ∧ dx i3 ∧ · · · ∧ dx im − ziα1 dx i1 ∧ dx i2 ∧ · · · ∧ dx im , η2 = dzα ∧ dx i1 ∧ dx i3 ∧ · · · ∧ dx im − ziα1 dx i1 ∧ dx i2 ∧ · · · ∧ dx im , .. .
(2.3)
σ = dziα1 i2 ...im−1 ∧ dx i1 ∧ dx i2 ∧ dx im · · · − F A (x a , uα , μ)ω, where ω = dx i1 ∧ dx i2 ∧ · · · ∧ dx im is the volume element. The Wahlquist–Estabrook process may be described as searching for a map ψ : J k−1 (M, N ) × J 0 (M, N ) → J 1 (M, N ) with the following coordinate presentation: ⎧ ⎪ xa = xa, ⎪ ⎪ ⎪ ⎪ ⎨ zα = zα , ψ := ⎪ zA = zA , ⎪ ⎪ ⎪ ⎪ ⎩ zA = Z A x, zα , . . . , zα a
a
i1 i2 ...im , z
(2.4)
(2.5)
.
Now the pullback of ψ maps the cotangent bundles, so it is reasonable that we insist that it maps the contact modules appropriately: ψ ∗ : p (J 1 ) = p (J k−1 × N ),
(2.6)
ψ ∗ (θ A ) = dzA − ZaA dx a .
We must therefore also insist that it be closed in the combined contact module, which creates requirements on the functions ZrA . dZaA ∧ dx a + dZbA ∧ dx b = fiA ηi + g A σ + ξ ∧ ζ,
i = 1, 2, . . . ,
(2.7)
where fi and g are arbitrary functions on J k−1 (M, N ) × J 0 (M, N ), ζ = dz − Za dx a − Zb dx b , and ξ is a 1-form on J k−1 (M, N ) × J 0 (M, N ). The presence of μ in σ induces a parameterization of the fiber bundle which we shall denote by μ so that A A ∧ dx a + dZb,μ ∧ dx b = fiA ηi + g A σμ + ξ ∧ ζ, μ : dZa,μ
i = 1, 2, . . . ,
(2.8)
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E. O. IFIDON
μ is the parameterized map ψ. Allowing μ vary from 0 to 1 then establishes a homotopy between the B¨acklund map 0 and the B¨acklund map 1 , 0 being the B¨acklund map corresponding to (2.1) when μ = 0 and 1 the B¨acklund map corresponding to (2.1) when μ = 1 which then is the B¨acklund map for the equation of interest. DEFINITION 1. Two maps 0 , 1 : J k−1 (M, N ) × J 0 (M, N ) → J 1 (M, N ) are said to be homotopic if there exists a map μ : J k−1 (M, N ) × J 0 (M, N ) × R → J 1 (M, N ) (called a homotopy from 0 to 1 ) such that μ J k−1 (M, N ) × J 0 (M, N ) × {0} = 0 and μ J k−1 (M, N ) × J 0 (M, N ) × {1} = 1 Eq. (2.8) establishes an implicit relationship between the map 0 and the map 1 . This relationship is made explicit as follows. Let g be the free Lie algebra associated with the system (2.1) and G = exp g the corresponding Lie group, then given a B¨acklund map ψ one can obtain another B¨acklund map ψ˜ from (2.5) by composing ψ with a C ∞ composition map λ of the group G = exp g, namely ψ˜ → λ ◦ ψ ◦ λ−1 . In particular, this mapping may be represented by Za → e+μAa Za e−μAa ,
Zb → e+μDb Zb e−μDb .
(2.9)
Conjugation is performed by means of the adjoint action of G which gives Za → Za + μ [Aa , Za ] +
μ2 [Aa , [Aa , Za ]] + · · · = eadμAa Za 2!
(2.10)
Zb → Zb + μ [Db , Zb ] +
μ2 [Db , [Db , Zb ]] + · · · = eadμDb Zb . 2!
(2.11)
and
3.
Robinson–Trautman type III solutions In this section, we consider an application of the method derived in the previous section to the study of nontrivial (twisting) gravitational wave equations occurring in nonlinear physics with a view to obtaining exact description of solutions of the vacuum Einstein field equations for algebraically degenerate Petrov types. Robinson and Trautman [7] introduced a class of metrics representing a very simple kind of spherical radiation. These metrics are algebraic special solutions assumed to have the property that they admit a shearfree, twistfree and diverging null congruence which is geodesic [8, 9]. They are indeed of great importance in the gravitational theory of spherical waves since they are known to be of good approximations to actual radiation fields at great distances from the source (see [9] and [7]) and more importantly, Einstein’s equations reduce to a single parabolic fourth-order equation in this class of metrics [8]. The general form of the metric which any Einstein space must have if it should permit such a repeated null direction is given in [10]. For this class of solutions, the field equations are reduced to the nonlinear partial differential equation 1 (3.12) uxy = (x + y)e−2u , 2
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MULTIPLE PROLONGATION STRUCTURES
where subscripts denote partial derivatives and all the symbols have their usual 3 meanings as in [6]. Only one rather trivial solution, u = log(x + y) 2 has been found till date [10]. More general solutions are difficult to construct. This has led several authors in search of generalized symmetries for this equation with a bid to generating new nontrivial solutions from a knowledge of its B¨acklund maps. Early attempts by Finley [6] to obtain nontrivial solutions to the Robinson–Trautman type III (RT) equations lead to the study of infinite-dimensional prolongation structures to (3.12) which were of contragradient type, specifically K2 . Unfortunately, Finley was unable to find a B¨acklund transformation for this problem as no realizations of K2 has yet been found. More recently, Bakas adopted two different approaches in a bid to study the prolongation structures of the equation. The first based on Toda theories [9], reproduces the results of Finley while the second approach involves a more complicated algebra depending on an additional parameter t (called spectator) which contains K2 in its zero modes [11]. As observed in [9], the ‘intertwining’ of these two different algebraic descriptions of the same equation deserves further attention. It is therefore worth-while to use prolongation in conjunction with other methods to study the prolongation structures of (3.12). The entire process begins by first introducing a parameter μ into (3.12) so that we have the pde 1 μ ∈ [0, 1] . (3.13) (x + μy)e−2u , 2 We take M = R2 (with coordinates x, y), N = R1 (with coordinates u), N = R1 (with coordinates v) the volume element ω = dx ∧ dy. The exterior differential system of 2-forms associated with (3.13) can be written as uxy =
σ = dux ∧ dx − duy ∧ dy + 2f ω, η1 = du ∧ dy − ux dx ∧ dy,
(3.14)
η2 = du ∧ dx + ux dx ∧ dy, where
1 f = (x + μy)e−2u . 2 The Wahlquist–Estabrook procedure requires that
(3.15)
dF ∧ dx + dG ∧ dy = lη1 + mη2 + nσ + ξ ∧ ζ,
(3.16)
where l, m and n are arbitrary functions on J (M, N ) × J (M, N ), ζ = dv − F dx − Gdy and ξ is a 1-form on J 1 (M, N ) × J 0 (M). Comparison of the terms in du ∧ dx, dux ∧ dx, dux ∧ dy, du ∧ dy, duy ∧ dy, duy ∧ dx gives 1
Fu = l,
Fp = m,
Gp = 0 = F q ,
Gu = n,
0
Gq = −Fp .
(3.17)
Comparison of the term dx ∧ dy gives
1 [F, G] = −pGu + qFu + Fy − Gx + (x + μy)e−2u Fp − Gq , 2
(3.18)
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E. O. IFIDON
where p = ux , q = uy . In view of (3.17) we set F = pZ + B,
G = −qZ + C,
Zu = 0,
(3.19)
where B = B(x, u) and C = C(y, u). We take advantage of the existence of the explicit gauge equivalence F = pZ + e+μA Be−μA ,
G = −qZ + e+μD Ce−μD ,
(3.20)
induced by an equivalence of pseudopotentials Za ∼ Za = λ ◦ Za ◦ λ−1 where λ is given explicitly by (2.9). In view of (2.10) and (2.11) the inferred vector fields take the form μ2 μ (3.21) [A, B] + [A, [A, B]] + · · · = pZ + eadμA B, 1! 2! μ2 μ (3.22) G = −qZ + C + [D, C] + [D, [D, C]] + · · · = −qZ + eadμD C. 1! 2! Inserting these forms into (3.17) yields a polynomial in p and q, so that the vanishing of all of the separate coefficients gives the following equations (where all x- and y- derivatives are ignored) (3.23) Z, eadμA B = +eadμA Bu , adμD adμD Z, e C = −e Cu , (3.24) ad (3.25) e μA B, eadμD C = (x + μy)e−2u Z. F = pZ + B +
Eqs. (3.23) and (3.24) can be integrated to give
eadμA B(x, u) = e+u(adZ ) eadμA R(x) and
eadμD C(y, u) = e−u(adZ ) eadμD S(y) ,
(3.26) (3.27)
where the x and y dependence has been written explicitly. Inserted into (3.25) give
+u(ad ) ad Z (3.28) e μA R(x) , e−u(adZ ) eadμD S(y) = (x + μy)e−2u Z. e Comparing coefficients of u gives a countable list of commutation relations to be satisfied, namely k k 1 k k−m (3.29) adm Z adμA R(x), adZ adμD S(y) = (x + μy)Z, 2 m m=0 where = (−1)m . Expansion of both sides of (3.29) in a power series in μ yields the following set of equations k k 1 k k−m 0 0 adm (3.30) Z adA R(x), adZ adD S(y) = xZ, m 2 m=0
59
MULTIPLE PROLONGATION STRUCTURES k k 1 1 k 1−l k−m l adm Z adA R(x), adZ adD S(y) = yZ, m 2 l=0 m=0 n=2,3.. k k 1 k n k−m l n−l adZ adD S(y), adm Z adA R(x) = 0. 2 m l l=0 m=0
(3.31)
(3.32)
Eqs. (3.30), (3.31) and (3.32) are the equations that generate all the requirements in the actual solution and are the most important part of the requirements of the prolongation commutators, it is in fact this set of equations that causes the major restrictions on the prolongation algebra. The lowest-order term in (3.31) implies [Z, [R, S]] = 0. Using this fact and the Jacobi induction hypothesis identity an then shows that the value of the commutator Rk−m,0 , Sm,0 is independent of m whenever 0 ≤ m ≤ k, where Sn,0 = adm Z S.
Rn,0 = (−1)n adnZ R,
(3.33)
The sum in (3.30) therefore reduces to the set of commutators Rk−m,0 , Sm,0 = xZ, ∀m = 0, 1, . . . , k.
(3.34)
The explicit existence of x and y in (3.29) implies that our prolongation vector fields must depend on these variables. However, since the equations are linear in these variables, and display themselves explicitly in that manner in (3.30) and (3.31), it is reasonable to assume that all of Rk−m,0 and Sm,0 are first-order polynomials in x and y, respectively. We would then be led to an infinitely generated set of solutions depending on x and y. Furthermore, since Rk−m,0 and Sm,0 satisfy the same commutation relations, namely (3.34), they may be regarded algebraically as representing the same elements. We may then have the homomorphism Rn,0 → −f1 − xf2 ,
Sm,0 → e2 + ye1 ,
(3.35)
whose coefficients satisfy the following set of commutation relations, [e2 , f2 ] = Z,
0 = [e1 , f1 ] = [e1 , f2 ] = [e2 , f1 ] .
(3.36)
Introducing the quantities p
q
p
Rp,q = (−1)p adZ adA R,
q
Sp,q = adZ adD S,
(3.37)
an the induction hypothesis, and the use of Jacobi identity shows that the values of Rm,1−l , Sm,l are independent of m when 0 ≤ m ≤ k. We may therefore write 1
Rk−m,1−l , Sm,l = yZ,
∀m = 0, 1, . . . , k.
(3.38)
l=0
Again we may regard all of Rm,1 and Sm,1 as algebraically representing the same elements since the right-hand side of (3.38) is independent of the values of m. As such we have the mapping Rm,1 → − [A, f1 ] − x [A, f2 ] ,
Sm,1 → [D, e2 ] + y [D, e1 ] .
(3.39)
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E. O. IFIDON
In view of these, (3.38) then gives the following set of four commutation relations: [[A, f1 ] , e2 ] + [f1 , [D, e2 ]] = 0, [[A, f1 ] , e1 ] + [f1 , [D, e1 ]] = −Z,
[[A, f2 ] , e2 ] + [f2 , [D, e2 ]] = 0, [[A, f2 ] , e1 ] + [f2 , [D, e1 ]] = 0.
Using the induction hypothesis, the sum in (3.32) reduces to the series n=2,3 n Rk−m,n−l , Sm,l = 0, ∀m = 0, 1, . . . , k. l l=0
(3.40)
(3.41)
The next stage involves identifying elements of the free algebra whose commutation relations are given by Eq. (3.36), (3.40) and the still infinite commutation relations (3.41) with the elements of an already studied algebra. The studied differential equation and the corresponding prolongation structure depend upon the parameter μ. The prolongation structures that are to be obtained from the complete set of equations (3.36), (3.40) and (3.41) are indeed valid only for μ > 0. In the case when μ = 1 the results of this example become the corresponding ones for the RT equations. 4.
Infinite-dimensional algebra Using similar notations as in [12], we consider the algebra A1 ⊕ A1 with Cartan Weyl basis: (n) (m+n) (m+n) Hi(m) , E±β = ±2δi1 δβ1 E±1 ± 2δi2 δβ2 E∓2 , (m) (n) (4.42) = δα1 δβ1 H1(m+n) (mod H2(m+n) ) , E−β E+α + δα2 δβ2 H2(m+n)
(mod H1(m+n) )
where m, n = 0, ±1, ±2, ±3, . . . ; i = 1, 2; α, β = 1, 2. We see that H1(m+n) is isomorphic to H1(m+n) + H2(m+n) in the first copy of A1 and H2(m+n) is isomorphic to H1(m+n) + H2(m+n) in the second copy of A1 . Making the substitutions Z = H1(0) + H2(0) , f1 = D (1) ,
e1 = E1(0) ,
(0) f2 = E−2 ,
e2 = E2(0) ,
(−1) A = E−1 ,
D=0
(4.43)
where D (1) is an element of the Virasoro algebra satisfying the commutation relation [13] (m) (n) D , Eα = nEα(m+n) (4.44) (m) (n) D ,D = (n − m)D (m+n) . We see that all of the constrain conditions (3.36), (3.40) and (3.41) are identically (n) , D (l) } constitute the Kac–Moody algebra satisfied. The set of vectors {Hi(m) , E±α A1 ⊕ A1 coupled with the Virasoro algebra. The Serre relations and dimensions of this algebra are given in [14]. A discussion of the growth of the algebra can be
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MULTIPLE PROLONGATION STRUCTURES
found in [15] as well as [16]. One may use the structure and representation theory of the algebra to obtain realizations in terms of finite matrices, which can then be used to construct Lax pairs for the RT equations. Thus far we have made specific choices for A and D which has enabled us identify these quantities as basis of A1 ⊕ A1 . We can however reproduce the results in [6] by making other choices for A and D. For instance, if we take A = 0, (3.38) becomes ∀m = 0, 1, . . . k, (4.45) Rk−m,0 , Sm,1 = yZ, and (3.41) reduces to
Rk−m,0 , Sm,n = 0,
∀n = 2, 3, . . . , k.
(4.46)
Following the previous arguments we could set Rk−m,0 = −f1 − xf2 ,
Sm,0 = e2 + ye1 ,
(4.47)
substitution in (3.34) reproduces (3.36). Substitution in (4.45) gives [[D, e1 ] , f1 ] = Z,
[[D, e2 ] , f1 ] = 0 = [[D, e2 ] , f2 ] = [[D, e1 ] , f2 ] .
The last of the constrain equations reduces to Rk−m,0 , Sm,n = 0, n = 2, 3, . . . .
(4.48) (4.49)
Since [D, S] depends on y we assume a reduction of the algebra of the form [D, S] = yD.
(4.50)
Thus we have [D, e2 ] = 0 and [D, e1 ] = D. Defining the isomorphism D → E1 ,
e2 → E2 ,
−f1 → F1 ,
we have the commutation relations [Z, Ei ] = Ei , [Z, Fi ] = −Fi , [E2 , F1 ] = 0 = [E1 , F2 ] , [E1 , F1 ] = Z, [E2 , F2 ] = Z,
−f2 → F2 ,
(4.51)
(4.52)
(4.52) are exactly the commutation relations given in [6] and constitute a description of Kac’s algebra K2 , which is the most elementary example of a simple Lie algebra with infinite growth (see [14]). The K2 algebra was also encountered as the prolongation structure of the Robinson–Trautman equation by Bakas [9], using a somewhat different approach based on Toda theories. 5.
Conclusion We have shown the connection between the prolongation structures of the Robinson–Trautman equation of Petrov type III and a coupled Kac–Moody, Virasoro algebra. This is an interesting result since the representation theory of this algebra are better understood than their contragradient counterpart K2 which grows exponentially fast. Since these algebras emanate from the same differential ideal, we may conclude
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E. O. IFIDON
that our method is capable of generating multiple prolongation structures belonging to the same differential ideal, thus enhancing the ability of the procedure to generate new, nontrivial, solutions to the RT equation. REFERENCES [1] C. Rogers and W. F. Shadwick: B¨acklund Transformations and Their Applications, Academic Press, New York 1982. [2] J. D. Finley, III: Estabrook–Wahlquist prolongations and infinite-dimensional algebras, in: Symmetry Methods in Physics, Vol. 1, Dubna 1995, p. 203–211. [3] R. K. Dodd and R. K. Bullough: Polynomial conserved densities for the sine-Gordon equations, in: Proc. R. Soc. Lond. A 352 (1977), 481–503. [4] B. Kent Harrison: The Differential Form Method for Finding Symmetries, in: Symmetry, Integrability and Geometry: Methods and Applications 1, (2005), 001. [5] J. D. Finley, III and John K. McIver: Infinite-dimensional Estabrook–Wahlquist prolongations for the sine-Gordon equation, J. Math. Phys. 36 (1995), 5707–5734. [6] J. D. Finley III: Robinson–Trautman Type III prolongation structure contains K2 , Commun. Math. Phys. 178 (1996), 375–390. [7] I. Robinson and A. Trautman: Some spherical gravitational waves in general relativity, in: Proceedings of the Royal Society of London. Series A, Mathematical and Physical, Vol. 265, No. 1323 (Feb. 6, 1962), pp. 463–473. [8] P. T. Chru´sciel: On the global structure of Robinson–Trautman space-times, in: Proceedings: Mathematical and Physical Sciences, Vol. 436, No. 1897 (Feb. 8, 1992). [9] I. Bakas: The algebraic structure of geometric flows in two dimensions, hep-th/0507284, CERN-PHTH/2005-134 (July 2005). [10] D. Kramer, H. Stephani, E. Herlt, M. MacCallum: Exact Solutions of Einstein’s Field Equations, Cambridge Univ. Press, Cambridge 1980. [11] I. Bakas: On the integrability of spherical gravitational waves in vacuum, gr-qc/0504130.CERN-PHTH/2005-069 (April 2005). [12] H. X. Yang and Y Li: Prolongation approach to B¨acklund transformation of Zhiber-Mikhailov-Shabat equation, J. Math. Phys. 7 (1996), 37. [13] M. Omote: Prolongation structures of nonlinear equations and infinite-dimensional algebras, J. Math. Phys. 27 (1986), 2853. [14] V. G. Kac: Infinite-Dimensional Lie Algebras, 3rd edition, Cambridge University Press, Cambridge, New York, 1990. [15] I. M. Gelfand and A. A. Kirillov: Fields associated with enveloping algebras of Lie algebras, Doklady 167 (1966), 407–409. [16] G. R. Krause and T. H. Lenagan: Growth of algebras and Gelfand–Kirillov dimension, Amer. Math. Soc. Grad. Studies in Math. 22 (2000).
Vol. 67 (2011)
No. 1
CONSTRUCTING MULTIPLE PROLONGATION STRUCTURES FROM HOMOTOPIC MAPS E. O. I FIDON Department of Mathematics, University of Benin, Benin-City, Nigeria (e-mail: ifi[email protected]) (Received May 4, 2010) In this paper, we show how multiple prolongation structures developed out of homotopy theory, can be constructed from a differential ideal corresponding to an exterior differential system. We use this method to construct multiple prolongation structures for the Robinson–Trautman equations of Petrov type III. It is found that the introduction of two arbitrary pseudo-potentials in the carrier space of the vector fields of this equation imposes nontrivial constraints on the prolongation structures which prevents the algebra from growing rapidly. Specific choices of the newly introduced pseudo-potentials result a coupled Kac–Moody A1 ⊕ A1 and Virasoro algebra as prolongation structure. Other choices of the potentials reproduce previously established results, namely the contragradient algebra K2 of infinite growth. The Lax pair and Riccati equations for pseudo-potentials can be formulated respectively from linear and nonlinear realizations of the prolongation structure. 1991 Mathematics Subject Classification: 17B80, 37K10, 17B66, 83C35. Keywords: B¨acklund transformations, Einstein equations, gravitational waves.
1.
Introduction
B¨acklund transformations have in the past two decades emerged as an important tool in the study of a wide range of nonlinear partial differential equations arising in mathematical physics. A collection of several of its applications is given in [1]. Perhaps till date, the most widely used method of deriving B¨acklund transformations is the prolongation procedure of Wahlquist and Estabrook which is based on both the notion of pseudo potentials and on Cartan’s method of describing differential equations via the construction of a ‘contact module’. The algebras obtained from these potentials are called ‘prolongation structures’. Appropriate modifications of the prolongation procedure proved to be sufficient in finding general (even nonlinear) pseudo-potentials and sometimes B¨acklund transformations (see for example [2–4] for various variants of the method). It has however been shown that the existence of various distinct subideals of the contact module causes different prolongation structures to be generated. A good example of this can be seen in [1] (p. 150), where two distinct (linearly independent) sub-ideals 1 , 3 are considered as effective for the sine-Gordon equation. However, the B¨acklund map which defines the classical B¨acklund transformations [53]
54
E. O. IFIDON
obtained from 3 , cannot be obtained by applying the same procedure to 1 . Another example can be found in [5] where all the nonisomorphic ideals of 2-forms that can be used to generate the sine-Gordon equation were considered. In that paper, it was demonstrated that these differences lead to major differences in the prolongation algebra generated. There are however situations in which the prolongation structures generated are independent of the choice of the generating ideal as each in-equivalent sub-ideals give rise to prolongation structures which are the same or are related by gauge transformations. Typical examples of this class of generating ideals can be found in [6]. In such situations, one may be lead to an algebra of infinite dimension as well as infinite growth with no known realizations, thus impeding the ability of the method to generate new (nontrivial) solutions or B¨acklund transformations. The prolongation technique is obviously not sufficient to guarantee manageable results that could eventually lead to the construction of B¨acklund maps. This suggest to us that we need to use prolongation in conjunction with other methods. It seems desirable therefore to seek methods that can generate more than one distinct prolongation structure from a given differential ideal with the hope that at least one of these prolongation structures can be represented by finite matrices in order to obtain the corresponding B¨acklund transformation which can then be used to generate new nontrivial solutions from a known trivial solution. Such a method can be regarded as a generalization of the Wahlquist–Estabrook procedure for which a means is devised whereby additional ‘pseudo-potentials’ is introduced into the usual Wahlquist–Estabrook prolongation space. The commutator of the newly introduced potentials with the Wahlquist– Estabrook pseudo-potentials are expected to impose nontrivial conditions on the prolongation algebra. A brief outline of the method is given in the next section. 2.
Outline of the method Our theory begins with the introduction of a parameter μ ∈ R in the defining equations of the system under consideration. Of particular interest would be equations of the form A a α μ ∈ [0, 1] , A = 1, 2, 3, . . . , (2.1) uA τ = F (x , u , μ), which are nonevolution equations with coordinates x a , a = 1, . . . , m, uα , α = 1, . . . , n, and the multi-index τ ∈ {i1 , i2 , . . . , ik−1 , ik }. We shall adopt Cartan’s approach (systematized by Wahlquist and Estabrook) for studying differential equations. We would assume in the first instance that μ is a fixed parameter, then define our ‘contact module’ and exterior differential system in terms of μ. (2.1) may be realized as a submanifold of the k-jet bundle Y k ⊂ J k (M, N ) where M ∈ Rm and N ∈ Rn . Cartan’s approach involves the construction of a ‘contact module’ generated by a set of 1-forms: ⎧ ⎪ θ α = dzα − ziα dx i , ⎪ ⎪ ⎨ α α α j k (M, N ) := θi = dzi − zij dx , (2.2) ⎪ . . . ⎪ ⎪ ⎩θα α α ik i1 i2 ...ik−1 = dzi1 i2 ...ik−1 − zi1 i2 ...ik−1 ik dx ,
55
MULTIPLE PROLONGATION STRUCTURES
where the summation convention has been used with respect to the repeated (one upper, one lower) indices. The ziα1 ...iq are symmetric in their subscripts. We let the contact module ‘know’ about the pde we want to study by restricting it to Y k . k The restricted contact module would be denoted by (M, N ). We prolong this space by ‘adjoining’ a space, N ∈ Rn , for the desired ‘potentials’ to live; these are additional functions of the underlying coordinates x a ∈ M . The exterior differential system of m-forms on J k−1 (M, N ) associated with (2.1) is then generated by η1 = dzα ∧ dx i2 ∧ dx i3 ∧ · · · ∧ dx im − ziα1 dx i1 ∧ dx i2 ∧ · · · ∧ dx im , η2 = dzα ∧ dx i1 ∧ dx i3 ∧ · · · ∧ dx im − ziα1 dx i1 ∧ dx i2 ∧ · · · ∧ dx im , .. .
(2.3)
σ = dziα1 i2 ...im−1 ∧ dx i1 ∧ dx i2 ∧ dx im · · · − F A (x a , uα , μ)ω, where ω = dx i1 ∧ dx i2 ∧ · · · ∧ dx im is the volume element. The Wahlquist–Estabrook process may be described as searching for a map ψ : J k−1 (M, N ) × J 0 (M, N ) → J 1 (M, N ) with the following coordinate presentation: ⎧ ⎪ xa = xa, ⎪ ⎪ ⎪ ⎪ ⎨ zα = zα , ψ := ⎪ zA = zA , ⎪ ⎪ ⎪ ⎪ ⎩ zA = Z A x, zα , . . . , zα a
a
i1 i2 ...im , z
(2.4)
(2.5)
.
Now the pullback of ψ maps the cotangent bundles, so it is reasonable that we insist that it maps the contact modules appropriately: ψ ∗ : p (J 1 ) = p (J k−1 × N ),
(2.6)
ψ ∗ (θ A ) = dzA − ZaA dx a .
We must therefore also insist that it be closed in the combined contact module, which creates requirements on the functions ZrA . dZaA ∧ dx a + dZbA ∧ dx b = fiA ηi + g A σ + ξ ∧ ζ,
i = 1, 2, . . . ,
(2.7)
where fi and g are arbitrary functions on J k−1 (M, N ) × J 0 (M, N ), ζ = dz − Za dx a − Zb dx b , and ξ is a 1-form on J k−1 (M, N ) × J 0 (M, N ). The presence of μ in σ induces a parameterization of the fiber bundle which we shall denote by μ so that A A ∧ dx a + dZb,μ ∧ dx b = fiA ηi + g A σμ + ξ ∧ ζ, μ : dZa,μ
i = 1, 2, . . . ,
(2.8)
56
E. O. IFIDON
μ is the parameterized map ψ. Allowing μ vary from 0 to 1 then establishes a homotopy between the B¨acklund map 0 and the B¨acklund map 1 , 0 being the B¨acklund map corresponding to (2.1) when μ = 0 and 1 the B¨acklund map corresponding to (2.1) when μ = 1 which then is the B¨acklund map for the equation of interest. DEFINITION 1. Two maps 0 , 1 : J k−1 (M, N ) × J 0 (M, N ) → J 1 (M, N ) are said to be homotopic if there exists a map μ : J k−1 (M, N ) × J 0 (M, N ) × R → J 1 (M, N ) (called a homotopy from 0 to 1 ) such that μ J k−1 (M, N ) × J 0 (M, N ) × {0} = 0 and μ J k−1 (M, N ) × J 0 (M, N ) × {1} = 1 Eq. (2.8) establishes an implicit relationship between the map 0 and the map 1 . This relationship is made explicit as follows. Let g be the free Lie algebra associated with the system (2.1) and G = exp g the corresponding Lie group, then given a B¨acklund map ψ one can obtain another B¨acklund map ψ˜ from (2.5) by composing ψ with a C ∞ composition map λ of the group G = exp g, namely ψ˜ → λ ◦ ψ ◦ λ−1 . In particular, this mapping may be represented by Za → e+μAa Za e−μAa ,
Zb → e+μDb Zb e−μDb .
(2.9)
Conjugation is performed by means of the adjoint action of G which gives Za → Za + μ [Aa , Za ] +
μ2 [Aa , [Aa , Za ]] + · · · = eadμAa Za 2!
(2.10)
Zb → Zb + μ [Db , Zb ] +
μ2 [Db , [Db , Zb ]] + · · · = eadμDb Zb . 2!
(2.11)
and
3.
Robinson–Trautman type III solutions In this section, we consider an application of the method derived in the previous section to the study of nontrivial (twisting) gravitational wave equations occurring in nonlinear physics with a view to obtaining exact description of solutions of the vacuum Einstein field equations for algebraically degenerate Petrov types. Robinson and Trautman [7] introduced a class of metrics representing a very simple kind of spherical radiation. These metrics are algebraic special solutions assumed to have the property that they admit a shearfree, twistfree and diverging null congruence which is geodesic [8, 9]. They are indeed of great importance in the gravitational theory of spherical waves since they are known to be of good approximations to actual radiation fields at great distances from the source (see [9] and [7]) and more importantly, Einstein’s equations reduce to a single parabolic fourth-order equation in this class of metrics [8]. The general form of the metric which any Einstein space must have if it should permit such a repeated null direction is given in [10]. For this class of solutions, the field equations are reduced to the nonlinear partial differential equation 1 (3.12) uxy = (x + y)e−2u , 2
57
MULTIPLE PROLONGATION STRUCTURES
where subscripts denote partial derivatives and all the symbols have their usual 3 meanings as in [6]. Only one rather trivial solution, u = log(x + y) 2 has been found till date [10]. More general solutions are difficult to construct. This has led several authors in search of generalized symmetries for this equation with a bid to generating new nontrivial solutions from a knowledge of its B¨acklund maps. Early attempts by Finley [6] to obtain nontrivial solutions to the Robinson–Trautman type III (RT) equations lead to the study of infinite-dimensional prolongation structures to (3.12) which were of contragradient type, specifically K2 . Unfortunately, Finley was unable to find a B¨acklund transformation for this problem as no realizations of K2 has yet been found. More recently, Bakas adopted two different approaches in a bid to study the prolongation structures of the equation. The first based on Toda theories [9], reproduces the results of Finley while the second approach involves a more complicated algebra depending on an additional parameter t (called spectator) which contains K2 in its zero modes [11]. As observed in [9], the ‘intertwining’ of these two different algebraic descriptions of the same equation deserves further attention. It is therefore worth-while to use prolongation in conjunction with other methods to study the prolongation structures of (3.12). The entire process begins by first introducing a parameter μ into (3.12) so that we have the pde 1 μ ∈ [0, 1] . (3.13) (x + μy)e−2u , 2 We take M = R2 (with coordinates x, y), N = R1 (with coordinates u), N = R1 (with coordinates v) the volume element ω = dx ∧ dy. The exterior differential system of 2-forms associated with (3.13) can be written as uxy =
σ = dux ∧ dx − duy ∧ dy + 2f ω, η1 = du ∧ dy − ux dx ∧ dy,
(3.14)
η2 = du ∧ dx + ux dx ∧ dy, where
1 f = (x + μy)e−2u . 2 The Wahlquist–Estabrook procedure requires that
(3.15)
dF ∧ dx + dG ∧ dy = lη1 + mη2 + nσ + ξ ∧ ζ,
(3.16)
where l, m and n are arbitrary functions on J (M, N ) × J (M, N ), ζ = dv − F dx − Gdy and ξ is a 1-form on J 1 (M, N ) × J 0 (M). Comparison of the terms in du ∧ dx, dux ∧ dx, dux ∧ dy, du ∧ dy, duy ∧ dy, duy ∧ dx gives 1
Fu = l,
Fp = m,
Gp = 0 = F q ,
Gu = n,
0
Gq = −Fp .
(3.17)
Comparison of the term dx ∧ dy gives
1 [F, G] = −pGu + qFu + Fy − Gx + (x + μy)e−2u Fp − Gq , 2
(3.18)
58
E. O. IFIDON
where p = ux , q = uy . In view of (3.17) we set F = pZ + B,
G = −qZ + C,
Zu = 0,
(3.19)
where B = B(x, u) and C = C(y, u). We take advantage of the existence of the explicit gauge equivalence F = pZ + e+μA Be−μA ,
G = −qZ + e+μD Ce−μD ,
(3.20)
induced by an equivalence of pseudopotentials Za ∼ Za = λ ◦ Za ◦ λ−1 where λ is given explicitly by (2.9). In view of (2.10) and (2.11) the inferred vector fields take the form μ2 μ (3.21) [A, B] + [A, [A, B]] + · · · = pZ + eadμA B, 1! 2! μ2 μ (3.22) G = −qZ + C + [D, C] + [D, [D, C]] + · · · = −qZ + eadμD C. 1! 2! Inserting these forms into (3.17) yields a polynomial in p and q, so that the vanishing of all of the separate coefficients gives the following equations (where all x- and y- derivatives are ignored) (3.23) Z, eadμA B = +eadμA Bu , adμD adμD Z, e C = −e Cu , (3.24) ad (3.25) e μA B, eadμD C = (x + μy)e−2u Z. F = pZ + B +
Eqs. (3.23) and (3.24) can be integrated to give
eadμA B(x, u) = e+u(adZ ) eadμA R(x) and
eadμD C(y, u) = e−u(adZ ) eadμD S(y) ,
(3.26) (3.27)
where the x and y dependence has been written explicitly. Inserted into (3.25) give
+u(ad ) ad Z (3.28) e μA R(x) , e−u(adZ ) eadμD S(y) = (x + μy)e−2u Z. e Comparing coefficients of u gives a countable list of commutation relations to be satisfied, namely k k 1 k k−m (3.29) adm Z adμA R(x), adZ adμD S(y) = (x + μy)Z, 2 m m=0 where = (−1)m . Expansion of both sides of (3.29) in a power series in μ yields the following set of equations k k 1 k k−m 0 0 adm (3.30) Z adA R(x), adZ adD S(y) = xZ, m 2 m=0
59
MULTIPLE PROLONGATION STRUCTURES k k 1 1 k 1−l k−m l adm Z adA R(x), adZ adD S(y) = yZ, m 2 l=0 m=0 n=2,3.. k k 1 k n k−m l n−l adZ adD S(y), adm Z adA R(x) = 0. 2 m l l=0 m=0
(3.31)
(3.32)
Eqs. (3.30), (3.31) and (3.32) are the equations that generate all the requirements in the actual solution and are the most important part of the requirements of the prolongation commutators, it is in fact this set of equations that causes the major restrictions on the prolongation algebra. The lowest-order term in (3.31) implies [Z, [R, S]] = 0. Using this fact and the Jacobi induction hypothesis identity an then shows that the value of the commutator Rk−m,0 , Sm,0 is independent of m whenever 0 ≤ m ≤ k, where Sn,0 = adm Z S.
Rn,0 = (−1)n adnZ R,
(3.33)
The sum in (3.30) therefore reduces to the set of commutators Rk−m,0 , Sm,0 = xZ, ∀m = 0, 1, . . . , k.
(3.34)
The explicit existence of x and y in (3.29) implies that our prolongation vector fields must depend on these variables. However, since the equations are linear in these variables, and display themselves explicitly in that manner in (3.30) and (3.31), it is reasonable to assume that all of Rk−m,0 and Sm,0 are first-order polynomials in x and y, respectively. We would then be led to an infinitely generated set of solutions depending on x and y. Furthermore, since Rk−m,0 and Sm,0 satisfy the same commutation relations, namely (3.34), they may be regarded algebraically as representing the same elements. We may then have the homomorphism Rn,0 → −f1 − xf2 ,
Sm,0 → e2 + ye1 ,
(3.35)
whose coefficients satisfy the following set of commutation relations, [e2 , f2 ] = Z,
0 = [e1 , f1 ] = [e1 , f2 ] = [e2 , f1 ] .
(3.36)
Introducing the quantities p
q
p
Rp,q = (−1)p adZ adA R,
q
Sp,q = adZ adD S,
(3.37)
an the induction hypothesis, and the use of Jacobi identity shows that the values of Rm,1−l , Sm,l are independent of m when 0 ≤ m ≤ k. We may therefore write 1
Rk−m,1−l , Sm,l = yZ,
∀m = 0, 1, . . . , k.
(3.38)
l=0
Again we may regard all of Rm,1 and Sm,1 as algebraically representing the same elements since the right-hand side of (3.38) is independent of the values of m. As such we have the mapping Rm,1 → − [A, f1 ] − x [A, f2 ] ,
Sm,1 → [D, e2 ] + y [D, e1 ] .
(3.39)
60
E. O. IFIDON
In view of these, (3.38) then gives the following set of four commutation relations: [[A, f1 ] , e2 ] + [f1 , [D, e2 ]] = 0, [[A, f1 ] , e1 ] + [f1 , [D, e1 ]] = −Z,
[[A, f2 ] , e2 ] + [f2 , [D, e2 ]] = 0, [[A, f2 ] , e1 ] + [f2 , [D, e1 ]] = 0.
Using the induction hypothesis, the sum in (3.32) reduces to the series n=2,3 n Rk−m,n−l , Sm,l = 0, ∀m = 0, 1, . . . , k. l l=0
(3.40)
(3.41)
The next stage involves identifying elements of the free algebra whose commutation relations are given by Eq. (3.36), (3.40) and the still infinite commutation relations (3.41) with the elements of an already studied algebra. The studied differential equation and the corresponding prolongation structure depend upon the parameter μ. The prolongation structures that are to be obtained from the complete set of equations (3.36), (3.40) and (3.41) are indeed valid only for μ > 0. In the case when μ = 1 the results of this example become the corresponding ones for the RT equations. 4.
Infinite-dimensional algebra Using similar notations as in [12], we consider the algebra A1 ⊕ A1 with Cartan Weyl basis: (n) (m+n) (m+n) Hi(m) , E±β = ±2δi1 δβ1 E±1 ± 2δi2 δβ2 E∓2 , (m) (n) (4.42) = δα1 δβ1 H1(m+n) (mod H2(m+n) ) , E−β E+α + δα2 δβ2 H2(m+n)
(mod H1(m+n) )
where m, n = 0, ±1, ±2, ±3, . . . ; i = 1, 2; α, β = 1, 2. We see that H1(m+n) is isomorphic to H1(m+n) + H2(m+n) in the first copy of A1 and H2(m+n) is isomorphic to H1(m+n) + H2(m+n) in the second copy of A1 . Making the substitutions Z = H1(0) + H2(0) , f1 = D (1) ,
e1 = E1(0) ,
(0) f2 = E−2 ,
e2 = E2(0) ,
(−1) A = E−1 ,
D=0
(4.43)
where D (1) is an element of the Virasoro algebra satisfying the commutation relation [13] (m) (n) D , Eα = nEα(m+n) (4.44) (m) (n) D ,D = (n − m)D (m+n) . We see that all of the constrain conditions (3.36), (3.40) and (3.41) are identically (n) , D (l) } constitute the Kac–Moody algebra satisfied. The set of vectors {Hi(m) , E±α A1 ⊕ A1 coupled with the Virasoro algebra. The Serre relations and dimensions of this algebra are given in [14]. A discussion of the growth of the algebra can be
61
MULTIPLE PROLONGATION STRUCTURES
found in [15] as well as [16]. One may use the structure and representation theory of the algebra to obtain realizations in terms of finite matrices, which can then be used to construct Lax pairs for the RT equations. Thus far we have made specific choices for A and D which has enabled us identify these quantities as basis of A1 ⊕ A1 . We can however reproduce the results in [6] by making other choices for A and D. For instance, if we take A = 0, (3.38) becomes ∀m = 0, 1, . . . k, (4.45) Rk−m,0 , Sm,1 = yZ, and (3.41) reduces to
Rk−m,0 , Sm,n = 0,
∀n = 2, 3, . . . , k.
(4.46)
Following the previous arguments we could set Rk−m,0 = −f1 − xf2 ,
Sm,0 = e2 + ye1 ,
(4.47)
substitution in (3.34) reproduces (3.36). Substitution in (4.45) gives [[D, e1 ] , f1 ] = Z,
[[D, e2 ] , f1 ] = 0 = [[D, e2 ] , f2 ] = [[D, e1 ] , f2 ] .
The last of the constrain equations reduces to Rk−m,0 , Sm,n = 0, n = 2, 3, . . . .
(4.48) (4.49)
Since [D, S] depends on y we assume a reduction of the algebra of the form [D, S] = yD.
(4.50)
Thus we have [D, e2 ] = 0 and [D, e1 ] = D. Defining the isomorphism D → E1 ,
e2 → E2 ,
−f1 → F1 ,
we have the commutation relations [Z, Ei ] = Ei , [Z, Fi ] = −Fi , [E2 , F1 ] = 0 = [E1 , F2 ] , [E1 , F1 ] = Z, [E2 , F2 ] = Z,
−f2 → F2 ,
(4.51)
(4.52)
(4.52) are exactly the commutation relations given in [6] and constitute a description of Kac’s algebra K2 , which is the most elementary example of a simple Lie algebra with infinite growth (see [14]). The K2 algebra was also encountered as the prolongation structure of the Robinson–Trautman equation by Bakas [9], using a somewhat different approach based on Toda theories. 5.
Conclusion We have shown the connection between the prolongation structures of the Robinson–Trautman equation of Petrov type III and a coupled Kac–Moody, Virasoro algebra. This is an interesting result since the representation theory of this algebra are better understood than their contragradient counterpart K2 which grows exponentially fast. Since these algebras emanate from the same differential ideal, we may conclude
62
E. O. IFIDON
that our method is capable of generating multiple prolongation structures belonging to the same differential ideal, thus enhancing the ability of the procedure to generate new, nontrivial, solutions to the RT equation. REFERENCES [1] C. Rogers and W. F. Shadwick: B¨acklund Transformations and Their Applications, Academic Press, New York 1982. [2] J. D. Finley, III: Estabrook–Wahlquist prolongations and infinite-dimensional algebras, in: Symmetry Methods in Physics, Vol. 1, Dubna 1995, p. 203–211. [3] R. K. Dodd and R. K. Bullough: Polynomial conserved densities for the sine-Gordon equations, in: Proc. R. Soc. Lond. A 352 (1977), 481–503. [4] B. Kent Harrison: The Differential Form Method for Finding Symmetries, in: Symmetry, Integrability and Geometry: Methods and Applications 1, (2005), 001. [5] J. D. Finley, III and John K. McIver: Infinite-dimensional Estabrook–Wahlquist prolongations for the sine-Gordon equation, J. Math. Phys. 36 (1995), 5707–5734. [6] J. D. Finley III: Robinson–Trautman Type III prolongation structure contains K2 , Commun. Math. Phys. 178 (1996), 375–390. [7] I. Robinson and A. Trautman: Some spherical gravitational waves in general relativity, in: Proceedings of the Royal Society of London. Series A, Mathematical and Physical, Vol. 265, No. 1323 (Feb. 6, 1962), pp. 463–473. [8] P. T. Chru´sciel: On the global structure of Robinson–Trautman space-times, in: Proceedings: Mathematical and Physical Sciences, Vol. 436, No. 1897 (Feb. 8, 1992). [9] I. Bakas: The algebraic structure of geometric flows in two dimensions, hep-th/0507284, CERN-PHTH/2005-134 (July 2005). [10] D. Kramer, H. Stephani, E. Herlt, M. MacCallum: Exact Solutions of Einstein’s Field Equations, Cambridge Univ. Press, Cambridge 1980. [11] I. Bakas: On the integrability of spherical gravitational waves in vacuum, gr-qc/0504130.CERN-PHTH/2005-069 (April 2005). [12] H. X. Yang and Y Li: Prolongation approach to B¨acklund transformation of Zhiber-Mikhailov-Shabat equation, J. Math. Phys. 7 (1996), 37. [13] M. Omote: Prolongation structures of nonlinear equations and infinite-dimensional algebras, J. Math. Phys. 27 (1986), 2853. [14] V. G. Kac: Infinite-Dimensional Lie Algebras, 3rd edition, Cambridge University Press, Cambridge, New York, 1990. [15] I. M. Gelfand and A. A. Kirillov: Fields associated with enveloping algebras of Lie algebras, Doklady 167 (1966), 407–409. [16] G. R. Krause and T. H. Lenagan: Growth of algebras and Gelfand–Kirillov dimension, Amer. Math. Soc. Grad. Studies in Math. 22 (2000).