Toda Equation Daniel Finley And

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GENERALIZED SYMMETRIES FOR THE SDIFF(2) TODA EQUATION

DANIEL FINLEY AND JOHN K. MCIVER University of New Mexico, Albuquerque, NM 87131, USA E-mail: ﬁ[email protected]

Self-dual Einstein metrics which admit one (rotational) symmetry vector are determined by solutions of the sDiff(2) Toda equation, which has also been studied in a variety of other physical contexts. Non- trivial solutions are difficult to obtain, with considerable effort in that direction recently. Therefore much effort has been involved with determining solutions with symmetries, and also with a specific lack of symmetries. The contact symmetries have been known for some time, and form an infinite- dimensional Lie algebra over the jet bundle of the equation. Generalizations of those symmetries to include derivatives of arbitrary order are often referred to as higher- or generalized-symmetries. Those symmetries are described, with the unexpected result that their existence also requires prolongations to “potentials” for the original dependent variables for the equation: potentials which are generalizations of those already usually introduced for this equation. Those prolongations are described, and the prolongations of the commutators for the symmetry generators are created. The generators so created form an infinite-dimensional, Abelian, Lie algebra, defined over these prolongations.

1 The sDiff(2) Toda equation, formulation for an h-space, a heaven, which and the standard Toda lattice is a 4-dimensional, complex manifold with a self-dual curvature tensor. He of course All self-dual vacuum solutions of the Einstein showed that such a space is determined ﬁeld equations that admit (at least) one rota- by a single function of 4 variables, Ω = tional Killing vector are determined by solu- Ω(p, p,¯ q, q), which must satisfy one con- tions of the sDiff(2) Toda equation, which straining pde, and then determines the met- may be written in various equivalent forms: ric via its second derivatives, as follows:

u,ss r g = 2(Ω,pp¯ dp dp¯ + Ω,pq dp dq u,qq + e = 0 ⇐⇒ r,qq + (e,s),s = 0 +Ω dq dp¯ + Ω dq dq) , v ,qp¯ ,qq ⇐⇒ v,qq + (e ),ss = 0 , v ≡ r,s ≡ u,ss . (1) Ω,pp¯Ω,qq − Ω,pqΩ,qp¯ = 1 . (2) How this comes about was shown by Charles Boyer and myself1 in 1982. [In Restricting attention to those complex spaces fact Plebańskiand I first wrote the equation that allow real metrics of Euclidean signa- down, for complex-valued, self-dual spaces in ture, there are only two possible “sorts” 1979.2] Since that time there has been con- of Killing vectors, “translations” and “ro- siderable interest in this equation, in general tations.” Noting that the covariant deriva- relativity, and also in some other fields of tive of any Killing tensor must be skew- physics and mathematics. Nonetheless, most symmetric, by virtue of Killing’s equations, currently known solutions describe metrics we may make this division more technical that also allow a translational Killing vec- by dividing the class of Killing vectors based tor. Such solutions do not provide much new, on this skew-symmetric tensor’s anti-self-dual real understanding of this equation since they part, which must be constant. For “trans- were susceptible to discovery by a much sim- lational” Killing vectors, this anti-self-dual pler route, as solutions of the 3-dimensional part vanishes, while it does not for the “rota- Laplace equation. tional” ones. The self-dual case—where the To understand how this process occurs, anti-self-dual part vanishes—has been com- we may begin with the standard Plebański3 pletely resolved.4,5 (In this case the con-

PlebTalkLatex: submitted to World Scientiﬁc on January 16, 2009 1 For Publisher’s use straining equation for Ω reduces simply to the to the hierarchy for the KP equation, which 3-dimensional Laplace equation.) contained operator realizations of sDiff(2). We continue by insisting that the space The name also emphasized its relationship to under study admit a rotational Killing vector, certain limits of the 2-dimensional Toda lat- ξe, we re-deﬁne the variables so that they are tice equations: adapted to it: a b b a K bu a a v u,xy = e or v,xy = K b e , e e ξ = i(p∂p − p∂¯ p¯) ≡ ∂φ , ξ(Ω) = 0 , a a b √ √ (v ≡ K b u ) , a, b = 1, 2, . . . , n , (7) p ≡ r eiφ , p¯ ≡ r e−iφ , (3) a where K b is the Cartan matrix for the Lie al- which changes the constraining equation as gebra which is also the generator of the sym- follows, construing Ω to now depend on the metries of these same Toda equations . variables {r, φ, q, q}: The hoped-for virtue of the relationship

(rΩ,r),rΩ,qq − r Ω,qrΩ,qr = 1 . (4) with the Toda lattice lay in the fact that the Toda lattice equations, when based on any It is however often more convenient to rewrite finite-dimensional, semi-simple algebra, has the constraining equation, and the metric, in symmetries which allow the determination terms of a new set of coordinates, obtained of Bäcklund transformations which generate from the original ones via a Legendre trans- new solutions from old ones. We first use form based on variables r and s ≡ rΩ,r. Tak- the gauge freedom in the original equations ing {s, q, q}, along with φ, as the new coor- to divide into two parts the unknown func- dinates, and v ≡ ln r as the function of these tions va ≡ aa + ba. Then for the case when coordinates that will generate the metric, we the Lie algebra is sl(n + 1), one finds that find the following new presentation, which the explicit first-order pde’s for the Bäcklund show the agreement with the sDiff(2) Toda transformation as the following, where the equation: wa = wa(x, y) are the “other” set of depen- −1 2 dent variables, i.e., the “pseudopotentials” g = V γ + V (dφ + ∼ω) , γ ≡ ds2 + 4 ev dq ∧ dq , (5) involved in the transformation: 1 ω i a a wa+ba wa−1+ba−1 V ≡ 2 v,s , ∼ ≡ 2 {v,qdq − v,qdq} , {w − a },x = −e + e , a a −wa+aa −wa+1+aa+1 and we still require that {w + b },y = e − e . (8) v v,qq + (e ),ss = 0 , (6) The zero-curvature conditions, for the 2 difference of the two cross-derivatives, then and ∗ (d∼ω) = −iV d(2s − 1/V ) . γ generate exactly the original Toda equations, The name I have used for Eq. (1) was first Eqs. (7), in the variables va, as desired, and used by Mikhail Saveliev,6 and also Kanehisa expected. Moreover, if one adds the two 7 a Takasaki and T. Takebe, emphasizing their cross-derivatives, inserts the form for b,xy understanding of its relationship to the alge- from the Toda equations, and changes to bra of all area-preserving diffeomorphisms of the new, translated pseudopotentials, `k ≡ a 2-surface. Saveliev and Vershik used8 this wk − wk+1 + ak+1 + bk, then these new de- equation as a non-trivial example of the use of pendent variables are also required to satisfy their development of continuum Lie algebras, the Toda equations, although for sl(n), since it having the symmetry group which was a there are only n − 1 of them: limit of An as n went to +∞. Takasaki and a a−1 a+1 ¡ ¢a b à = 2e` −e` − e` = e` , Takebe created a (double) hierarchy of equa- ,xy Kn−1 b tions connected with this equation, analogous a, b = 1, . . . , n − 1 . (9)

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2 The continuous limit of the Toda pseudopotentials wa, we acquire the following lattice equations limiting forms for the “proposed” Bäcklund transformations: Following the straight-forward existence of (W − A) = −∂ e+(W +B) , both soliton-type solutions and Bäcklund ,z s −(W −A) transformations of the Toda lattice equa- (W + B),z = −∂se . (13) tions, we, earlier, studied limits of these equa- However, the integrability conditions of tions to the continuous case, with the intent these equations are not what was desired: of course that these limits would carry over V = −∂ eV ∂ V = −∂2eV , (14) to the existence of Bäcklund transformations ,zz s s n s o V for our equation with three independent vari- and L,zz = −∂s e ∂sL , (15) ables. To accomplish the change from dis- L ≡ 2W + B − A. crete indices to functions of a (new) continuous variable, we begin with a new function, The first of these equations is of course V = V (z, z, s), that depends on a third con- what we expect; but the second is not. This tinuous variable, s, which varies, say, from 0 particular pair of equations is just the system 10 to β. We then superpose on these values for s of equations that LeBrun requires to de- a lattice of n points, a distance δ apart, fill in termine his “weak heavens,” which have only the space between the lattice points by tak- self-dual conformal curvature, and therefore ing the limit as n → ∞, with β fixed, which a possibly non-zero matter tensor. There is is the same as taking the limit as δ → 0, and quite a lot of interesting work on the complete following earlier work of Park,9 re-scale the resolution of this pair of equations; perhaps other continuous variables so that appropri- we ought to look at it as a system and “try ate differences of the exponentials of the va’s again”? Nonetheless, it certainly does not will create second derivatives with respect to create a Bäcklund equation for the original s: problem. ¯ ¯ At this point no progress has been V (z, z, s)¯ ≡ va(z/δ, z/δ) , s=aδ made toward the advertised goal, namely a = 1, . . . , n , δ = β/(n − 1) , (10) a method to “buy” some new solutions to [s/δ] the original sDiff(2) Toda equation from =⇒ V (z, z, s) ≡ lim v (z/δ, z/δ) , δ→0 old, previously-known ones, or, stated dif- where the square brackets indicate the inte- ferently, how to obtain families of metrics ger part of the quotient within them. This that are solutions to the stated problem in works well, in fact, although the ub’s need a general relativity using ones that were al- scaling of their own, to create their second ready known. It is also worth noting that derivatives: the approach via limits seemed desirable and ¯ ¯ interesting because the Estabrook-Wahlquist U(z, z, s)¯ ≡ δ2 {ua(z/δ, z/δ)} . (11) s=aδ method of finding prolongations, pseudopo- Assuming sufficient continuity of our func- tentials, Bäcklund transformations, etc.— tions, we may now take limits of the Toda which is our method of choice—has yet to equations, which give the desired results: find a truly effective generalization to prob- lems with more than two independent vari- U = e−U,ss ,V = −∂2eV . (12) ,zz ,zz s ables. It had indeed been hoped that a so- However, when we take the same lim- lution to this problem would allow an un- its on the prolongation equations themselves, derstanding of the 3-variable problem suffi- agreeing to treat the gauged parts, aa and cient to create a good generalization; unfor- ba, the same as their sum va, and also the tunately this has not (yet) occurred.

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Before continuing, this is perhaps a good been found by Atiyah,14 originally studying point to review some of the studies of this monopole solutions of the Yang-Mills equa- equation that have been made by several tions; this was then elaborated in some de- other groups. We have of course already men- tail by Olivier.15 The advantage of such a tioned Mikhail Saveliev and A.M. Vershik large symmetry group is that the pde’s are and their theory of continuum Lie algebras.8 reduced to ordinary differential equations in where the elements in the algebra might be just one independent variable. A some- labelled by continuous variables, as “indices,” what different approach, via (3-dimensional) instead of the more usual discrete indices. Einstein-Weyl spaces, has brought Tod,16 Their continuum approach to, say, the Lie and co-workers,17,18 to looking at other re- algebra A∞, would use “test functions” from ductions to simply ordinary differential equa- some appropriate function space to label the tions. Olivier’s equations involve elliptic elements of the algebra instead of discrete la- functions as solutions, while Tod’s go one bels. The result would then be the following step higher and involve Painlevétranscen- where X0(f) are elements of the (Abelian) dents. Cartan subalgebra, i.e., with grade 0, while Yet another approach is to simply look X±1(f) are elements of the first and minus- for ansätze that give non-trivial results. first grades, often referred to as Ei and Fj: Plebańskiand myself already put forward some very simple ansätzefor this equation, [X (f),X (g)] = ± X ((fg)0) , (16) 0 ±1 ±1 which did indeed demonstrate all possible [X+1(f),X−1(g)] = X0(fg) . (17) heavenly Petrov types; nonetheless, they This approach enabled them to write were not particularly inspired. In fact inter- down a form for a “general solution” for an esting ansätzeare rather difficult to acquire, initial-value problem for our equation, involv- since too much symmetry is easily acquired, ing choices of functions of two variables; un- reducing the problem to one that also con- fortunately, at least as we see it, this form is tains a self-dual Killing vector. However, one rather too formal, and has not yet been made plausible ansatz is obtained by requiring the practically useful. second term in the v-form of the equation to From different directions, R. be independent of s. This imposes the condi- v S. Ward,11,12,13 and K. Takasaki7 have cre- tion that e be a second-order polynomial in ated objects they refer to as Lax pairs for s, and creates a problem that can be resolved this equation, using Poisson brackets instead by simply solving some Liouville equations. of the usual commutators. However, the Lax This generates the result pairs involved to not seem to involve pseu- evC = +[s + G(q)][s + H(q)]α dopotentials, i.e., realizations of the group of A0(q)B0(q) = −[s + G(q)][s + H(q)] . symmetry involving new dependent variables. (A + B)2 We have therefore been unable to use them to (18) generate Bäcklund transformations, although they surely do generate an infinite hierarchy where α is the general solution of the Liou- of associated equations, in the spirit of the ville equation, and the four functions shown KP hierarchy. are arbitrary functions of one variable. It Although Boyer and myself showed that should of course be pointed out that the equa- the metrics did not admit just two, rota- tion has conformal symmetry so that two of tional symmetry vectors, it is certainly true these functions could be absorbed into new that there are metrics which admit an entire definitions of the original independent vari- sl(2) of such vectors. These have originally ables. By different methods this solution has

PlebTalkLatex: submitted to World Scientific on January 16, 2009 4 For Publisher’s use been published some 3 different times in the equations defining the co-coordinates. We 19 last few years. Calderbank and Tod found may then look for generators, ϕv, of symme- it (first) by imposing restrictions on the asso- tries involving any (finite) number of deriva- ciated Einstein-Weyl spaces. Martina, Shef- tives of the original variables, which must tel and Winternitz20 also found it by asking a satisfy the standard equation, which we take question related to an allusion above, namely from Vinogradov’s approach:21 by looking specifically for solutions without n v£ excess invariances. DqDq + e DsDs + 2 vs Ds o 2 ¤ +(vss + Ωs) ϕv = 0 . (20) 3 Generalized Symmetries of the Equation We were rather perplexed when explicit com- putations showed that there were no such ob- Eventually, following some helpful comments jects involving derivatives higher than first by M. Dunajski, we began an alternative at- order. (We actually did expect a symme- tempt to find a method to obtain new so- try algebra built on sDiff(2).) These first- lutions, which will indeed be the principal order ones were of course just the ordinary point of this talk. Unable to find proper (Lie) symmetries for the equation, published prolongation algebras for the equation, we at various times before:22 changed tactics and looked at the question ¯ of finding the algebra of generalized symme- ϕv = A(q) vq + A(q) vq + (α s + β)vs tries over the infinite jet over the pde. We +A,q(q) + A¯,q(q) − 2α , (21) begin our hunt for the generalized symmetries, as usual by considering the pde, in with the two arbitrary functions of 1 variable, the form with v = v(q, q, s), as a variety A(q) and A(q). When q and q are restricted in the second jet bundle, with coordinates to be complex conjugates of one another, originating from the original geometry of a {q, q, s, v, vq, vq, vs, vqq, vqs, vss, vq,s, vq,q} and Euclidean signature metric, then this pair co- co-coordinate defining the surface, vqq, determined in terms of the coordinates from the ordinate and define the well-understood con- pde. We then prolong that bundle to the infi- formal transformations of that underlying 2- nite jet, where the co-coordinates are chosen space. We record here their commutators: to be all “derivatives” of v that involve at {ϕα, ϕβ} = ϕβ , {ϕA, ϕA¯} = 0 , least one q and also one q, resolved from the equations created from all possible deriva- {ϕα, ϕA} = {ϕβ, ϕA} = 0 , tives of the original pde. On this infinite jet {ϕβ, ϕA¯} = {ϕα, ϕA¯} = 0 , (22) we use the usual total derivative operators in {ϕ , ϕ } = ϕ , each direction, of the form, for example, A1 A2 A1A2,q −A2A1,q {ϕ ¯ , ϕ ¯ } = ϕ ¯ ¯ ¯ ¯ . A1 A2 A1A2,q −A2A1,q Dq = ∂q + vq∂v + v,qq∂vq + vg,qq∂vq + v,qs∂vs The lack of any genuine symmetries at higher +v,qqq∂vqq + v,qqs∂vqs + vqss∂vss order than the first jet was eventually re- +v ∂ + v ∂ + ..., (19) qqs vqs qqq vqq solved by the introduction of “potentials” where the overbar on the derivative opera- into the jet bundle. This is perhaps not truly tor reminds us that this generic operator has surprising since our original presentation of been restricted to live on the variety which our equation gave not only the v-form of the defines the pde. Because of this, we then use equation, currently being considered, but also the “over-tilde” to indicate that this coeffi- two forms involving two different potential cient is to be determined from the constraint functions, r and u, defined such that r,s ≡ v

PlebTalkLatex: submitted to World Scientiﬁc on January 16, 2009 5 For Publisher’s use and u,ss ≡ v, believing them to be “equiva- ond option would say that, since this symme- lent” equations. Indeed, when we prolonged try generator involves a potential, and a pro- our jet bundle for v in these “integral di- longation of the jet bundle in that direction, rections,” such symmetry generators did in then we must also prolong the total deriva- fact appear. We intend now to ﬁrst write tive operators that appear in Eq. (23). This, down generators at the next two levels. How- indeed, is the correct choice, if we replace ever, to simplify the discussion, we note that, those total derivative operators, Di, by new modulo the well-understood conformal trans- ones, 1Dbi, prolonged with appropriate addi- formations depending on arbitrary functions, tional terms, then Q2 will indeed satisfy that the Lie symmetries could be thought of as be- prolonged version of the equation. We de- ing generated by the two transformations in note these prolonged operators also with a s and simply by vq and vq. Therefore, when pre-script 1 since there will be more: discussing the generalized ones we will also Db − D = r ∂ + r ∂ + r ∂ + ... think of them modulo those conformal sym- 1 q q q r qq rq qqq rqq +rf ∂ + rg∂ + ..., metries and therefore as generated by two se- qq rq qqq rqq quences of generators, beginning with vq and b 1Dq − Dq = r,q∂r + rqq∂rq + rqqq∂rqq + ... vq, respectively, which we will label as Q1 and +rfqq∂rq + rgqqq∂rqq + ..., (24) Q1. By explicit calculation we found the next Db − D = v∂ + v ∂ + v ∂ + ... two pairs, and display those three pairs here: 1 s s r q rq qq rqq

+vq∂rq + vqq∂rqq + .... −v v Q1 = vq = (rq)s = e (e )q , The same sort of thing happens, again, Q = v , 1 q of course, when we attempt to find the gen- 1 2 Q2 = rqq + rq vq = [uqq + 2 (rq) ]s eralized symmetry, Q3, which involves u in −v v its definition. We again prolong the underly- = e [e rq]q , 1 2 ing jet bundle and also the total derivative, Q2 = rqq + rq vq = [uqq + 2 (rq) ]s this time creating the quantities 2Dbi, with −v v = e [e rq]q , (23) coefficients that involve the various q- and q- −v v 2 Q3 = e [e (uqq + (rq) )]q , derivatives of u, but no mixed ones, and no s- −v v 2 derivatives, either, for they are expressible in Q3 = e [e (uqq + (rq) )]q . terms of the r’s which are already in the pro- It is probably important to emphasize longed bundle. As before Q3 is a symmetry at this point that, for instance, Q1 satisfies only for this re-definition of the requirements. the linearization equation, Eq. (23), just as it stands. However, Q2 does not, because it 4 Commutators for the Symmetry involves a potential for the pde. In principle, Generators one could imagine two different sorts of generalizations to that equation which might be The first pair of generalized symmetries re- appropriate for Q2. The first option would quired a potential at the level of a first “inte- say that, since it involves the potential r, we gral”; the the second pair needed a potential should just start the problem over again and at the second level of integration. These were look for symmetries of the defining pde for easy because they were already understood. r, and expect that this is what we should On the other hand, one needs yet a third level obtain. This is definitely not true. That of integration to acquire another pair of gen- pde suffers exactly the same deficit of gener- erators. The question immediately arises as alized symmetries, in its own right, as did our to how to choose these higher levels of po- original, equivalent equation for v. The sec- tentials. This question is of course closely

PlebTalkLatex: submitted to World Scientific on January 16, 2009 6 For Publisher’s use related to similar questions that occur in the X∞ b b −n study of the KP equation, for example, where −B = {+vn,q + B vn,s}L . (29) the standard (Japanese school) approach in- 1 volves an infinite hierarchy of dependent vari- The quantities B and Bb are the following ables all satisfying more- and more-involved short, finite series, while the Poisson bracket equations as one climbs upward in the hier- is in the variables s and the logarithm of the archy. Therefore we used as a guide the hier- spectral variable, p ≡ ln λ: archical approach to this equation taken by ev B = λ + u , Bb = , (30) Takasaki and Takebe, for which we now give 0 λ 7 a (very) brief description. They created a {A, B} ≡ λA,λB,s − λB,λA,s . pair of hierarchies that are associated with Comparing powers of λ gives several infinite the sDiff(2) Toda equation, which involve 4 sequences of pde’s, involving the u ’s, the infinite sequences of functions dependent on j v ’s, u = r and v. As expected the ear- our 3 independent variables, which we may k 0 q liest members of these sequences repeat the label as u , v , ub and vb , as i goes from 0 i i i i original pde’s, while we acquire more identifi- to +∞. The first pair are involved with the cations, such as u = u and v = u . More quantities that will create symmetries in the 1 qq 1 q specifically, they give the s- and q-derivatives q variables, while the second pair are involved of the u ’s in terms of derivatives of lower- with symmetries in the q variables. They are j order u ’s, and the s- and q-derivatives of then inserted as coefficients into two series k the v ’s in terms of the u ’s and lower-order in powers of a “spectral” variable, λ, which m k v ’s. These can be arranged to determine a act as generating functions for the entire se- n hierarchy of equations. quence of pde’s, written in a Poisson-bracket Since all the higher-numbered quantities format, as follows, where we write only the in these hierarchies are essentially potentials ones for the q variables, with the q ones be- for the lower ones, there were not unique ing completely analogous: choices for the desired extension. It turned −1 −2 −3 out that Takasaki’s quantities v seem to be a L ≡ λ + u0 + u1λ + u2λ + u3λ + ... j X∞ very good choice, however. We therefore have −i = λ + uiλ , u0 ≡ rq = vqs , (25) taken a renormalization of them for an infi- 0 nite sequence of potentials in the q-direction, ∞ X and a similar choice involving thev ˆ ’s for po- M ≡ qL + s + v L−n (26) k n tentials in the q-direction. There does not 1 −1 seem to be a single choice appropriate to both = qλ + s + q u0 + (qu1 + v1)λ + .... directions at once, as there was in the begin- These series must satisfy the following ning when we were using r and u. However, Poisson brackets (analogous to commutator this does not increase, noticeably, the total brackets), which generate infinite sequences size of the prolonged jet bundle. The rea- of relationships, equating powers of λ: son for this is that while, for instance, for u as a potential, we had to also append all b {B, L} = L,q , {B, L} = L,q , its q-derivatives and all its q-derivatives (but {L, M} = L , (27) not the mixed ones, nor the ones involving

{B, M} = M,q ⇐⇒ s-derivatives), for these new objects we need X∞ only one set, and not the others. More par- −n L − λ = {−vn,q + λ vn,s}L , (28) ticularly, since v1 may be identiﬁed with uq, 1 we really begin with v2. Labelling the ele- {Bb, M} = M,q ⇐⇒ ments of this sequence of potentials by xi,

PlebTalkLatex: submitted to World Scientific on January 16, 2009 7 For Publisher’s use and the corresponding q-type potentials by other hand, they do have several other rather yj, we have the following: unexpected properties. Each of the general- ½ 1 2 ized symmetry generators can be written as a 1 x2,s = uqq + 2 (rq) , x2 ≡ 2 v2 =⇒ v (31) perfect s-derivative; moreover, each of them x2,q = −rq e . ½ 1 2 may also be written in two different ways in 1 y2,s = uqq + 2 (rq) , y2 ≡ 2 vˆ2 =⇒ v (32) terms of second derivatives of the correspond- y2,q = −r e . q ingly numbered new potential: As can be seen we already know both the −v s-derivatives and the q-derivatives of x2, −e DqDq xj = Qj = Ds{Ds(xj)} ; (39) so that only the infinite sequence of q- Therefore, we may write, for each j, the cor- derivatives, and the q-derivatives of y2, must responding “linear” pde: be appended to the list of coordinates for the prolonged infinite jet bundle. v xj,qq + e xj,ss . (40) We now note a few more of the q-forms of these potentials, and then explain reasons That the symmetry generators may be writ- why they are good choices: ten in terms of second derivatives of poten- ½ 1 3 tials is, after the fact, not too surprising, for x3,s = x2,q + rquqq + 3 (uq) , x3 : 2 v (33) reasons which will be explained shortly. On x3,q = −[u,qq + (rq) ] e ,  2 the other hand, that each of those potentials  x4,s = x3,q + rqx2,q + uqq(rq) satisfies this linear equation similar to the Le- x : + 1 (u )2 + 1 (r )4 , (34) 4  2 qq 4 q Brun monopole equation was certainly unex- x = −[x + 2r u + (r )3] ev . 4,q 2,q q qq q pected. (The statement that it is linear is Now, of course, after having added appropri- of course slightly misleading insofar as the ate dimensions to the jet bundle as already qj’s are potentials for the unknown function discussed, and after having additional appro- v that also appears within the equation.) priate terms to the yet-again-prolonged total A last comment relevant to the choice derivative operators, these potentials must of potentials for this problem returns to be suitable to generate additional generalized Takasaki’s approach to the hierarchy of de- symmetries for our equation: indeed we may pendent functions and corresponding pde’s write them in terms of these quantities, one that they satisfy. In this hierarchical ap- new generator for each new potential: proach it is usual to also introduce an addi- −v v 3 Q4 = e {e [x2,q + 2rquqq + (rq) ]}q , (35) tional infinite sequence, of independent vari- −v v 3 ables, on which the various functions may Q4 = e {e [y2,q + 2rquqq + (rq) ]}q , (36) depend. As the equations in the hierarchy Q = e−v{ev[x + 2r x + (u )2 5 3,q q 2,q qq may be satisfied simultaneously they consti- 2 4 +3 uqq(rq) + (rq) ]}q , (37) tute distinct, commuting flows over the solu- −v v Q6 = e {e [x4,q + 2 rq x3,q tion manifold, so that these new independent 2 2 variables may be thought of as the flow pa- +(2uqq + 3rq )x2,q + 3(uqq) rq 3 5 rameters along the curves described by the +4uqq(rq) + (rq) ]}q . (38) flows. Takasaki and Takebe refer to these ... additional independent variables by qm and Each of these satisfies the appropriate prolon- qm, for m = 1..∞, and give generalizations of gation of the Vinogradov equation for sym- the Poisson-bracket equations above that ap- metry generators. They have quite interest- ply to them. It is then also common in such ing structure, and we presume that a recur- descriptions to determine a τ-function that sion process may be defined for them, al- depends on the entire infinite set of indepen- though this has not yet been found. On the dent variables, and allows one to determine

PlebTalkLatex: submitted to World Scientific on January 16, 2009 8 For Publisher’s use all the other dependent variables from that. bly just 0, that we refer to as the commutator In their description, the various derivatives of of the two solutions because the vector field the logarithm of that τ-function, with respect that it generates is the vector field commuta- to the variables qi and qj are just these quan- tor of the two vector fields generated by the tities vi andv ˆj; i.e., ∂(log τ)/∂qi ∝ vi ∝ xi, initial pair of symmetries. This commutator making it seem rather more reasonable that is specified by a Poisson-bracket sort of re- these functions would indeed to “potentials” lationship: given two symmetry generators, to describe the desired properties of the solu- φ and ψ, then the one they determine is η, tion space. given by

To return to the symmetries now, and µ µ µ µ consider why it is not too surprising that η = {φ, ψ} ≡ Zφ(ψ ) − Zψ(φ ) . (42) these expressions may be written as perfect s- For symmetry generators on the va- derivatives, we must first re-visit the notion riety over our jet bundle defined by the that they are generators for symmetries. If pde and all its derivatives, the coordi- the symmetries were written in terms of their nates in use are the sets {v, vq, vqq,...}, associated vector fields, over the jet bundle, {vq, vqq,...}, {vs, vss,...}, {vsq,...}, and then we would expect to consider the stan- {vsq ,...}. Therefore, for a function, say Q, dard commutators, i.e., Lie brackets, of two defined over this variety, we can say that a of them, and insist that they close onto them- more explicit form for the Z operator will be selves. Since we are describing the symme- n 2 tries in terms of their generators, instead of ZQ = Q∂v + {Dq(Q)}∂vq + {Dq(Q)}∂vqq their vector fields, there must be an asso- + ... ciated mapping of the generators that ac- 2 +{D (Q)}∂ + {D (Q)}∂ + ... complishes the same thing, i.e., a realiza- q vq q vqq tion of the Lie bracket in the underlying, ab- +{Ds(Q)}∂vs + {DsDq(Q)}∂vsq stract algebra. Our approach to this realiza- 2 +{DsDq(Q)}∂vsq + {Ds(Q)}∂vss tion is via the universal linearization opera- o tor, which was also used to create the (Vino- + ... . (43) gradov) equation that must be satisfied by a However, for our situation there are not very symmetry generator. We define a linear op- many interesting places to apply this since erator for functions on the infinite jet bundle we have only just Q and Q as symmetry and then restrict it to the variety defined by 1 1 generators defined on the variety over the some system of pde’s, F , which are resolved original jet bundle. All the other symmetry by some system of functions uν = uν (xa): n generators require this potentialization of the ν ν ≡ φ ∂ ν + {D (φ )}∂ ν bundle, described above. As already noted, Zφ u a ua o this potentialization requires a prolongation ν +{D D (φ )}∂ ν + ... a b uab of the total derivative operator. However, it X(∞) n o also requires a prolongation of the lineariza- ν ≡ D (φ ) ∂uν , (41) tion operator, since the functions involved are (σ) (σ) σ=0 now defined over a rather larger space. where the sum is over all “multi-indices.” A convenient approach to determine how The Vinogradov equation, which determines this should be done is accomplished by first a system defining a symmetry φν of F , is retreating somewhat, and “deriving” the ex- simply that Zφ(F ) = 0. In general, given pression involving the linearization operator, two such solutions, i.e., two such symmetries, using the Frechét(or Gateaux) derivative on then they determine a third solution, possi- function spaces. If σ is a function over some

PlebTalkLatex: submitted to World Scientific on January 16, 2009 9 For Publisher’s use space of functions, and φ and ψ are func- −1 v, i.e., v = Ds(r), or r ≡ Ds (v). There- tions in that space, then we may of course fore the prolongation of the linearization op- talk about σ(φ) or σ(φ + ²ψ). The Frechét b erator Zψ, which we would denote by Zψ, derivative, of σ in the direction ψ is then the −1 should have an extra term {D ψ}∂ . How- function on the function space, σ0[ψ], which s r ever, the existence of this new coordinate, v, is that part of σ(φ + ²ψ) that is linear in ², also generates its ordinary derivatives as well; divided by ²: σ(φ + ²ψ) = σ(φ) + ²σ0[ψ](φ) + therefore, we also need new coordinates on O(²2). Applying this now to functions over the jet of the form D r, D r, etc., while D r our infinite jet, we usually think of σ as de- q qq s would simply be v, and therefore not gener- pending on every one of the coordinates on a ate any new jet coordinate. The prolonga- the jet, {x , v, va, vab, vabc,... }. Now, when tion b may then be expressed in the following we imagine translating v by some very small Z way, where we use Q as a reasonable exam- amount, in some direction, say by ²ψ, deter- 2 ple function for it, and we use the pre-script 1 mined by ψ, some other function on the jet, to indicate that this is simply the first of sev- we need to have a method to determine how eral prolongations that we will have to make: this translation affects each of the other jet coordinates. To do this we consider an opera- b −1 1ZQ2 = ZQ2 + {Ds Q2} ∂r tor that “creates” the appropriate jet variable −1 2 −1 +{DqDs Q2} ∂rq + {DqDs Q2} ∂rqq from q, such as qab = Dxa Dxb q, and then let that operate on the translated version of q: + ... −1 2 −1 +{DqDs Q2} ∂rq + {DqDs Q2} ∂rqq σ(q, q , q , q ,... ) −→ a ab abc + .... (45) σ(q + ²ψ, qa + ²Daψ, qab + ²DaDbψ, Going on toward the symmetry Q3, we q + ²D D D ψ, . . . ) abcn a b c have introduced yet a new potential, u = −2 = σ(q, qa,...) + ² (ψ) σq + (Daψ) σqa Ds (v), and, as discussed earlier, all of its o unmixed q- and q-derivatives. We may then +(D D ψ) σ + ... + O(²2) a b qab use the same approach as above for the next

≡ σ(q, . . .) + ²Zψ(σ) + ..., (44) prolongation of the linearization operator: −2 Zb = Zb + {D Q } ∂ where the last equality follows from the def- 2 Q3 1 Q3 s 3 u −2 2 −2 inition of the linearization operator given +{DqDs Q3} ∂uq + {DqDs Q3} ∂uqq above. This allows us to see that the Frechét + ... derivative, of a function over a function space, −2 2 −2 +{D D Q } ∂ + {D D Q } ∂ is the complete analogue of the lineariza- q s 3 uq q s 3 rqq tion operator acting on functions over a jet + .... (46) bundle. However, our “philosophical” under- Proceeding onward with the strings of poten- standing of the one concept can assist us in tials that we need, from Section III, the next determining the correct prolongation of the one is considerably more complicated, being other. This is especially because our intro- −1 1 2 nonlinear: x2 = Ds (uqq + 2 (rq) ). This duction of various potentials into the larger time the explicit operator that acts on v to jet bundle requires us to now consider func- create x2 is nonlinear. Nonetheless, we create tions that also depend on some of these “in- it, replace v by v+²Ψ, and then find the first- tegrals” of jet coordinates. The simplest case term in ² in this process; i.e., we linearize it is just the one where our function, like the by finding the first functional derivative: symmetry generator Q2, depends on the first n h i2o −1 −2 2 1 −1 integral, r, of the original dependent variable x2 = Ds Ds Dq v + 2 Ds Dq v

PlebTalkLatex: submitted to World Scientiﬁc on January 16, 2009 10 For Publisher’s use

2 ≡ x2(v) , (47) +{DqY3(ψ)}∂y3,qq + .... (52) =⇒ x (v + ²ψ) − x (v) n2 2 h io Having now prolonged the linearization −3 2 −1 −1 = ² Ds Dqψ + Ds rqDs Dqψ operator to include these new, nonlinear po- + O(²2) tentials as well, the calculation of a commutator of two symmetries requires that both ≡ ²X (ψ) + O(²2) . (48) 2 of those symmetries admit these various in- We also recall that this potential only tegrals themselves: a first, and a second, inte- needs us to introduce the set of all of its gral, with respect to s, and then accept rather q-derivatives, as additional jet variables, all more complicated things in the cases for x2 other “derivatives” already being functions of and x3. This is surely not the case for ev- coordinates on the bundle. However, as com- ery function on the jet bundle; however, we pensation for this, at this level, we also need are really only interested in applying them −1 to introduce the potential y2 = Ds (uqq + to the various symmetry generators, such as 1 2 2 (rq) ), and all of its q-derivatives. Qj and Qj. This of course now, finally, tells At the next level, we proceed similarly: us why we should have not have been sur- n −2 −2 3 −1 prised to learn that each of the symmetry x = D D D v + 2(D D v) 3 s s q s q generators could be conceived of as a per- −1 2 −2 2 (Ds D v) + (Dq v)[Ds D v fect s-derivative, and even a perfect second q o q −1 2 s-derivative. Since this is true, let us use the +(Ds Dq v) ] ≡ x3(v) , (49) symbol ηj for the first s-integrals of the sym- =⇒ x (v + ²ψ) − x (v) metry generators, x , and we can write out 3 n 3 j −2 −2 3 −2 2 some preliminary general cases of the action = ²Ds Ds D ψ + 2rqDs D ψ q q of the higher-level operators on the symmetry −2 +2r D D ψ generators, and also some examples: h qq s q −2 2 −2 2 +vq Ds Dqψ + 2rqDs Dqψ Qj = Ds(ηj) ≡ Ds(xj) , (53) io −1 2 2 X (Q ) = D {x + r η } , (54) +(uqq + (rq) )Dqψ + O(² ) 2 j sn j,qq q j,q −2 2 X3(Qj) = Ds xj,qqq + 2(rqηj,q)q ≡ ²X3(ψ) + O(² ) . (50) +vq(xj,qq + 2rqηj,q) For the analogous operations based on the yj o 2 variables, and relevant to the Qj symmetry +(uqq + (rq) )Qj,x ; (55) generators, using q-derivatives, we will use ..., the notation Y , analogous to the X oper- j j X (Q ) = x , ators above. Then we may explicitly write 2 1 2,q 1 2 down the form of the further prolongations X2(Q2) = x3,q − 2 (uqq) , of the linearization operator: X2(Q3) = x4,q − uqqx2,q ,..., (56)

b b X3(Q1) = x3,q ,.... (57) 3Zψ = 2Zψ + X2(ψ)∂x2 + {DqX2(ψ)}∂x2,q 2 This structure is important in under- +{DqX2(ψ)}∂x2,qq + ... standing that everything displayed is actu- +Y2(ψ)∂y + {DqY2(ψ)}∂y 2 2,q ally what was wanted, namely generalized 2 +{DqY2(ψ)}∂y2,qq + ..., (51) symmetries. In addition, we have succeeded b b in understanding how to “drag along” the 4Zψ = 3Zψ + X3(ψ)∂x3 + {DqX3(ψ)}∂x3,q 2 prolongations of various important operators +{DqX3(ψ)}∂x3,qq + ... when we prolong the standard inﬁnite jet

+Y3(ψ)∂y3 + {DqY3(ψ)}∂y3,q bundle. Nonetheless, there are two rather

PlebTalkLatex: submitted to World Scientific on January 16, 2009 11 For Publisher’s use distressing difficulties with it. All the dis- 2. J.D. Finley, III and J.F. Plebański, J. played symmetries are part of a commut- Math. Phys.20, 1938 (1979). ing hierarchy; i.e., they commute as vector 3. J.F. Plebański, J. Math. Phys.12, 2395 fields, so that all the commutators are in fact (1975). zero. This structure then asserts that it has 4. K.P. Tod and R.S. Ward, Proc. Roy. been created correctly, but it does not help Soc. A 368, 411 (1979). you in finding more details, since it contains 5. G.W. Gibbons and Malcolm J. Perry, no recursion operators. One must retreat to Phys. Rev. D 22, 313 (1980). the generating equations again, presumably, 6. R.M. Kashaev, M.V. Saveliev, S.A. to determine algorithms that concisely show Savelieva, and A.M. Vershik in Ideas what the form of the n-th symmetry genera- and Methods in Mathematical Analysis, tor is. Or, there may be some other approach Stochastic, and Applications, ed. S. Al- to finding details of recursion operators. beverio, J.E. Fenstad, H. Holden and T. A second, current difficulty is that the Lindstrom, Vol. 1, (Cambridge Univ. intended purpose of calculating the symme- Press, Cambridge, UK, 1992). try generators is to use them as a tool, or 7. Kanehisa Takasaki and Takashi Takebe, guide, to methods to reduce the difficulty of Lett. Math. Phys.23, 205 (1991). determining solutions of the original pde. For 8. M.V. Saveliev and A.M. Vershik, Phys. example, taking the first pair of symmetry Lett. A143, 121 (1990) is a good place generators, vq and vq, we may use the vanish- to begin. ing of one of them alone, or a linear relation 9. Q.H. Park, Phys. Lett. B236, 429 between them to lower the dimension of the (1990). searched-for solution space. [Any of those re- 10. C.R. LeBrun, J. Diff. Geom. 34, 223 quirements are equivalent, under the confor- (1991). mal symmetries of the equation.] One then 11. R.S. Ward in Springer Lecture Notes in hopes that similar use of the generalized sym- Physics Vol. 280 (1987). metries would also help solve the original pde. 12. R.S. Ward, Phys. Lett. B234, 81 (1990). So far, we have been unable to find anything 13. R.S. Ward, J. Geom. Phys. 8, 317 new there. It is true that one can locate, (1992). again, the solution of the form of a quadratic 14. M.F. Atiyah and N.J. Hitchin, Phys. polynomial in s for ev, but this is not par- Lett. A107, 21 (1985). ticularly exciting. Lastly, in the study of the 15. D. Olivier, Gen. Rel. Grav.23, 1349 KP hierarchy there are methods to determine (1991). solutions that originate in nice behavior of 16. K.P. Tod, Cl. Qu. Grav.12, 1535 (1995). the τ-function. When looked at fairly care- 17. Ian A.B. Strachan, Math. Proc. Cambr. fully none of those methods for the Toda lat- Philos. Soc.115, 513 (1994). tice equations have reasonable limits for this 18. H. Pedersen and Y. Poon, Cl. Qu. problem. As well, Takasaki’s τ-function ap- Grav.7, 1007 (1990). pears to be simply our potential function, u, 19. David M.J. Calderbank and Paul Tod, which satisfies an equation equivalent to the Diff. Geom. Appl. 14, 199 (2001). original pde, which also does not help. 20. L. Martina, M.B. Sheftel and P. Winter- nitz, J. Phys. A 34, 9243 (2001). 21. A.M. Vinogradov, Acta Appl. Math.2, References 21 (1984). 22. D. Levi and P. Winternitz, Phys. Lett. 1. C.P. Boyer and J.D. Finley, III, J. Math. A 152, 335 (1991). Phys.23, 1126 (1982).

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