The supersymmetric Camassa-Holm equation and geodesic flow on the superconformal group

Transcription

1 J. Math. Phys. # Third revision, October 000 The supersymmetric Camassa-Holm equation and geodesic flow on the superconformal group Chandrashekar Devchand 1, and Jeremy Schiff 3 1 Max-Planck-Institut für Mathematik Postfach 780, 5307 Bonn, Germany Max-Planck-Institut für Mathematik in den Naturwissenschaften Inselstraße -6, Leipzig, Germany devchand@mpim-bonn.mpg.de 3 Department of Mathematics and Computer Science Bar Ilan University, Ramat Gan 5900, Israel schiff@math.biu.ac.il Abstract We study a family of fermionic extensions of the Camassa-Holm equation. Within this family we identify three interesting classes: a equations, which are inherently hamiltonian, describing geodesic flow with respect to an H 1 metric on the group of superconformal transformations in two dimensions, b equations which are hamiltonian with respect to a different hamiltonian structure and c supersymmetric equations. Classes a and b have no intersection, but the intersection of classes a and c gives a system with interesting integrability properties. We demonstrate the Painlevé property for some simple but nontrivial reductions of this system, and also discuss peakon-type solutions.

2 I Introduction Recently there has been substantial interest in the Camassa-Holm CH equation [1, ]: u t νu xxt = κu x 3uu x + νuu xxx + u x u xx. 1 This equation has been proposed as a model for shallow water waves. It is believed to be integrable, having bihamiltonian structure, as was first observed by Fokas and Fuchssteiner [3] 1 years prior to Camassa and Holm s work. Due to the nonlinear dispersion term, uu xxx, it exhibits more general wave phenomena than other integrable water wave equations such as KdV. In particular, when κ = 0 it admits a class of nonanalytic weak solutions known as peakons, as well as finite time blow-up of classical solutions [1]. Geometrically, the relationship of CH to KdV is rather deeper: Both are regularisations of the Euler equation for a one dimensional compressible fluid Monge or inviscid Burgers equation, u t = 3uu x. A solution to this equation describes a geodesic on the group of diffeomorphisms of the circle DiffS 1 [4] with respect to a right-invariant metric induced by an L norm, u dx, on the associated Lie algebra. If the group is centrally extended to the Bott-Virasoro group, the KdV equation arises [5, 6, 7, 8]. On the other hand, if the metric is changed to one induced by an H 1 norm, u +νu xdx, the CH equation arises [9, 10, 11]. Both these deformations have a regularising effect on solutions of, which exhibit discontinuous shocks. Thus KdV and CH arise in a unified geometric setting; both are geodesic flows which are integrable systems. Here, and henceforth in this paper, when we refer to a geodesic flow we mean the evolutionary PDE which can be formally associated in the manner we will see in section II with any inner product on the Lie algebra of a diffeomorphism group, and which, at least in the cases mentioned above, is known to describe geodesic flow, in the usual sense of the phrase, with respect to the correpsonding right-invariant metric on the group. In the case of a general inner product, the existence of the corresponding geodesic flow, in the usual sense of the phrase, is highly non-trivial. The following important question arises: What features of the underlying geometry give rise to integrability? In general, geodesic flows are not integrable: the Euler equation for fluid flow in more than one spatial dimension is an example [4]. Indeed, for the latter, Arnold has suggested a relationship between negative sectional curvatures and non-predictability of the flow. We feel that it ought to be possible to identify some other geometric property that causes integrability. In a remarkable recent paper [1], Fringer and Holm have shown that certain features usually considered to be 1

3 hallmarks of integrable systems, such as elastic scattering and asymptotic sorting according to height, in fact appear in geodesic flows on DiffS 1 with respect to a large class of metrics. Thus, there may well be a hierarchy of geometric structures corresponding to various degrees of integrability. One further example of an integrable bihamiltonian system arising as a geodesic flow has been discussed by Ovsienko and Khesin [5]. Using the superconformal group with an L type metric, they obtained the so-called kuperkdv system of Kupershmidt [13]. This is a fermionic extension of KdV: it describes evolution of functions valued in the odd or even parts of a grassmann algebra. In fact, as we will see below, taking a general L type metric on the superconformal group gives rise to a one parameter family of fermionic extensions of KdV, which includes not only kuperkdv, but also the superkdv system of Mathieu and Manin-Radul [14, 15]. The latter is integrable: it has only a single hamiltonian structure, but unlike kuperkdv it is supersymmetric, a property which is widely believed to contribute to integrability. It remains a mystery as to why, of the one parameter family of geodesic flows associated with L type metrics on the superconformal group, only two specific choices of the parameter give rise to integrable systems. The main purpose of this paper is to investigate geodesic flows obtained from H 1 type norms on the superconformal group; more generally we consider the following family of fermionic extensions of CH: u t νu xxt = κ 1 u x + κ u xxx + β 1 uu x + β u x u xx + β 3 uu xxx + γ 1 ξξ xx + γ ξ x ξ xxx + γ 3 ξξ xxxx ξ t µξ xxt = σ 1 ξ x + σ ξ xxx + ǫ 1 u x ξ + ǫ uξ x + ρ 1 uξ xxx + ρ u x ξ xx + ρ 3 u xx ξ x + ρ 4 u xxx ξ. 3 Here ux, t and ξx, t are fields valued, respectively, in the even and odd parts of a grassmann algebra, and {ν, µ, κ 1, κ, β 1, β, β 3, γ 1, γ, γ 3, σ 1, σ, ǫ 1, ǫ, ρ 1, ρ, ρ 3, ρ 4 } are parameters. By rescaling u and ξ it is possible to set β 1 = 3 and γ 1 = assuming that they are nonzero, and we shall do this throughout. In addition it is possible to eliminate up to two further parameters by rescaling the coordinates x, t. We derive three interesting classes of systems of the form 3. In section, we consider geodesic flows on the superconformal group with an H 1 type metric; the resulting systems have a natural hamiltonian structure, or more precisely, since the fields are grassmann algebra valued, a graded hamiltonian structure. In section 3 we identify a class of systems having a different hamiltonian structure. Unfortunately the latter has no intersection with the class of section, so there does not seem to be a bihamiltonian fermionic extension of CH. In section 4 we consider systems of the form 3 that are invariant under supersymmetry

4 transformations between u and ξ. This class has nontrivial intersections with both the classes of sections and 3. In particular there is a unique supersymmetric geodesic flow which is a candidate for being a new integrable system. We call this equation superch. In section 5 we show that two reductions of superch have the Painlevé property, which is positive evidence for integrability. In section 6 we look for peakon-type solutions of superch; as for CH, multipeakon solutions arise from the solutions of a system of ODEs, but the integrability of this unfortunately remains unclear. SuperCH is a supersymmetric geodesic flow whose bosonic part is integrable. While in this paper we do not fully establish integrability of superch, we regard it as an interesting test case to determine whether in general supersymmetric geodesic flows with integrable bosonic parts must be integrable. A trivial integrable CH system of the form 3, which is not incorporated in the classes of sections,3, and 4, and which we shall not discuss further, is the odd linearisation of the bosonic CH system 1 u t νu xxt = κu x 3uu x + νuu xxx + u x u xx, ξ t νξ xxt = κξ x 3ξu x + νξu xxx + uξ xxx + ξ x u x x. 4 Replacing u by u + κ and considering the limit ν 0, κ, with νκ = 3, yields the 3 system u t = 3uu x + u xxx, 5 ξ t = 3ξu x + ξ xxx. This trivial fermionic extension of KdV has appeared often in the literature see e.g. [14]. II Geodesic flows on the superconformal group An inner product.,. on a Lie algebra g determines a right or a left invariant metric on the corresponding Lie group G. The equation of geodesic motion on G with respect to this metric is determined as follows [4]. Define a bilinear operator B : g g g by [V, W], U = W, BU, V, W g. 6 Then geodesics are determined by solutions of the geodesic flow U t = BU, U, 7 3

5 In our case, g is the NSR superconformal algebra, consisting of triples ux, ϕx, a, where u is a bosonic field, ϕ is a fermionic field and a is a constant. The Lie bracket is given by [ u, ϕ, a, v, ψ, b ] = uv x u x v+ 1ϕψ, uψ x 1u xψ ϕ x v+ 1ϕv x, dxc 1 u x v xx +c uv x +c 1 ϕ x ψ x + c 4 ϕψ, where c 1, c are constants. On this algebra, an H 1 type inner product is given by u, ϕ, a, v, ψ, b = = dx uv + νu x v x + αϕ 1 x ψ + αµϕ xψ + ab 8 dx u 0 v + ϕ 1 ψ + ab, 9 where 0 = 1 ν x, 1 = α 1 x µ x, 10 and µ, ν, α are further constants, all assumed nonzero. See [5] for the definition of the natural fermionic extension of the standard L inner product, to which the above reduces if µ = ν = 0. The natural fermionic extension of the standard H 1 inner product is constructed, as for pure bosonic systems, by taking the sum of the L inner product for the functions involved with the L inner product for the derivatives of the functions involved. Writing U=u, ϕ, a, V =v, ψ, b, we find BU, V = B 0, B 1, 0, where 0 B 0 U, V = v x 0 u + v 0 u x + 3 ψ x 1 ϕ + 1 ψ 1ϕ x + ac1 v xxx c v x, 1 B 1 U, V = 3 v x 1 ϕ + v 1 ϕ x + 1 ψ 0u + ac 1 ψ xx c 4 ψ. 11 The geodesic flows are therefore conveniently written in the form 0 u t = 0 B 0 U, U 0 ϕ t = 1 B 1 U, U 1 a t = 0. Writing ϕ = λξ x, where λ is a constant satisfying λ = 4, this yields the system 3α u t νu xxt = κ 1 u x + κ u xxx 3uu x + νuu xxx + u x u xx + ξξ xx + µ 3 ξ xξ xxx, ξ t µξ xxt = κ 1 4α ξ x + κ α ξ xxx 3 u xξ α uξ x + µuξ xxx + 3µ u xξ xx + ν α u xxξ x. 13 Here κ 1, κ are independent parameters determined by a, c 1, c. This is evidently a 5 parameter class of systems of type 3. 4

6 Setting ξ to zero in 13 yields the CH result of [9, 10, 11]. If instead we choose µ, ν to vanish, the H 1 norm becomes an L norm; then choosing κ 1 to be zero and rescaling κ to 1 we obtain the following 1 parameter fermionic extension of KdV: u t = u xxx 3uu x + ξξ xx, ξ t = 1 α ξ xxx 3 u xξ α uξ x. 14 Modulo rescalings, the superkdv of Mathieu and Manin-Radul is obtained by taking α = 1. The kuperkdv system arises by taking α = 1, the choice made in [5]. Other values of the 4 parameters give systems which are not believed to be integrable see however [16]. III Hamiltonian equations Like KdV, CH has bihamiltonian structure, and this accounts for its integrability. We might hope that for some choices of parameters the system 13 should also have a bihamiltonian structure. One hamiltonian structure follows automatically from the geometric origins of the system [4]. Explicitly, introducing new variables, m = u νu xx and η = ξ µξ xx, 13 takes the form δh mt δm = P δh η t δη 15 where κ x 3 P = 1 1 x x m m x xη + η x x η 1η 3 x 4α κ κ x 3 8α 16 and the hamiltonian functional is given succinctly by the H 1 inner product on the algebra, H = 1 U, U = 1 dx u + νu x ξ xξ + µξ xx ξ x. 17 This generalises the so-called second Hamiltonian structure of KdV and its fermionic extensions [13, 14]. Checking 15 is straightforward: the Euler-Lagrange derivatives δh, δh are δm δη defined by δh δh = dx δm δm + δh δη δη, 18 from which it follows immediately that δh δm = u and δh δη = 4 3 ξ x. To investigate the possibility of systems amongst 13 having another hamiltonian form, we look at systems of the form mt η t δh1 δm = P 1 δh 1 δη 5, 19

7 where x 1 ν x 0 P 1 = 0 ǫ 1 1 µ x. 0 Here ǫ 1 is a constant and H 1 is a functional generalising the KdV first Hamiltonian, H 1 = dx 1 u3 β 3 uu x κ u x + κ 1 u + σ 1 ǫ 1 ξξ x + σ ǫ 1 ξξ xxx +uξξ x + γ γ 3 uξ x ξ xx + γ 3 uξξ xxx. 1 This is the most general functional of this type, up to rescalings of u and ξ. Since δm=1 ν xδu, we have 1 ν x δh 1 = δh 1, and similarly 1 δm δu µ x δh 1 = δh 1. Thus equations δη δξ 19 take the simple form u t νu xxt = δh1 x δu = κ 1 u x + κ u xxx 3uu x + β 3 u x u xx + uu xxx + ξξ xx + γ ξ x ξ xxx + γ 3 ξξ xxxx ξ t µξ xxt = ǫ δh1 1 δξ = σ 1 ξ x + σ ξ xxx + ǫ 1 u x ξ + uξ x + ǫ 1 γ 3 γ uξ xxx + 3 ǫ 1γ 3 γ u x ξ xx + 1 ǫ 14γ 3 γ u xx ξ x + 1 ǫ 1γ 3 u xxx ξ. This is a 10 parameter class of systems of the form 3. Comparing with 13, we see that the only bihamiltonian systems occur when {µ=ν=β 3 =γ =γ 3 =0, ǫ 1 = 3, σ 1=κ 1, σ =4κ }, which is equivalent to 13 with {µ=ν=0, α= 1 }, i.e. the kuperkdv system. Thus, no new 4 bihamiltonian systems arise. We note that the systems can be obtained from a Lagrangian. Introducing a potential f defined by u=f x, they are Euler-Lagrange equations for the functional L = dx 1 f x νf xxx f t + 1 ǫ 1 ξ µξ xx ξ t + 1 f3 x + β 3 f xfxx + κ fxx κ 1 fx σ 1 ǫ 1 ξξ x σ ǫ 1 ξξ xxx f x ξξ x + γ 3 γ f x ξ x ξ xx γ 3 f x ξξ xxx. 3 IV Supersymmetric equations Define a fermionic superfield Φx, ϑ = sξ + ϑu and superderivative D = ϑ + ϑ x, where s is a nonzero parameter and ϑ is an odd coordinate. The most general superfield equation having component content of the form 3 is the 8 parameter system, 1 νd 4 Φ t = κ 1 D Φ + κ D 6 Φ s ΦD 3 Φ + s 3 DΦD Φ + γ 3 s +β 3 DΦD 6 Φ γ 3 s ΦD 7 Φ + β 3 + γ 3 γ s D ΦD 5 Φ + β β 3 + γ γ 3 s D 3 ΦD 4 Φ, 4 6

8 where {ν, s, κ 1, κ, β, β 3, γ, γ 3 } are parameters. The component equations are, u t νu xxt = κ 1 u x + κ u xxx 3uu x + β u x u xx + β 3 uu xxx + ξξ xx + γ ξ x ξ xxx + γ 3 ξξ xxxx, ξ t νξ xxt = κ 1 ξ x + κ ξ xxx s u x ξ + s 3 uξ x + γ 3 s + β 3 uξxxx + β β 3 + γ γ 3 s ux ξ xx + γ 3 γ s + β 3 uxx ξ x γ 3 s u xxx ξ. 5 These systems are by construction invariant under the supersymmetry transformations δu = τξ x, δξ = τu s, 6 where τ is an odd parameter. The superkdv limit, namely {ν, β, β 3, γ, γ 3, κ 1 } all zero, yields, modulo rescalings, the one-parameter family of systems studied by Mathieu [14]. By comparing 5 and it is straightforward to extract systems which are both supersymmetric and have hamiltonian form 19,0. Taking s = in 5, {ν=µ, σ 1 =κ 1, σ =κ, ǫ= 1} in, and {β =β 3, β 3 = γ 5 γ 3} in both, we obtain the systems, u t νu xxt = κ 1 u x + κ u xxx 3uu x + γ 5 γ 3u x u xx + uu xxx +ξξ xx + γ ξ x ξ xxx + γ 3 ξξ xxxx, ξ t νξ xxt = κ 1 ξ x + κ ξ xxx u x ξ uξ x + γ γ 3 uξ xxx + 3 γ γ 3 u x ξ xx + 1 γ 4γ 3 u xx ξ x 1 γ 3u xxx ξ. 7 These may be expressed in superfield form 4 with the above choice of parameters. The manifestly supersymmetric hamiltonian form is given by with M t = P 1 δĥ1 δm, M = Φ νd4 Φ, 8 P 1 = D1 νd 4, 9 Ĥ 1 = dxdϑ κ 1 ΦDΦ κ D ΦD 3 Φ 1 ΦDΦ γ 3ΦD 3 Φ γ γ 3 DΦ D 4 Φ. 30 Since the KdV reduction of 7 with κ 1 = γ = γ 3 = 0 is not believed to be integrable, we have not explored this class of systems further. In a similar fashion, we may look for choices of parameter sets for which the geodesic flows of section are also supersymmetric. Comparing 13 with 5, we see that the choice 7

9 {µ=ν, α=1, κ 1 =0} in the former and {s = 4 3, β =ν, β 3 =ν, γ = ν 3, γ 3=κ 1 =0} in the latter, yields the two-parameter system of supersymmetric geodesic flows: u t νu xxt = κ u xxx 3uu x + ξξ xx + νuu xxx + u x u xx + ν 3 ξ xξ xxx, ξ t νξ xxt = κ ξ xxx 3 uξ x + νuξ xxx + 3 u xξ xx + 1 u xxξ x. 31 We shall call this system, with κ =0 and ν 0, the supersymmetric Camassa-Holm equation superch. The system 31 reduces to superkdv, upon setting ν to zero, and to CH, upon setting ξ to zero and translating u. Not surprisingly, the systems 31 arise as geodesic flows precisely when the metric 9 on the NSR superconformal algebra is supersymmetric. Then, the calculations of section can be performed using superfields. Specifically, writing U = u + ϑφ and V = v + ϑψ, the bracket 8 takes the form [ ] U, a, V, b = UD V VD U + 1DUDV, c 1 and the inner product 9 may be written U, a, V, b = dxdϑd UD 3 V 3 dxdϑ UD 1 V + νd UDV + ab. 33 The superspace bilinear operator B is given by B U, a, V, b = B 0, 0, where B 0 satisfies 1 νd 4 D 1 B 0 = c 1 ad 5 V 3 D V1 νd 4 D 1 U 1 DV1 νd4 U V1 νd 4 DU. 34 Writing c 1 a=κ and U = DΦ, the geodesic flows U t, a t = B U, a, U, a yield 1 νd 4 Φ t = κ D 6 Φ 3 ΦD3 Φ+DΦD Φ+ν DΦD 6 Φ + 1 D ΦD 5 Φ + 3 D3 ΦD 4 Φ. 35 We thus recover the subsystem of 4 having component content 31. Equation 35 has superfield hamiltonian formulation, with M t = P δĥ δm, M = Φ νd4 Φ, 36 P = κ D 5 1DMD D M MD, 37 Ĥ = 1 DΦ, 0, DΦ, 0 = 1 dxdϑ ΦDM. 38 8

10 V Painlevé integrability of superch systems In this section we investigate, in more detail, the supersymmetric geodesic flow 31 with ν=1 and κ = 0, m t = mu x um x + ηξ + 3 η xξ x, m = u u xx, η t = 3 ηu x 1 mξ x uη x, η = ξ ξ xx. 39 We shall consider the two simplest possible choices for the grassmann algebra in which the fields are valued, viz. algebras with one or two odd generators. Taking the algebra to be finite dimensional is a very convenient tool for preliminary investigations of systems with grassmann algebra-valued fields. Manton [17] recently studied some simple supersymmetric classical mechanical systems in this way and he introduced the term deconstruction to denote a component expansion in a grassmann algebra basis. In [18] we investigate fermionic extensions of KdV in a similar fashion. V-i First deconstruction of superch We first consider the superch system 39 with fields taking values in the simplest grassmann algebra with basis {1, τ}, where τ is a single fermionic generator. In this case the fermionic fields may be expressed as ξ = τξ 1, η = τη 1, where ξ 1 and η 1 are standard i.e. commuting, c-number functions, as are u and m in this simple case. Since τ = 0, the fermionic bilinear terms do not contribute and we are left with the system m t = mu x um x, m = u u xx η 1t = 3 η 1u x 1 mξ 1x uη 1x, η 1 = ξ 1 ξ 1xx. 40 Further analysis is simplified by changing coordinates as described in [19]. Writing m=p, the first equation of 40 takes the form p t = pu x, which suggests new coordinates y 0, y 1 defined via dy 0 = p dx pu dt, dy 1 = dt, 41 or dually, via = p x y 0, = t y 1 pu y 0. 4 Implementing this coordinate change and eliminating the functions u and ξ 1, the remaining equations for p and q η 1 are: p ṗ pṗp + ṗ p + ṗp p 3 p ṗ = 0, 43 9

11 .q 3p.q 3ṗ 4p p p q + p p p 1 15p.q ṗ + 3ṗ p p p p q +3 ṗp + p ṗ p 4ṗp p q = p 3 Here the dot and prime denote differentiations with respect to y 1 and y 0 respectively. We note: a thanks to supersymmetry 6, if p is a solution of 43, then q=p is a solution of 44; and b under the substitution q = p 3/ r, 44 takes the substantially simpler form ṙ + p 4p p p 1 p The system 43,44 passes the WTC Painlevé test. ṙ p r 3p 4 r = Proof: Equation 43 is a rescaled version of the Associated Camassa-Holm equation of [19]. Consideration of solutions with py 0, y 1 p 0 y 0, y 1 φy 0, y 1 n near φy 0, y 1 = 0, for some n 0, yields n = or n = 1 as the possible leading orders of Laurent series solutions. We need to perform the WTC Painlevé test [0] for both these types of series. The first type, namely, Laurent series solutions exhibiting double poles on the singular manifold φy 0, y 1 = 0, have already been considered in [1]. These take the form. p = φ φ. φ φ φ + p + p 3 φ + p 4 φ +..., 46 where φ, p, p 4 are arbitrary functions of y 0, y 1, and p 3 = 1 φ. φ. φ φṗ. + φ φ p.. φ φ φ φ. φ. + φ φ. p. φ φ φ.. φ.. φ φ. φφ... φ + φφ.. φ.. 47 We have, at present, no explanation of the remarkable symmetry of these expressions under interchange of the independent variables. The second type of solutions have a simple zero on the singular manifold φy 0, y 1 = 0. They take the form p = ± φ φ + p φ + p 3 φ , 48 where φ, p, p 3 are arbitrary functions. The verification of the consistency of both these types of expansions is straightforward. This completes the WTC test for equation 43. It remains to look at the equation 44. Although linear in q, it is not automatically Painlevé. The movable poles and zeros in p give rise to movable poles in the coefficient functions of the linear equation for q, and we need to examine the resulting singularities of 10

12 q. If p has a pole on φ=0, then near φ=0 we have p. φφ /φ, and equation 44 takes the form.q + 6φ φ +... q. 3.. φ + φ +... q + 4φ φ +... q. + 11φ φ q + O q = 0. φ φ Thus the equation has a solution with q φ n if nn 1n +9nn 1+15n = 0, giving n= 4,, 0. It follows that in the case when p is given by the series 46, no inconsistencies will arise near the double poles of p if 44 has a series solution of the form q = q 0 φ 4 + q 1 φ 3 + q φ + q 3 φ + q with q 0, q, q 4 arbitrary. The consistency of such a solution can easilly be verified using a symbolic manipulator. Using MAPLE we find that The explicit expression for q 3 is too lengthy to be given here. q 1 = φ q0 φ q 0 φ. 50 Suppose now that p has a zero on φ=0. Near this, p ±φ/φ and equation 44 has the structure.q 3φ φ +... q. 3.. φ φ +... q + 3φ φ +... q+. 15φ φ. φ +... q 1φ φ +... q = 0. φ 3 Thus 44 has a solution with q φ n if nn 1n 9 nn 1+1 n 1 = 0, giving,, 4. The appearance of a half-integer here is not considered a violation of the Painlevé n= 3 test see e.g. []. The half integer value of n gives rise to a series solution of 44, near a zero of p, of the form q = q 0 φ 3 + q 1 φ 5 + q φ with q 0 arbitrary, and q 1, q,... determined by q 0 and the arbitrary functions arising in the series 48 for p. The two integer values of n tell us that we need to check the consistency of solutions of 44 taking the form q = Q 0 φ + Q 1 φ 3 + Q φ with two arbitrary functions Q 0 and Q. This is indeed consistent; using MAPLE we obtain Q 1 = ±φ Q 0 p 1 3φ. φ φ. Q 0 + φ. φq0 + φ. φq 0 + 4φ. φ Q0, 53 11

13 with the choice of ± depending on the choice in 48. The general solution of 44 near a zero of p, with three arbitrary functions, is a linear combination of the series 51 and 5. Thus the system 43,44 passes the WTC test. The WTC test is evidence for the complete integrability of the system 43,44. This in turn suggests that superch indeed has some integrable content. V-ii Second deconstruction of superch We now consider the system 39 with fields taking values in a grassmann algebra with two anticommuting fermionic generators, τ 1, τ. Expanding in the basis {1, τ 1, τ, τ 1 τ }, u = u 0 + τ 1 τ u 1, ξ = τ 1 ξ 1 + τ ξ, m = m 0 + τ 1 τ m 1, η = τ 1 η 1 + τ η, 54 where the functions u 0, u 1, m 0, m 1, ξ 1, ξ, η 1, η are all standard, we obtain the system: m 0t = m 0 u 0x u 0 m 0x, m 0 = u 0 u 0xx, 55 η it = 3 u 0xη i 1 m 0ξ ix u 0 η ix, η i = ξ i ξ ixx, i = 1,, 56 m 1t = m 1 u 0x m 0 u 1x u 0 m 1x u 1 m 0x +η 1 ξ η ξ η 1xξ x η x ξ 1x, m 1 = u 1 u 1xx. 57 Supersymmetry 6 tells us that given a solution u 0, m 0 of 55, we can solve the remaining equations by taking ξ i = α i u 0, η i = α i m 0 i=1,, u 1 = βu 0x and m 1 = βm 0x, where α 1, α, β are arbitrary constants. We handle the system following the procedure of the previous section. Writing m 0 =p and changing coordinates to y 0, y 1, the system can be written: 1. ṗ u 0 =, u 0 = p p, 58 p p ξ i = 3η iṗ p 4 m1 p.. η i 3ηi ṗ, ξ p 3 i = η i + p p 3 = u 1 p + 8. η1 η. η η 1 3p 3 +. η i p, i = 1,, 59 4η 1 η η η 1 3p 3 m 1 = u 1 ppu Applying the WTC Painlevé test to this is a mammoth task, so instead we consider the Galilean-invariant reduction and apply the Painlevé test at this level. The Galilean-invariant 1.,

14 reduction is obtained, as usual, by restricting all functions to depend on the single variable z=y 0 vy 1 alone. Evidently the first equations of both 58 and 60 can be integrated once immediately. Then eliminating u 0 from 58, ξ i from 59 and m 1 from 60, we obtain, p = p p v + c 1 p 1 p, 61 η i 9p 11p p η i + v 5c 1 p p η p p i 3p p p v 3c 1 p + 3 +p η p p i = 0, i=1,, 6 u 1 + p p p u 1 + v 1 u p 1 = d p 3η 1η η η 1, 63 where c 1, d 1 are integration constants. The equation for pz may be integrated again after multiplying both sides by p /p; this gives p = 1 c 1 p + c p v p3, 64 where c is another integration constant. This equation is well known in KdV theory. Its general solution can be written in terms of the Weierstrass -function, pz = v z c v, 65 where the periods of are determined by the coefficients c 1, c, v. Using 64, the coefficients in 6 can be simplified. Further, we know from supersymmetry that this equation has a solution η i = p. Substituting η i = p q i the equation becomes a second order equation for q i : q i + 3p p q i + 3p v 3 p + c q i = 0, i = 1,. 66 Supersymmetry 6 allows a reduction of the order of 63 as well. It implies that u 1 = p /p, η i =p is a solution. So, writing u 1 =rp /p, η i =p q i in 63 yields a first order equation for r : r + c p 4p v 1 r = p d1 + 4pq p p p 1 q q q Multiplying by the integrating factor p /p and integrating, we obtain where d is a further constant of integration. r = p d p 1 p + d + 4 q 1 q q q 1pp dz, 68 Thus the Galilean-invariant reduction of the second deconstruction of superch takes the form of the three equations 64,66,68, to which we now apply the Painlevé test. All substitutions hitherto have been ones which do not interfere with the test. Equation 64 13

15 has movable double poles and movable simple zeros. Near a double pole at z 0, the series solution contains only even powers of z z 0, pz = v z z 0 + c v 6 + 1c 1 c v z z v + c3 v 18c 1 c 304 z z and near a simple zero at z 0, pz = ±z z 0 1 c 1z z 0 ± 1 6 c z z v + c 1c z z At both the zeros and poles of p, equation 66, which is just a linear third order ODE, has regular singular points. Checking Painlevé property for this reduces to doing the necessary Frobenius-Fuchs analysis at these regular singular points to check that no logarithmic singularities in the solutions q i arise. Finally, equation 68 gives an explicit formula for r involving two quadratures. Here the necessary analysis involves wrtiting series expansions for the integrands near the zeros and poles of p, and checking for the absence of 1/z z 0 terms, which would give rise to logarithms on integration. We do not present all these calculations in detail; with the aid of a symbolic manipulator they are quite straightforward. We conclude that the Galilean-invariant reduction of the second deconstruction of superch has the Painlevé property. We note, in conclusion, that two of the equations we have encountered are interesting variants of the Lamé equation: In 66, the substitution q i = p 3/4 h i yields h i p v c 6 + c 1 p 7 h p i = 0, 71 and similarly, on writing u 1 = p 1/ k, the homogeneous part of 63 takes the form, k + 3p v c 4 3 k = p By the arguments above, the latter is integrable by quadratures. VI Superpeakon solutions As mentioned in the introduction, one of the intriguing features of the CH equation is the existence of peakon solutions. One would hope that superch shares this property. However, peakon solutions are weak solutions, with a discontinuity in the first derivative; and the action of supersymmetry on such functions, for a general underlying grassmann algebra, yields objects which are not regular enough to be considered as weak solutions. So, CH peakon 14

16 solutions do not admit a general supersymmetrisation. The above argument does not hold in the first deconstruction, because if there is only one fermionic generator, the supersymmetry transformation 6 does not involve an x-derivative. So such supersymmetrised peakon solutions of the superch system 39 do exist if the fields are restricted to take values in a grassmann algebra with only one fermionic generator. Consider the equations 40 of the first deconstruction. Supersymmetry implies that if u, m is a solution of the first equation in 40, then ξ 1 = cu, η 1 = cm where c is a constant gives a solution of the second equation. Thus, for example, the speed v travellingwave peakon solution of CH, u = v exp x vt, can be supersymmetrised, as can any multipeakon solution. In fact, there also exist more general superpeakons. The superposition ansatz ux, t = ξ 1 x, t = N i=1 N i=1 p i t exp x q i t 73 r i t exp x q i t 74 gives a solution of the system 40 provided the functions q i t, p i t, r i t i = 1,..., N satisfy the ODE system N q it = p j exp q i q j 75 p it = r it = 1 j=1 N j=1 N sgnq i q j p i p j exp q i q j 76 j=1 sgnq i q j p i r j + p j r i exp q i q j, 77 where the primed sums range over values of j i. Equations 75 and 76 are the conditions which determine u of the form 73 to be a multipeakon solution of CH. They describe geodesic motion on an N-dimensional surface with coordinates q i [1] and form an integrable hamiltonian system [3]. The further equations 77 are linear equations for the functions r i. Clearly, taking the r i = cp i for some constant c gives a solution, these being the supersymmetrised multipeakon solutions discussed before. More general solutions certainly exist. Since the system is integrable, integrability of the additional N linear equations 77 depends on the existence of N 1 independent conserved quantities depending on the r i. We have not settled this question in general, but we note that N i=1 r i is a conserved quantity, just as the total momentum N i=1 p i is also conserved. This suffices for integrability when N=, in which case the remaining equation for r 1 r can be integrated explicitly. 15

17 Note that unlike the existence of the superpeakons which arise in virtue of supersymmetry transformations of CH peakons, the existence of this extra conserved quantity depends critically on the coefficients of the η 1 evolution equation in 40. Even if the full superpeakon system 77 proves not to be fully integrable, the geodesy and supersymmetry conditions have certainly picked out an equation with some integrability properties c.f. [1]. VII Outlook In this paper we have examined fermionic extensions of the Camassa-Holm equation. In particular we have identified the superch system 39, which, for low dimensional grassmann algebras displays some integrability properties and has peakon type solutions. Further investigation is needed to determine whether the superch system is fully integrable. Our work provides a further instance of integrability properties arising in the context of geodesic flows on a group manifold, and in particular provides some evidence that supersymmetric geodesic flows whose bosonic part is integrable must also be integrable. We note in closing that the KP and superkp systems have yet to be presented as geodesic flows. If such a presentation exists, it would have a bearing on the question of whether there is a KP-type higher dimensional generalisation of Camassa-Holm arising in a way similar to that in which KP generalises KdV. Acknowledgements We thank the MPI für Mathematik in den Naturwissenschaften for hospitality to J.S. during a visit to Leipzig and the Department of Mathematics and Computer Science of Bar-Ilan University for hospitality to C.D. during a visit to Israel. We acknowledge encouragement to add section 6 from one of the anonymous referees of the paper. References [1] R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons, patt-sol/930500, Phys. Rev. Lett [] R. Camassa, D.D. Holm, and J.M. Hyman, A new integrable shallow water equation with peaked solutions, in Advances in Applied Mechanics 31, Ed. J. Hutchinson and T. Wu, Academic Press 1994,

18 [3] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D [4] V.I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag [5] V.Yu. Ovsienko and B.A. Khesin, Korteweg-de Vries superequation as an Euler equation, Funct. Anal. Appl [6] G. Misiolek, Conjugate points in the Bott-Virasoro group and the KdV equation, Proc. AMS [7] V.I. Arnold and B.A. Khesin, Topological methods in hydrodynamics, Springer-Verlag [8] P.W. Michor and T.S. Ratiu, On the geometry of the Virasoro-Bott group, math.dg/ , J. Lie Theory [9] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys [10] S. Shkoller, Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics, math.ap/ , J. Funct. Anal [11] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, math-ph/980701, J. Math. Phys [1] O.B. Fringer and D.D. Holm, Integrable vs nonintegrable geodesic soliton behavior, solv-int/ [13] B.A. Kupershmidt, A super Korteweg de Vries equation: an integrable system, Phys. Lett. A [14] P. Mathieu, Supersymmetric extension of the Korteweg de Vries equation, J. Math. Phys [15] Yu.I. Manin and A.O. Radul, A supersymmetric extension of the Kadomtsev Petviashvili hierarchy, Commun. Math. Phys [16] D. Zhang, Integrability of fermionic extensions of the Korteweg-de Vries equation, Phys. Lett. A [17] N.S. Manton, Deconstructing supersymmetry, hep-th/ , J. Math. Phys

19 [18] C. Devchand and J. Schiff, Deconstructing superkdv, in preparation. [19] J. Schiff, The Camassa-Holm equation: a loop group approach, solv-int/ , Physica D [0] J. Weiss, M. Tabor and G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys [1] A.N.W. Hone, The associated Camassa-Holm equation and the KdV equation, solv-int/ , J. Phys. A L307-L314. [] A. Ramani, B. Grammaticos and T. Bountis, The Painlevé property and singularity analysis of integrable and non-integrable systems, Phys. Rep [3] F. Calogero, An integrable hamiltonian system, Phys. Lett. A ; F. Calogero, A solvable hamiltonian system, J. Math. Phys ; F. Calogero and J.P. Francoise, A completely integrable hamiltonian system, J. Math. Phys ; O. Ragnisco and M. Bruschi, Peakons, R-Matrix and Toda lattice, solv-int/950901, Physica A