Transcription

1 Max-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig The supersymmetric Camassa{Holm equation and geodesic ow on the superconformal group by Chandrashekar Devchand and Jeremy Schi Preprint-Nr:

2

3 solv-int/986 The supersymmetric Camassa-Holm equation and geodesic ow on the superconformal group Chandrashekar Devchand 2 and Jeremy Schi 3 Max-Planck-Institut fur Mathematik Gottfried-Claren-Strae 26, Bonn, Germany 2 Max-Planck-Institut fur Mathematik in den Naturwissenschaften Inselstrae 22-26, 43 Leipzig, Germany devchand@mismpgde 3 Department of Mathematics and Computer Science Bar{Ilan University, Ramat Gan 529, Israel schi@mathbiuacil 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 ow with respect to an H metric on the group of superconformal transformations in two dimensions, (b) equations which are hamiltonian with respect to a dierent hamiltonian structure and (c) supersymmetric ow equations Classes (a) and (b) have nointersection, but the intersection of classes (a) and (c) gives a candidate for a new supersymmetric integrable system We demonstrate the Painleve property for some simple but nontrivial reductions of this system

4 Introduction Recently there has been substantial interest in the Camassa-Holm (CH) equation [, 2]: u t ; u xxt = u x ; 3uu x + (uu xxx +2u x u xx ) : () This equation has been proposed as a model for shallow water waves It is believed to be integrable (having bihamiltonian structure), but 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 it admits a class of nonanalytic weak solutions known as peakons, as well as nite time blow-up of solutions Geometrically, the relationship of CH to KdV is rather deeper: Both are regularisations of the Euler equation for a one dimensional compressible uid (Monge or inviscid Burgers equation), u t = ;3uu x : (2) This latter equation describes geodesic motion on the group of dieomorphisms of the circle Di(S ) [3] with respect to a metric induced by an L 2 R norm, u 2 dx, on the associated algebra If the group is centrally extended to the Bott-Virasoro group, the KdV equation arises [4, 5, 6, 7] On the other hand, if the metric is changed to one induced by anh R norm, (u2 + u 2 x)dx, the CH equation arises [8, 9, ] Both these `deformations' have a regularising eect on solutions of (2), which exhibit discontinuous shocks Thus KdV and CH arise in a unied geometric setting both are integrable systems which describe geodesic ows This raises an important question: What features of the underlying geometry give rise to integrability? In general, geodesic ows of this type are not integrable: the Euler equation for uid ow in more than one spatial dimension is an example [3] Indeed, for the latter, Arnold has suggested a relationship between negative sectional curvatures and nonpredictability of the ow Is integrability also geometrically determined? One further example of an integrable bihamiltonian system arising from a geodesic ow has been discussed in the pioneering paper of Ovsienko and Khesin [4] Using the superconformal group with an L 2 type metric, they obtained the so-called kuperkdv system of Kupershmidt [] 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 2 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 [2, 3] 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 ow equations associated with L 2 type metrics on the superconformal group, only two specic choices of the parameter give rise to integrable systems The main purpose of this paper is to investigate geodesic ow equations obtained from H type norms on the superconformal group more generally we consider the following family of fermionic

5 extensions of CH: u t ; u xxt = u x + 2 u xxx + uu x + 2 u x u xx + 3 uu xxx + xx + 2 x xxx + 3 xxxx t ; xxt = x + 2 xxx + u x + 2 u x + u xxx + 2 u x xx + 3 u xx x + 4 u xxx : (3) Here u(x t)and(x t) are elds valued, respectively, in the even and odd parts of a grassmann algebra, and f g are parameters By rescaling u and it is possible to set = ; 3 and =2 (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 2, we consider geodesic ow on the superconformal group with an H type metric the resulting systems have a natural hamiltonian structure, or more precisely, since the elds are grassmann algebra valued, a graded hamiltonian structure In section 3 we identify a class of systems having a dierent hamiltonian structure Unfortunately the latter has no intersection with the class of section 2, 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 transformations between u and This class has nontrivial intersections with both the classes of sections 2and3 In particular, there is a unique supersymmetric geodesic ow system, which isapromising candidate for being integrable In section 5 we showthattwo reductions of this system have the Painleve property A trivial integrable CH system of the form (3), which is not incorporated in the classes of sections 2,3, and 4, and which we shall not discuss further, is the odd linearisation of the bosonic CH system () u t ; u xxt = u x ; 3uu x + (uu xxx +2u x u xx ) t ; xxt = x ; 3(u) x + (u xxx + u xxx +2( x u x ) x ) : (4) Replacing u by u + 3 and considering the limit,,with = 3, yields the system u t = ;3uu x + u xxx t = ;3(u) x + xxx : (5) This trivial fermionic extension of KdV has appeared often in the literature (see eg [2]) 2 Geodesic ows on the superconformal group An inner-product h: :i 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 [3] Dene a bilinear operator B : g g g by D E [V W] U = D E W B(U V ) 8 W 2 g : (2) 2

6 The geodesic ow equation is then simply U t = B(U U) : (22) In our case, g is the NSR superconformal algebra, consisting of triples (u(x) '(x) a), where u is a bosonic eld, ' is a fermionic eld and a is a constant The Lie bracket is given by h i (u ' a) (v b) = uv x ;u x v+ ' u 2 x; u 2 x ;' x v+ 'v 2 x dx(c u x v xx +c 2 uv x +c ' x x + c 2 4 ' ) where c c 2 are constants On this algebra, an H inner product is given by D E (u ' a) (v b) = dx uv + u x v x + '@ x ; + ' x where = + ab (23) dx (u v + ' ) + ab (24) = x 2 x ; x (25) and are further constants, all assumed nonzero Writing U=(u ' a) V=(v b), wend B(U V )=(B B ), where B (U V ) = ; 2v x u + v u x + 32 x ' + 2 ' x + a(c v xxx ; c 2 v x ) (26) 3 B (U V ) = ; v 2 x ' + v ' x + 2 u + a(c xx ; c 2 ) : 4 The geodesic equations are therefore conveniently written in the form u t = B (U U) ' t = B (U U) (27) a t = : Writing ' = x, where is a constant satisfying 2 = 4, this yields the system 3 u t ; u xxt = u x + 2 u xxx ; 3uu x + (uu xxx +2u x u xx )+2 xx x xxx t ; xxt = 4 x + 2 xxx ; 3 2 u x ; ( + 2 )u x + u xxx u x xx + 2 u xx x : Here 2 are independent parameters determined by a c c 2 This is evidently a 5 parameter class of systems of type (3) Setting to zero in (28) yields the CH result of [8, 9, ] If instead we choose to vanish, the H norm becomes an L 2 norm then choosing to be zero and rescaling 2 to we obtain the following parameter fermionic extension of KdV: u t = u xxx ; 3uu x +2 xx t = xxx ; 3 2 u x ; ( + 2 )u x : Modulo rescalings, the superkdv of Mathieu and Manin-Radul is obtained by taking = The kuperkdv system arises by taking =, the choice made in [4] Other values of the parameters 4 give systems which are not believed to be integrable (see however [4]) 3 (28) (29)

7 3 Hamiltonian ows Like KdV, CH has bihamiltonian structure, and this accounts for its integrability We might hope that for some choices of parameters the system (28) should also have a bihamiltonian structure One hamiltonian structure follows automatically from the geometric origins of the system [3] Explicitly, introducing new variables, m = u ; u xx where m t t = P 2 H 2 m H 2 and = ; xx, (28) takes the form P 2 = 2@ x 3 x x m ; m@ 2 x x ;@ x 3 2 x 4 ( + 4 2@ x) 2 ; 3m 8 and the hamiltonian functional is given succinctly by theh inner product on the algebra, H 2 = 2 D E U U = 2 dx u 2 + u 2 x + 4( 3 x + xx x ) (3) (32) : (33) This generalises the so-called second Hamiltonian structure of KdV and its fermionic extensions [, 2] Checking (3) is straightforward: the Euler-Lagrange derivatives H 2 m H 2 H 2 = dx H2 m m + H 2 from which it follows immediately that H 2 m = u and H 2 = 4 3 x are dened by (34) To investigate the possibility of systems amongst (28) having another hamiltonian form, we look at systems of the form where m t t H m = P H (35) P x(;@ x) 2 ; : (36) 2 (;@2 x) Here is a constant and H is a functional generalising the KdV rst Hamiltonian, H = dx ; 2 u3 ; 3 2 uu2 x ; 2 2 u2 x + 2 u2 + x + 2 xxx +2u x +( 2 ; 3 )u x xx + 3 u xxx : (37) This is the most general functional of this type, up to rescalings of u and Since m=(;@ 2 x)u, we have ( x) 2 H m = H u x) H = H Thus equations (35) take the simple form u t ; u xxt H x u = u x + 2 u xxx ; 3uu x + 3 (2u x u xx + uu xxx )+2 xx + 2 x xxx + 3 xxxx t ; xxt = H = x + 2 xxx + (u x +2u x )+ (2 3 ; 2 )u xxx (2 3 ; 2 )u x xx + 2 (4 3 ; 2 )u xx x u xxx : (38) 4

8 This is a parameter class of systems of the form (3) Comparing with (28), we see that the only bihamiltonian systems occur when f== 3 = 2 = 3 =, =; 3 2, =, 2 =4 2 g,whichis equivalent to (28) with f== = g, ie the kuperkdv system Thus, no new bihamiltonian 4 systems arise We note that the systems (38) can be obtained from a Lagrangian Introducing a potential f dened by u=f x, they are Euler-Lagrange equations for the functional L = dx 2 (f x ; f xxx )f t + ( ; xx ) t + 2 f 3 x f xf 2 xx f 2 xx ; 2 f 2 x ; x ; 2 xxx ; 2f x x +( 3 ; 2 )f x x xx ; 3 f x xxx : (39) 4 Supersymmetric ows Dene a fermionic supereld (x #) = s + #u and superderivative + #@ x, where s is a nonzero parameter and # is an odd coordinate The most general supereld equation having component content of the form (3) is the 8 parameter system, ; D 4 t = D D 6 ; 2 D ; 3 DD s 2 s 2 s 2 3 DD 6 ; 3 D 7 + s ; 2 D 2 D 5 + s 2 2 ; 3 + 2; 3 D 3 D 4 (4) s 2 where f s g are parameters The component equations are, u t ; u xxt = u x + 2 u xxx ; 3uu x + 2 u x u xx + 3 uu xxx +2 xx + 2 x xxx + 3 xxxx t ; xxt = x + 2 xxx ; 2 u s 2 x + 2 ; 3 s ; 3 + 2; 3 s 2 u x xx + 3 ; 2 s 2 u x + 3 s u xxx + 3 u xx x ; 3 s 2 u xxx : (42) These systems are by construction invariant under the supersymmetry transformations u = x = u s 2 (43) where is an odd parameter The superkdv limit, namely f g all zero, yields, modulo rescalings, the one-parameter family of systems studied by Mathieu [2] By comparing (42) and (38) it is straightforward to extract systems which are both supersymmetric and have hamiltonian form (35),(36) Taking s 2 =2 in (42), f=, =, 2 = 2, =;g in (38), and f 2 =2 3, 3 = 2 ; 5 2 3g in both, we obtain the systems, u t ; u xxt = u x + 2 u xxx ; 3uu x +( 2 ; 5 2 3)(2u x u xx + uu xxx ) +2 xx + 2 x xxx + 3 xxxx t ; xxt = x + 2 xxx ; u x ; 2u x +( 2 ; 2 3 ) u xxx ( 2 ; 2 3 ) u x xx + 2 ( 2 ; 4 3 ) u xx x ; 2 3u xxx : (44) 5

9 These may be expressed in supereld form (4) with the above choice of parameters The manifestly supersymmetric hamiltonian form is given by with M t = b P b H M M =; D4 (45) bp = D( ; D 4 ) (46) bh = dxd# 2 D ; 2 2 D2 D 3 ; 2 (D) (D 3 ) ( 2 ; 2 3 )(D) 2 D 4 : (47) Since the KdV reduction of (44) (with = 2 = 3 = ) 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 ow equations of section 2 are also supersymmetric Comparing (28) with (42), we see that the choice f=, = =g in the former and fs 2 = 4, 3 2=2, 3 =, 2 = 2, 3 3= =g in the latter, yields the two-parameter system of supersymmetric geodesic ow equations: u t ; u xxt = 2 u xxx ; 3uu x +2 xx + (uu xxx +2u x u xx )+ 2 3 x xxx t ; xxt = 2 xxx ; 3 2 (u) x + (u xxx u x xx + 2 u xx x ) : (48) We shall call this system, with 2 = and 6=, the supersymmetric Camassa-Holm equation (superch) In section 5, we present some evidence for its integrability The system (48) reduces to superkdv, upon setting to zero, and to CH, upon setting to zero and translating u Not surprisingly, the systems (48) arise as geodesic ow equations precisely when the metric (24) on the NSR superconformal algebra is supersymmetric Then, the calculations of section 2 can be performed using superelds Specically, writing U = u + # and V = v + #, the bracket (23) takes the form h i (U a) (V b) = UD 2 V;VD 2 U + DUDV c 2 dxd# D 2 UD 3 V (49) and the inner product (24) may be written D E (U a) (V b) = dxd# UD ; V + D 2 UDV + ab : (4) The superspace bilinear operator b B is given by B b (U a) (V b) =( B b ), where B b satises ( ; D 4 )D ; b B = c ad 5 V; 3 2 D2 V(;D 4 )D ; U; 2 DV(;D4 )U;V(;D 4 )DU : (4) Writing c a= 2 and U = D, the geodesic ow equations (U t a t )= B b (U a) (U a) ( ; D 4 ) t = 2 D 6 ; 3 2 (D3 +DD 2 ) + yield DD D2 D D3 D 4 : (42) 6

10 We thus recover the subsystem of (4) having component content (48) supereld hamiltonian formulation, Equation (42) has with M t = b P 2 b H 2 M M =; D4 (43) bp 2 = 2 D 5 ; 2 DMD ; D2 M ; MD 2 (44) bh 2 = 2 D E (D ) (D ) = 2 dxd# DM : (45) 5 Painleve integrability of superch systems In this section we investigate, in more detail, the supersymmetric geodesic ow system (48) with = and 2 =, m t = ;2mu x ; um x x x m = u ; u xx t = ; 3 2 u x ; 2 m x ; u x = ; xx : (5) We shall consider the two simplest possible choices for the grassmann algebra in which the elds are valued, viz algebras with one or two odd generators Taking the algebra to be nite dimensional is a very convenient tool for preliminary investigations of systems with grassmann algebra-valued elds Manton [5] 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 [6] we investigate fermionic extensions of KdV in a similar fashion 5 First deconstruction of superch We rst consider the superch system (5) with elds taking values in the simplest grassmann algebra with basis f g, where is a single fermionic generator In this case the fermionic elds may be expressed as =, =, where and are standard (ie commuting, c-number) functions, as are u and m in this simple case Since 2 =,the fermionic bilinear terms do not contribute and we are left with the system m t = ;2mu x ; um x m = u ; u xx t = ; 3 2 u x ; 2 m x ; u x = ; xx : (52) Further analysis is simplied by changing coordinates as described in [7] Writing m=p 2, the rst equation of (52) takes the form p t =(;pu) x, which suggests new coordinates y y dened via dy = pdx; pu dt dy = dt (53) or dually, : (54) 7

11 Implementing this coordinate change and eliminating the functions u and, the remaining equations for p and q are: p 2ṗ ; p(ṗp + ṗ p )+ṗp2 ; 2p 3 p ; ṗ = (55) q ; 3p q ; 3ṗ p 2p q + 4p2 p 2 ; 2p p ; q + p pṗ 2p 2 pp +2p ṗ p 2 ; 3ṗ p ; p q 2 ; 4ṗp2 ; p q = : (56) p 3 Here the dot and prime denote dierentiations with respect to y and y respectively We note: (a) thanks to supersymmetry (43), if p is a solution of (55), then q=p 2 is a solution of (56) and (b) under the substitution q = p 3=2 r, (56) takes the substantially simpler form r + p2 4p ; p 2 2p ; r ; p p 2 2 r ; 3p r =: (57) 4 The system (55),(56) passes the WTC Painleve test Proof: Equation (55) is a rescaled version of the Associated Camassa-Holm equation of [7] Consideration of solutions with p(y y ) p (y y )(y y ) n near (y y )=,forsomen 6=, yields n = ;2 or n = as the possible leading orders of Laurent series solutions We need to perform the WTC Painleve test [8] for both these types of series The rst type, namely, Laurent series solutions exhibiting double poles on the singular manifold (y y )=,have already been considered in [9] These take the form p = 2 2 ; where p 2 p 4 are arbitrary functions of y y,and p 3 = ; p 2 + p 3 + p ::: (58) 2 ṗ p 2 ; 2 ; p 2 ; ; ; + : (59) 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 (y y )= They take the form p = + p p ::: (5) where p 2 p 3 are arbitrary functions The verication of the consistency of both these types of expansions is straightforward This completes the WTC test for equation (55) It remains to look at the equation (56) Although linear in q, itisnot automatically Painleve The movable poles and zeros in p give rise to movable poles in the coecient functions of the linear 8

12 equation for q, and we need to examine the resulting singularities of q If p has a pole on =, then near = we have p 2 = 2, and equation (56) takes the form q ::: q 3 + ::: A q + + ::: A q + ::: A q O q =: 2 Thus the equation has a solution with q n if n(n;)(n;2)+9n(n;)+5n =, giving n= ; 4 ;2 It follows that in the case when p is given by the series (58), no inconsistencies will arise near the double poles of p if (56) has a series solution of the form q = q 4 + q 3 + q q 3 + q 4 + ::: (5) with q q 2 q 4 arbitrary The consistency of such a solution can easilly be veried using a symbolic manipulator Using MAPLE we nd that The explicit expression for q 3 is too lengthy tobegiven here q = 2 q ; q 2 : (52) Suppose now that p has a zero on = Near this, p = and equation (56) has the structure q ; 3 + ::: q ;@ ::: A q 32 + ::: A 5 + ::: A q ;@ 22 + ::: A q =: Thus (56) has a solution with q n if n(n;)(n;2); 9n(n;)+ 2n;2 =, giving n= The appearance of a half-integer here is not considered a violation of the Painleve test (see eg [2]) The half integer value of n gives rise to a series solution of (56), near a zero of p, of the form q = q q q ::: (53) with q arbitrary, andq q 2 ::: determined by q (and the arbitrary functions arising in the series (5) for p) The two integer values of n tell us that we needtocheck the consistency of solutions of (56) taking the form q = Q 2 + Q 3 + Q ::: (54) with two arbitrary functions Q and Q 2 This is indeed consistent using MAPLE we obtain Q = 2 Q p 2 ; Q +2 Q + Q +4 Q (55) with the choice of depending on the choice in (5) The general solution of (56) near a zero of p, with three arbitrary functions, is a linear combination of the series (53) and (54) Thus the system (55),(56) passes the WTC test The WTC test is strong evidence for the complete integrability of the system (55),(56) This in turn demonstrates that superch indeed has some integrable content 9

13 52 Second deconstruction of superch We now consider the system (5) with elds taking values in a grassmann algebra with two anticommuting fermionic generators, 2 Expanding in the basis f 2 2 g, u = u + 2 u = m = m + 2 m = (56) where the functions u u m m 2 2 are all standard, we obtain the system: m t = ;2m u x ; u m x m = u ; u xx (57) it = ; 3u 2 x i ; m 2 ix ; u ix i = i ; ixx i = 2 (58) m t = ;2m u x ; 2m u x ; u m x ; u m x +2( 2 ; 2 )+ 2( 3 x 2x ; 2x x ) m = u ; u xx : (59) Supersymmetry (43) tells us that given a solution u m of (57), we can solve the remaining equations by taking i = i u, i = i m (i= 2), u = u x and m = m x, where 2 are arbitrary constants We handle the system (57)-(59) following the procedure of the previous section Writing m =p 2 and changing coordinates to y y, the system can be written: u = p u = p 2 ; p p (52) p i = 3 ip ; 2 i p 4 p 3 i = i + p 3 ip p 3 ; 2 i p 2 i = 2 (52) m p 2 = ;(2u p) + 8( 2 ; 2 ) + 3p 3 4( 2 ; 2 ) 3p 3 m = u ; p(pu ) : (522) Applying the WTC Painleve test to this is a mammoth task, so instead we consider the Galileaninvariant reduction and apply the Painleve test at this level The Galilean-invariant reduction is obtained, as usual, by restricting all functions to depend on the single variable z=y ;vy alone Evidently the rst equations of both (52) and (522) can be integrated once immediately Then eliminating u from (52), i from (52) and m from (522), we obtain, i p p = ; p v + c p ; p 2 (523) ; 9p 2p i + p 2v ;5c p + 4 p 2 +3p2 2p 2 i; 3p p 2p v ;3c p + 3 p 2 +p2 p 2 i = i= 2 (524) u + p 2p p u + v ; u p 2 = d + 4 p ( 3 2 ; 2) (525)

14 where c d are integration constants The equation for p(z) maybeintegrated again after multiplying both sides by p =p this gives p 2 =; 2c p + c 2 p 2 ; 2 v p3 (526) where c 2 is another integration constant This equation is well known in KdV theory Its general solution can be written in terms of the Weierstrass }-function, p(z) =;2v}(z)+ 6 c 2v (527) where the periods of } are determined by the coecients c c 2 v Using (526), the coecients in (524) can be simplied Further, we know from supersymmetry that this equation has a solution i = p 2 Substituting i = p 2 q i the equation becomes a second order equation for q i : q i + 3p 2p q i + ; 3p 2v ; 3 2p + c 2 q 2 i = i = 2 : (528) 2 Supersymmetry (43) allows a reduction of the order of (525) as well It implies that u = p =p, i =p 2 is a solution So, writing u =rp =p i =p 2 q i in (525) yields a rst order equation for r : r + c 2 p ; 4p2 v ; p r p = p p d +4p(q q 2 ; q 2 q) : (529) Multiplying by the integrating factor p 2 =p and integrating, we obtain where d 2 is a further constant of integration r = p d p 2 p + d 2 +4 (q q 2 ; q 2 q)pp dz (53) Thus the Galilean-invariant reduction of the second deconstruction of superch takes the form of the three equations (526),(528),(53), to which we now apply the Painleve test All substitutions hitherto have been ones which do not interfere with the test Equation (526) has movable double poles and movable simple zeros Near a double pole at z, the series solution contains only even powers of (z ; z ), p(z) = ; 2v (z ; z ) + c 2v c ; c 2 2 v (z ; z ) v + c3 2 v ; 8c c (z ; z ) 4 + ::: (53) and near a simple zero at z, p(z) = (z ; z ) ; c 2 (z ; z ) 2 c 6 2(z ; z ) 3 ; ( v c c 2 )(z ; z ) 4 + ::: : (532) At both the zeros and poles of p, equation (528), which is just a linear third order ODE, has regular singular points Checking Painleve 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 (53) 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 =(z ; z ) terms, which would give rise to logarithms

15 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 Painleve property We note, in conclusion, that two of the equations we have encountered are interesting variants of the Lame equation: In (528), the substitution q i = p;3=4 h i h i yields p v ; c c p ; 7 h 2p 2 i = (533) and similarly, on writing u = p ;=2 k, the homogeneous part of (525) takes the form, By the arguments above, the latter is integrable by quadratures 3p k + v ; c 2 4 ; 3 k = : (534) 4p 2 6 Outlook In this paper we have examined fermionic extensions of the Camassa-Holm equation In particular we have identied the superch system (5), which, for low dimensional grassmann algebras displays some integrability properties Further investigation is needed in order to determine whether the superch system is integrable irrespective ofthechoice of grassmann algebra, and especially for the eld theoretically interesting case with innitely many odd generators We have not been able to nd a Lax pair for superch We also note that the peakon solutions of the Camassa-Holm equation do not admit supersymmetrisation (except when the grassmann algebra has just one fermionic generator) the peakon solutions are weak solutions, with a discontinuity in the rst derivative, and the action of the supercharge on such functions gives objects with insucient regularity properties to be considered as weak solutions Despite these open questions, our work provides a further instance of integrability arising in the setting of geodesic ow on a group manifold It remains a pressing open problem to understand integrability from this geometric viewpoint In this context, we should mention that the KP (and super KP) systems have yet to be presented as geodesic ow equations 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 fur Mathematik in den Naturwissenschaften for hospitality to JS during a visit to Leipzig and the Department of Mathematics and Computer Science of Bar-Ilan University for hospitality to CD during a visit to Israel 2

16 References [] R Camassa and DD Holm, An integrable shallow water equation with peaked solitons, patt-sol/9352, Phys Rev Lett 7 (993) 66 [2] R Camassa, DD Holm, and JM Hyman, A new integrable shallow water equation with peaked solutions, Adv Appl Mech 3 (994) -33 [3] VI Arnold, Mathematical methods of classical mechanics, Springer-Verlag (978) [4] VYu Ovsienko and BA Khesin, Korteweg-de Vries superequation as an Euler equation, Func Anal Appl 2 (987) [5] G Misiolek, Conjugate points in the Bott-Virasoro group and the KdV equation, Proc AMS 25 (997) [6] VI Arnold and BA Khesin, Topological methods in Hydrodynamics, Springer-Verlag (998) [7] PW Michor and TS Raitu, On the geometry of the Virasoro-Bott group, mathdg/985 [8] G Misiolek, A shallow water equation as a geodesic ow on the Bott-Virasoro group, J Geom Phys 24 (998) [9] S Shkoller, Geometry and curvature of dieomorphism groups with H metric and mean hydrodynamics, mathap/98778 [] S Kouranbaeva, The Camassa-Holm equation as a geodesic ow on the dieomorphism group, math-ph/9872 [] BA Kupershmidt, A super Korteweg-de Vries equation: an integrable system, PhysLett 2 A (984) [2] PMathieu, Supersymmetric extension of the Korteweg-de Vries equation, JMathPhys 29 (988) [3] YuI Manin and AO Radul, A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Commun Math Phys 98 (988) [4] Dhang, Integrability of fermionic extensions of the Korteweg-de Vries equation, PhysLettA 234 (997) [5] NS Manton, Deconstructing supersymmetry, hep-th/98677 [6] C Devchand and J Schi, The deconstruction of fermionic KdV systems, in preparation [7] J Schi, The Camassa-Holm equation: a loop group approach, solv-int/979, Physica D 2 (998) [8] JWeiss, MTabor and GCarnevale, The Painleve property for partial dierential equations, JMathPhys 24 (983) [9] ANW Hone, Exact solutions of the associated Camassa-Holm equation, preprint [2] A Ramani, B Grammaticos and TBountis, The Painleve property and singularity analysis of integrable and non-integrable systems, Phys Rep 8 (989)