solv-int/9811016

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

Chandrashekar Devchand ^{1,2} and Jeremy Schiff ^{3}

1 Max-Planck-Institut für Mathematik Gottfried-Claren-Straße 26, 53225 Bonn, Germany

2 Max-Planck-Institut für Mathematik in den Naturwissenschaften Inselstraße 22-26, 04103 Leipzig, Germany

devchand@mis.mpg.de

3 Department of Mathematics and Computer Science Bar–Ilan University, Ramat Gan 52900, 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 flow equations. Classes (a) and (b) have no intersection, but the intersection of classes (a) and (c) gives a candidate for a new supersymmetric integrable system. We demonstrate the Painlevé property for some simple but nontrivial reductions of this system.

1 Introduction

Recently there has been substantial interest in the Camassa-Holm (CH) equation [1, 2]:

u_{t} − νu_{xxt} = κu_{x} − 3uu_{x} + ν(uu_{xxx} + 2u_{x}u_{xx}) .  (1.1)

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 finite 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 fluid (Monge or inviscid Burgers equation),

u_{t} = −3uu_{x} .  (1.2)

This latter equation describes geodesic motion on the group of diffeomorphisms of the circle Diff(S^{1})

[3] with respect to a metric induced by an L^{2} 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 an H^{1} norm, ∫ (u^{2} + νu_{x}^{2})dx , the CH equation arises [8, 9, 10]. Both these ‘deformations’ have a regularising effect on solutions of (1.2), which exhibit discontinuous shocks.

Thus KdV and CH arise in a unified geometric setting; both are integrable systems which describe geodesic flows. This raises an important question: What features of the underlying geometry give rise to integrability? In general, geodesic flows of this type are not integrable: the Euler equation for fluid flow 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 flow. Is integrability also geometrically determined?

One further example of an integrable bihamiltonian system arising from a geodesic flow 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 [11]. 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 [12, 13]. 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 flow equations associated with L^{2} 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 flow equations 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_{2}u_{xxx} + β_{1}uu_{x} + β_{2}u_{x}u_{xx} + β_{3}uu_{xxx} + γ_{1}ξξ_{xx} + γ_{2}ξ_{x}ξ_{xxx} + γ_{3}ξξ_{xxxx}
ξ_{t} − μξ_{xxt} = σ_{1}ξ_{x} + σ_{2}ξ_{xxx} + ϵ_{1}u_{x}ξ + ϵ_{2}uξ_{x} + ρ_{1}uξ_{xxx} + ρ_{2}u_{x}ξ_{xx} + ρ_{3}u_{xx}ξ_{x} + ρ_{4}u_{xxx}ξ . (1.3)

Here u(x, t) and ξ(x, t) are fields valued, respectively, in the even and odd parts of a grassmann algebra, and {ν, μ, κ_{1}, κ_{2}, β_{1}, β_{2}, β_{3}, γ_{1}, γ_{2}, γ_{3}, σ_{1}, σ_{2}, ϵ_{1}, ϵ_{2}, ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}} are parameters. By rescaling u and ξ it is possible to set β_{1}= − 3 and γ_{1}=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 (1.3). In section 2, we consider geodesic flow 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 2, so there does not seem to be a bihamiltonian fermionic extension of CH. In section 4 we consider systems of the form (1.3) that are invariant under supersymmetry transformations between u and ξ. This class has nontrivial intersections with both the classes of sections 2 and 3. In particular, there is a unique supersymmetric geodesic flow system, which is a promising candidate for being integrable. In section 5 we show that two reductions of this system have the Painlevé property.

A trivial integrable CH system of the form (1.3), 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 (1.1)

\begin{aligned}u_{t} − νu_{xxt} & = & κu_{x} − 3uu_{x} + ν(uu_{xxx} + 2u_{x}u_{xx}) , \\ ξ_{t} − νξ_{xxt} & = & κξ_{x} − 3(ξu)_{x} + ν(ξu_{xxx} + uξ_{xxx} + 2(ξ_{x}u_{x})_{x}) . \end{aligned}  (1.4)

Replacing u by u + \frac{κ}{3} and considering the limit ν → 0 , κ → ∞, with νκ = 3, yields the system

\begin{aligned}u_{t} & = & −3uu_{x} + u_{xxx} , \\ ξ_{t} & = & −3(ξu)_{x} + ξ_{xxx} . \end{aligned}  (1.5)

This trivial fermionic extension of KdV has appeared often in the literature (see e.g. [12]).

2 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 [3]. Define a bilinear operator B : 𝔤 × 𝔤 → 𝔤 by

⟨[V, W] , U⟩ = ⟨W , B(U, V)⟩ ,  ∀ W ∈ 𝔤 .  (2.1)

The geodesic flow equation is then simply

U_{t} = B(U, U) .  (2.2)

In our case, 𝔤 is the NSR superconformal algebra, consisting of triples (u(x), φ(x), a) , where u is a bosonic field, φ is a fermionic field and a is a constant. The Lie bracket is given by

\begin{matrix}[(u, φ, a) , (v, ψ, b)] \\ = (uv_{x}−u_{x}v+\frac{1}{2}φψ , uψ_{x}−\frac{1}{2}u_{x}ψ−φ_{x}v+\frac{1}{2}φv_{x} , ∫ dx(c_{1}u_{x}v_{xx}+c_{2}uv_{x}+c_{1}φ_{x}ψ_{x}+\frac{c_{2}}{4} φψ)) , \end{matrix} (2.3)

where c_{1}, c_{2} are constants. On this algebra, an H^{1} inner product is given by

⟨(u, φ, a) , (v, ψ, b)⟩ = ∫ dx (uv + νu_{x}v_{x} + αφ∂_{x}^{−1}ψ + αμφ_{x}ψ) + ab
= ∫ dx (u Δ_{0} v + φ Δ_{1} ψ) + ab ,  (2.4)

where

Δ_{0} = 1 − ν∂_{x}^{2} , Δ_{1} = α (∂_{x}^{−1} − μ∂_{x}) ,  (2.5)

and μ, ν, α are further constants, all assumed nonzero. Writing U=(u, φ, a) , V =(v, ψ, b) , we find B(U, V) = (B_{0}, B_{1}, 0) , where

Δ_{0}B_{0}(U, V) = − (2v_{x}Δ_{0}u + vΔ_{0}u_{x} + \frac{3}{2}ψ_{x}Δ_{1}φ + \frac{1}{2}ψΔ_{1}φ_{x}) + a(c_{1}v_{xxx} − c_{2}v_{x}) ,
Δ_{1}B_{1}(U, V) = − (\frac{3}{2}v_{x}Δ_{1}φ + vΔ_{1}φ_{x} + \frac{1}{2}ψΔ_{0}u) + a(c_{1}ψ_{xx} − \frac{c_{2}}{4} ψ) .

(2.6)

The geodesic equations are therefore conveniently written in the form

\begin{aligned}Δ_{0} u_{t} & = & Δ_{0} B_{0}(U, U) \\ Δ_{0} φ_{t} & = & Δ_{1} B_{1}(U, U) \\ a_{t} & = & 0 . \end{aligned}  (2.7)

Writing φ = λξ_{x}, where λ is a constant satisfying λ^{2} = \frac{4}{3α} , this yields the system

\begin{aligned}u_{t} − νu_{xxt} & = & κ_{1}u_{x} + κ_{2}u_{xxx} − 3uu_{x} + ν(uu_{xxx} + 2u_{x}u_{xx}) + 2ξξ_{xx} + \frac{2μ}{3}ξ_{x}ξ_{xxx} , \\ ξ_{t} − μξ_{xxt} & = & \frac{κ_{1}}{4α}ξ_{x} + \frac{κ_{2}}{α} ξ_{xxx} − \frac{3}{2}u_{x}ξ − (1 + \frac{1}{2α})uξ_{x} + μuξ_{xxx} + \frac{3μ}{2}u_{x}ξ_{xx} + \frac{ν}{2α}u_{xx}ξ_{x} . \end{aligned} (2.8)

Here κ_{1}, κ_{2} are independent parameters determined by a, c_{1}, c_{2} . This is evidently a 5 parameter class of systems of type (1.3).

Setting ξ to zero in (2.8) yields the CH result of [8, 9, 10]. If instead we choose μ, ν to vanish, the H^{1} norm becomes an L^{2} norm; then choosing κ_{1} to be zero and rescaling κ_{2} to 1 we obtain the following 1 parameter fermionic extension of KdV:

\begin{aligned}u_{t} & = & u_{xxx} − 3uu_{x} + 2ξξ_{xx} , \\ ξ_{t} & = & \frac{1}{α}ξ_{xxx} − \frac{3}{2}u_{x}ξ − (1 + \frac{1}{2α})uξ_{x} . \end{aligned}  (2.9)

Modulo rescalings, the superKdV of Mathieu and Manin-Radul is obtained by taking α = 1 . The kuperKdV system arises by taking α = \frac{1}{4} , the choice made in [4]. Other values of the parameters give systems which are not believed to be integrable (see however [14]).

3 Hamiltonian flows

Like KdV, CH has bihamiltonian structure, and this accounts for its integrability. We might hope that for some choices of parameters the system (2.8) 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 − νu_{xx} and η = ξ − μξ_{xx}, (2.8) takes the form

( \begin{matrix}m_{t} \\ η_{t} \end{matrix} ) = P_{2} ( \frac{δH_{2}}{δm} ; \frac{δH_{2}}{δη} )  (3.1)

where

P_{2} = ( \begin{matrix}κ_{2}∂_{x}^{3} + κ_{1}∂_{x} − ∂_{x}m − m∂_{x} & \frac{1}{2}∂_{x}η + η∂_{x} \\ −∂_{x}η − \frac{1}{2}η∂_{x} & \frac{3}{4α}(\frac{κ_{1}}{4} + κ_{2}∂_{x}^{2}) − \frac{3m}{8} \end{matrix} )  (3.2)

and the hamiltonian functional is given succinctly by the H^{1} inner product on the algebra,

H_{2} = \frac{1}{2} ⟨U , U⟩ = \frac{1}{2} ∫ dx (u^{2} + νu_{x}^{2} + \frac{4}{3}(ξ_{x}ξ + μξ_{xx}ξ_{x})) .  (3.3)

This generalises the so-called second Hamiltonian structure of KdV and its fermionic extensions [11, 12]. Checking (3.1) is straightforward: the Euler-Lagrange derivatives \frac{δH_{2}}{δm} , \frac{δH_{2}}{δη} are defined by

δH_{2} = ∫ dx (\frac{δH_{2}}{δm} δm + \frac{δH_{2}}{δη} δη) ,  (3.4)

from which it follows immediately that \frac{δH_{2}}{δm} = u and \frac{δH_{2}}{δη} = \frac{4}{3}ξ_{x}.

To investigate the possibility of systems amongst (2.8) having another hamiltonian form, we look at systems of the form

( \begin{matrix}m_{t} \\ η_{t} \end{matrix} ) = P_{1} ( \frac{δH_{1}}{δm} ; \frac{δH_{1}}{δη} ) ,  (3.5)

where

P_{1} = ( \begin{matrix}∂_{x}(1−ν∂_{x}^{2}) & 0 \\ 0 & −\frac{ϵ_{1}}{2} (1−μ∂_{x}^{2}) \end{matrix} ) .  (3.6)

Here ϵ_{1} is a constant and H_{1} is a functional generalising the KdV first Hamiltonian,

H_{1} = ∫ dx (−\frac{1}{2}u^{3} − \frac{β_{3}}{2} uu_{x}^{2} − \frac{κ_{2}}{2} u_{x}^{2} + \frac{κ_{1}}{2} u^{2} + \frac{σ_{1}}{ϵ_{1}} ξξ_{x} + \frac{σ_{2}}{ϵ_{1}} ξξ_{xxx}
+2uξξ_{x} + (γ_{2} − γ_{3})uξ_{x}ξ_{xx} + γ_{3}uξξ_{xxx}) .  (3.7)

This is the most general functional of this type, up to rescalings of u and ξ. Since δm=(1−ν∂_{x}^{2})δu, we have (1 − ν∂_{x}^{2})\frac{δH_{1}}{δm} = \frac{δH_{1}}{δu} , and similarly (1 − μ∂_{x}^{2})\frac{δH_{1}}{δη} = \frac{δH_{1}}{δξ} . Thus equations (3.5) take the simple form

u_{t} − νu_{xxt} = ∂_{x} (\frac{δH_{1}}{δu} )
= κ_{1}u_{x} + κ_{2}u_{xxx} − 3uu_{x} + β_{3}(2u_{x}u_{xx} + uu_{xxx}) + 2ξξ_{xx} + γ_{2}ξ_{x}ξ_{xxx} + γ_{3}ξξ_{xxxx}
ξ_{t} − μξ_{xxt} = ϵ_{1} (\frac{δH_{1}}{δξ} )
= σ_{1}ξ_{x} + σ_{2}ξ_{xxx} + ϵ_{1}(u_{x}ξ + 2uξ_{x}) + ϵ_{1}(2γ_{3} − γ_{2})uξ_{xxx} + \frac{3}{2}ϵ_{1}(2γ_{3} − γ_{2})u_{x}ξ_{xx}
+\frac{1}{2}ϵ_{1}(4γ_{3} − γ_{2})u_{xx}ξ_{x} + \frac{1}{2}ϵ_{1}γ_{3}u_{xxx}ξ .  (3.8)

This is a 10 parameter class of systems of the form (1.3). Comparing with (2.8), we see that the only bihamiltonian systems occur when \{μ=ν=β_{3}=γ_{2}=γ_{3}=0 , ϵ_{1}=− \frac{3}{2} , σ_{1}=κ_{1} , σ_{2}=4κ_{2}\}, which is equivalent to (2.8) with \{μ=ν=0 , α=\frac{1}{4}\}, i.e. the kuperKdV system. Thus, no new bihamiltonian systems arise.

We note that the systems (3.8) 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 (\frac{1}{2}(f_{x} − νf_{xxx})f_{t} + \frac{1}{ϵ_{1}} (ξ − μξ_{xx})ξ_{t} + \frac{1}{2}f_{x}^{3} + \frac{β_{3}}{2} f_{x}f_{xx}^{2} + \frac{κ_{2}}{2} f_{xx}^{2} − \frac{κ_{1}}{2} f_{x}^{2}
−\frac{σ_{1}}{ϵ_{1}} ξξ_{x} − \frac{σ_{2}}{ϵ_{1}} ξξ_{xxx} − 2f_{x}ξξ_{x} + (γ_{3} − γ_{2})f_{x}ξ_{x}ξ_{xx} − γ_{3}f_{x}ξξ_{xxx}) .  (3.9)

4 Supersymmetric flows

Define a fermionic superfield Φ(x, ϑ) = sξ + ϑu and superderivative D = \frac{∂}{∂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 (1.3) is the 8 parameter system,

(1 − νD^{4}) Φ_{t} = κ_{1}D^{2}Φ + κ_{2}D^{6}Φ − \frac{2}{s^{2}} ΦD^{3}Φ + (\frac{2}{s^{2}} − 3) DΦD^{2}Φ + (\frac{γ_{3}}{s^{2}} +β_{3}) DΦD^{6}Φ
−\frac{γ_{3}}{s^{2}} ΦD^{7}Φ + (β_{3}+\frac{γ_{3}−γ_{2}}{s^{2}} ) D^{2}ΦD^{5}Φ + (β_{2}−β_{3}+\frac{γ_{2}−γ_{3}}{s^{2}} ) D^{3}ΦD^{4}Φ , (4.1)

where {ν, s, κ_{1}, κ_{2}, β_{2}, β_{3}, γ_{2}, γ_{3}} are parameters. The component equations are,

u_{t} − νu_{xxt} = κ_{1}u_{x} + κ_{2}u_{xxx} − 3uu_{x} + β_{2}u_{x}u_{xx} + β_{3}uu_{xxx} + 2ξξ_{xx} + γ_{2}ξ_{x}ξ_{xxx} + γ_{3}ξξ_{xxxx} ,
ξ_{t} − νξ_{xxt} = κ_{1}ξ_{x} + κ_{2}ξ_{xxx} − \frac{2}{s^{2}} u_{x}ξ + (\frac{2}{s^{2}} − 3) uξ_{x} + (\frac{γ_{3}}{s^{2}} + β_{3}) uξ_{xxx}
+ (β_{2} − β_{3} + \frac{γ_{2}−γ_{3}}{s^{2}} ) u_{x}ξ_{xx} + (\frac{γ_{3}−γ_{2}}{s^{2}} + β_{3}) u_{xx}ξ_{x} − \frac{γ_{3}}{s^{2}} u_{xxx}ξ .  (4.2)

These systems are by construction invariant under the supersymmetry transformations

δu = τξ_{x} , δξ = \frac{τu}{s^{2}} ,  (4.3)

where τ is an odd parameter. The superKdV limit, namely {ν, β_{2}, β_{3}, γ_{2}, γ_{3}, κ_{1}} all zero, yields, modulo rescalings, the one-parameter family of systems studied by Mathieu [12].

By comparing (4.2) and (3.8) it is straightforward to extract systems which are both supersymmetric and have hamiltonian form (3.5),(3.6). Taking s^{2}=2 in (4.2), {ν=μ, σ_{1}=κ_{1} , σ_{2}=κ_{2} , ϵ=−1\} in (3.8), and \{β_{2}=2β_{3} , β_{3} = γ_{2}−\frac{5}{2}γ_{3}\} in both, we obtain the systems,

u_{t} − νu_{xxt} = κ_{1}u_{x} + κ_{2}u_{xxx} − 3uu_{x} + (γ_{2} − \frac{5}{2}γ_{3})(2u_{x}u_{xx} + uu_{xxx})
+2ξξ_{xx} + γ_{2}ξ_{x}ξ_{xxx} + γ_{3}ξξ_{xxxx} ,
ξ_{t} − νξ_{xxt} = κ_{1}ξ_{x} + κ_{2}ξ_{xxx} − u_{x}ξ − 2uξ_{x} + (γ_{2} − 2γ_{3}) uξ_{xxx}
+\frac{3}{2} (γ_{2} − 2γ_{3}) u_{x}ξ_{xx} + \frac{1}{2} (γ_{2} − 4γ_{3}) u_{xx}ξ_{x} − \frac{1}{2}γ_{3}u_{xxx}ξ .  (4.4)

These may be expressed in superfield form (4.1) with the above choice of parameters. The manifestly supersymmetric hamiltonian form is given by

M_{t} = P̂_{1} \frac{δĤ_{1}}{δM} , M = Φ − νD^{4}Φ ,  (4.5)

with

P̂_{1} = D(1 − νD^{4}) ,  (4.6)
Ĥ_{1} = ∫ dxdϑ (\frac{κ_{1}}{2} ΦDΦ − \frac{κ_{2}}{2} D^{2}ΦD^{3}Φ − \frac{1}{2}Φ(DΦ)^{2}
+\frac{1}{4}γ_{3}Φ(D^{3}Φ)^{2} + \frac{1}{4}(γ_{2} − 2γ_{3})(DΦ)^{2}D^{4}Φ) .  (4.7)

Since the KdV reduction of (4.4) (with κ_{1} = γ_{2} = 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 flow equations of section 2 are also supersymmetric. Comparing (2.8) with (4.2), we see that the choice \{μ=ν, α=1 , κ_{1}=0\} in the former and \{s^{2}=\frac{4}{3} , β_{2}=2ν, β_{3}=ν, γ_{2}=\frac{2ν}{3} , γ_{3}=κ_{1}=0\} in the latter, yields the two-parameter system of supersymmetric geodesic flow equations:

u_{t} − νu_{xxt} = κ_{2}u_{xxx} − 3uu_{x} + 2ξξ_{xx} + ν(uu_{xxx} + 2u_{x}u_{xx}) + \frac{2ν}{3} ξ_{x}ξ_{xxx} ,
ξ_{t} − νξ_{xxt} = κ_{2}ξ_{xxx} − \frac{3}{2}(uξ)_{x} + ν(uξ_{xxx} + \frac{3}{2}u_{x}ξ_{xx} + \frac{1}{2}u_{xx}ξ_{x}) .  (4.8)

We shall call this system, with κ_{2}=0 and ν≠0 , the supersymmetric Camassa-Holm equation (superCH). In section 5, we present some evidence for its integrability. The system (4.8) reduces to superKdV, upon setting ν to zero, and to CH, upon setting ξ to zero and translating u.

Not surprisingly, the systems (4.8) arise as geodesic flow equations precisely when the metric (2.4) on the NSR superconformal algebra is supersymmetric. Then, the calculations of section 2 can be performed using superfields. Specifically, writing U = u + ϑϕ and V = v + ϑψ, the bracket (2.3) takes the form

[(U, a) , (V, b)] = (UD^{2}V − VD^{2}U + \frac{1}{2}DUDV , c_{1} ∫ dxdϑ D^{2}UD^{3}V)  (4.9)

and the inner product (2.4) may be written

⟨(U, a) , (V, b)⟩ = ∫ dxdϑ (UD^{−1}V + νD^{2}UDV) + ab .  (4.10)

The superspace bilinear operator B̂ is given by B̂((U, a), (V, b)) = (B̂_{0}, 0) , where B̂_{0} satisfies

(1 − νD^{4})D^{−1}B̂_{0} = c_{1}aD^{5}V − \frac{3}{2}D^{2}V(1−νD^{4})D^{−1}U − \frac{1}{2}DV(1−νD^{4})U − V(1−νD^{4})DU . (4.11)

Writing c_{1}a=κ_{2} and U = DΦ, the geodesic flow equations (U_{t}, a_{t}) = B̂((U, a), (U, a)) yield

(1 − νD^{4})Φ_{t} = κ_{2}D^{6}Φ − \frac{3}{2}(ΦD^{3}Φ + DΦD^{2}Φ) + ν (DΦD^{6}Φ + \frac{1}{2}D^{2}ΦD^{5}Φ + \frac{3}{2}D^{3}ΦD^{4}Φ) . (4.12)

We thus recover the subsystem of (4.1) having component content (4.8). Equation (4.12) has superfield hamiltonian formulation,

M_{t} = P̂_{2} \frac{δĤ_{2}}{δM} , M = Φ − νD^{4}Φ ,  (4.13)

with

P̂_{2} = κ_{2}D^{5} − \frac{1}{2}DMD − D^{2}M − MD^{2} ,  (4.14)
Ĥ_{2} = \frac{1}{2} ⟨(DΦ, 0) , (DΦ, 0)⟩ = \frac{1}{2} ∫ dxdϑ ΦDM .  (4.15)

5 Painlevé integrability of superCH systems

In this section we investigate, in more detail, the supersymmetric geodesic flow system (4.8) with ν=1 and κ_{2} = 0,

\begin{aligned}m_{t} & = & −2mu_{x} − um_{x} + 2ηξ + \frac{2}{3}η_{x}ξ_{x} , & m = u − u_{xx} , \\ η_{t} & = & −\frac{3}{2}ηu_{x} − \frac{1}{2}mξ_{x} − uη_{x} , & η = ξ − ξ_{xx} . \end{aligned}  (5.1)

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 [15] 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 [16] we investigate fermionic extensions of KdV in a similar fashion.

5.1 First deconstruction of superCH

We first consider the superCH system (5.1) 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^{2} = 0 , the fermionic bilinear terms do not contribute and we are left with the system

\begin{aligned}m_{t} & = & −2mu_{x} − um_{x} , & m = u − u_{xx} \\ η_{1t} & = & −\frac{3}{2}η_{1}u_{x} − \frac{1}{2}mξ_{1x} − uη_{1x} , & η_{1} = ξ_{1} − ξ_{1xx} . \end{aligned}  (5.2)

Further analysis is simplified by changing coordinates as described in [17]. Writing m=p^{2} , the first equation of (5.2) 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 ,  (5.3)

or dually, via

\frac{∂}{∂x} = p\frac{∂}{∂y_{0}} , \frac{∂}{∂t} = \frac{∂}{∂y_{1}} − pu\frac{∂}{∂y_{0}} .  (5.4)

Implementing this coordinate change and eliminating the functions u and ξ_{1} , the remaining equations for p and q ≡ η_{1} are:

p^{2} ._{p}′′ − p(._{p}p′′ + ._{p}′p′) + ._{p}p^{′2} − 2p^{3}p′ − ._{p} = 0 ,  (5.5)

._{q}′′ − \frac{3p′}{p} ._{q}′ − \frac{3._{p}}{2p}q′′ + (\frac{4p^{′2}}{p^{2}} − \frac{2p′′}{p} − \frac{1}{p^{2}} ) ._{q} + (\frac{15p′._{p}}{2p^{2}} − \frac{3._{p}′}{p} − \frac{p}{2}) q′

+3 ( \frac{._{p}p′′ + 2p′._{p}′}{p^{2}} − \frac{4._{p}p^{′2}}{p^{3}} − p′) q = 0 .  (5.6)

Here the dot and prime denote differentiations with respect to y_{1} and y_{0} respectively. We note:

(a) thanks to supersymmetry (4.3), if p is a solution of (5.5), then q=p^{2} is a solution of (5.6); and (b) under the substitution q = p^{3/2}r , (5.6) takes the substantially simpler form

._{r}′′ + (\frac{p^{′2}}{4p^{2}} − \frac{p′′}{2p} − \frac{1}{p^{2}} ) ._{r} − \frac{p}{2}r′ − \frac{3p′}{4} r = 0 .  (5.7)

The system (5.5),(5.6) passes the WTC Painlevé test.

Proof: Equation (5.5) is a rescaled version of the Associated Camassa-Holm equation of [17]. Consideration of solutions with p(y_{0}, y_{1}) ∼ p_{0}(y_{0}, y_{1})ϕ(y_{0}, y_{1})^{n} near ϕ(y_{0}, y_{1}) = 0, for some n ≠ 0, yields n = −2 or n = 1 as the possible leading orders of Laurent series solutions. We need to perform the WTC Painlevé test [18] for both these types of series. The first type, namely, Laurent series solutions exhibiting double poles on the singular manifold ϕ(y_{0}, y_{1}) = 0, have already been considered in [19]. These take the form

p = \frac{2ϕ′._{ϕ}}{ϕ^{2}} − \frac{._{ϕ}′}{ϕ} + p_{2} + p_{3}ϕ + p_{4}ϕ^{2} + …,  (5.8)

where ϕ, p_{2}, p_{4} are arbitrary functions of y_{0}, y_{1}, and

p_{3} = _{\frac{−1}{2ϕ^{′2} ._{ϕ}^{2}}} (ϕ^{′2} ._{ϕ}._{p}_{2} + ϕ′._{ϕ}^{2}p′_{2} − (ϕ^{′2} .._{ϕ} − 2ϕ′._{ϕ}._{ϕ}′ + ϕ′′._{ϕ}^{2}) p_{2}
− (ϕ′._{ϕ}.._{ϕ}′′ − ϕ′ .._{ϕ}._{ϕ}′′ − ._{ϕ}ϕ′′ .._{ϕ}′ + .._{ϕ}ϕ′′._{ϕ}′)) .  (5.9)

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 ϕ(y_{0}, y_{1}) = 0. They take the form

p = ±\frac{ϕ}{ϕ′} + p_{2}ϕ^{2} + p_{3}ϕ^{3} + …,  (5.10)

where ϕ, p_{2}, 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 (5.5).

It remains to look at the equation (5.6). 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 q. If p has a pole on ϕ=0 , then near ϕ=0 we have p ∼ 2 ._{ϕ}ϕ′/ϕ^{2} , and equation (5.6) takes the form

._{q}′′ + (\frac{6ϕ′}{ϕ} + …) ._{q}′ + (\frac{3 ._{ϕ}}{ϕ}+ …) q′′ + (\frac{4ϕ^{′2}}{ϕ^{2}} + …) ._{q} + (\frac{11ϕ′._{ϕ}}{ϕ^{2}}+ …) q′ + (O (\frac{1}{ϕ^{2}} )) q = 0 .

Thus the equation has a solution with q ∼ ϕ^{n} if n(n−1)(n−2)+9n(n−1)+15n = 0 , giving n= − 4, −2, 0 . It follows that in the case when p is given by the series (5.8), no inconsistencies will arise near the double poles of p if (5.6) has a series solution of the form

q = \frac{q_{0}}{ϕ^{4}} + \frac{q_{1}}{ϕ^{3}} + \frac{q_{2}}{ϕ^{2}} + \frac{q_{3}}{ϕ} + q_{4} + …  (5.11)

with q_{0}, q_{2}, q_{4} arbitrary. The consistency of such a solution can easilly be verified using a symbolic manipulator. Using MAPLE we find that

q_{1} = \frac{2ϕ′′q_{0} − ϕ′q′_{0}}{ϕ^{′2}} .  (5.12)

The explicit expression for q_{3} is too lengthy to be given here.

Suppose now that p has a zero on ϕ=0. Near this, p ∼ ±ϕ/ϕ′ and equation (5.6) has the structure

._{q}′′−(\frac{3ϕ′}{ϕ} + …) ._{q}′−(\frac{3 ._{ϕ}}{2ϕ} + …) q′′+(\frac{3ϕ^{′2}}{ϕ^{2}} + …) ._{q}+(\frac{15ϕ′._{ϕ}}{2ϕ^{2}} + …) q′−(\frac{12ϕ^{′2} ._{ϕ}}{ϕ^{3}} + …) q = 0 .

Thus (5.6) has a solution with q ∼ ϕ^{n} if n(n−1)(n−2)−\frac{9}{2}n(n−1)+\frac{21}{2}n−12 = 0 , giving n=\frac{3}{2}, 2, 4. The appearance of a half-integer here is not considered a violation of the Painlevé test (see e.g. [20]). The half integer value of n gives rise to a series solution of (5.6), near a zero of p, of the form

q = q_{0}ϕ^{\frac{3}{2}} + q_{1}ϕ^{\frac{5}{2}} + q_{2}ϕ^{\frac{7}{2}} + …  (5.13)

with q_{0} arbitrary, and q_{1}, q_{2}, … determined by q_{0} (and the arbitrary functions arising in the series (5.10) for p). The two integer values of n tell us that we need to check the consistency of solutions of (5.6) taking the form

q = Q_{0}ϕ^{2} + Q_{1}ϕ^{3} + Q_{2}ϕ^{4} + …  (5.14)

with two arbitrary functions Q_{0} and Q_{2}. This is indeed consistent; using MAPLE we obtain

Q_{1} = ±2ϕ′Q_{0}p_{2} − \frac{1}{3ϕ^{′2} ._{ϕ}} (2ϕ^{′2} ._{Q_{0}} + 2ϕ′′._{ϕ}Q_{0} + ϕ′._{ϕ}Q′_{0} + 4ϕ′._{ϕ}′Q_{0}) ,  (5.15)

with the choice of ± depending on the choice in (5.10). The general solution of (5.6) near a zero of p, with three arbitrary functions, is a linear combination of the series (5.13) and (5.14). Thus the system (5.5),(5.6) passes the WTC test.

The WTC test is strong evidence for the complete integrability of the system (5.5),(5.6). This in turn demonstrates that superCH indeed has some integrable content.

5.2 Second deconstruction of superCH

We now consider the system (5.1) with fields taking values in a grassmann algebra with two anticommuting fermionic generators, τ_{1} , τ_{2} . Expanding in the basis {1, τ_{1}, τ_{2}, τ_{1}τ_{2}},

\begin{matrix}u = u_{0} + τ_{1}τ_{2} u_{1} , & ξ = τ_{1}ξ_{1} + τ_{2}ξ_{2} , \\ m = m_{0} + τ_{1}τ_{2} m_{1} , & η = τ_{1}η_{1} + τ_{2}η_{2} , \end{matrix}  (5.16)

where the functions u_{0}, u_{1}, m_{0}, m_{1}, ξ_{1}, ξ_{2}, η_{1}, η_{2} are all standard, we obtain the system:

m_{0t} = −2m_{0}u_{0x} − u_{0}m_{0x} ,  m_{0} = u_{0} − u_{0xx} ,  (5.17)
η_{it} = −\frac{3}{2}u_{0x}η_{i} − \frac{1}{2}m_{0}ξ_{ix} − u_{0}η_{ix} ,  η_{i} = ξ_{i} − ξ_{ixx} ,  i = 1, 2 , (5.18)
m_{1t} = −2m_{1}u_{0x} − 2m_{0}u_{1x} − u_{0}m_{1x} − u_{0}m_{1x}
+2(η_{1}ξ_{2} − η_{2}ξ_{1}) + \frac{2}{3}(η_{1x}ξ_{2x} − η_{2x}ξ_{1x}) ,  m_{1} = u_{1} − u_{1xx} .  (5.19)

Supersymmetry (4.3) tells us that given a solution u_{0}, m_{0} of (5.17), we can solve the remaining equations by taking ξ_{i} = α_{i}u_{0} , η_{i} = α_{i}m_{0} (i=1, 2), u_{1} = βu_{0x} and m_{1} = βm_{0x}, where α_{1}, α_{2}, β are arbitrary constants.

We handle the system (5.17)-(5.19) following the procedure of the previous section. Writing m_{0}=p^{2} and changing coordinates to y_{0}, y_{1}, the system can be written:

u′_{0} = (\frac{1}{p}). ,  u_{0} = p^{2} − p ( \frac{._{p}}{p})′ ,  (5.20)

ξ′_{i} = \frac{3η_{i}._{p}}{p^{4}} − \frac{2._{η}_{i}}{p^{3}} ,  ξ_{i} = η_{i} + p (\frac{3η_{i}._{p}}{p^{3}} − \frac{2._{η}_{i}}{p^{2}} )′ ,  i = 1, 2 ,  (5.21)

(\frac{m_{1}}{p^{2}} ). = −(2u_{1}p)′ + (\frac{8(._{η}_{1}η_{2} − ._{η}_{2}η_{1})}{3p^{3}})′ + (\frac{4(η′_{1}η_{2} − η′_{2}η_{1})}{3p^{3}}). ,
m_{1} = u_{1} − p(pu′_{1})′ .  (5.22)

Applying the WTC Painlevé test to this is a mammoth task, so instead we consider the Galileaninvariant reduction and apply the Painlevé test at this level. The Galilean-invariant 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 (5.20) and (5.22) can be integrated once immediately. Then eliminating u_{0} from (5.20), ξ_{i} from (5.21) and m_{1} from (5.22), we obtain,

(\frac{p′}{p} )′ = −\frac{p}{v} + \frac{c_{1}}{p} − \frac{1}{p^{2}} ,  (5.23)
η′′′_{i} − \frac{9p′}{2p} η′′_{i} + (\frac{11p}{2v}−\frac{5c_{1}}{p} +\frac{4}{p^{2}} +\frac{13p^{′2}}{2p^{2}} ) η′_{i}−\frac{3p′}{p} (\frac{2p}{v}−\frac{3c_{1}}{p} +\frac{3}{p^{2}} +\frac{p^{′2}}{p^{2}} ) η_{i} = 0, i=1, 2, (5.24)
u′′_{1} + \frac{p′}{p} u′_{1} + (\frac{2p}{v} − \frac{1}{p^{2}} ) u_{1} = d_{1} + \frac{4}{p^{3}} (η_{1}η′_{2} − η_{2}η′_{1}) ,  (5.25)

where c_{1}, d_{1} are integration constants. The equation for p(z) may be integrated again after multiplying both sides by p′/p; this gives

p^{′2} = 1 − 2c_{1}p + c_{2}p^{2} − \frac{2}{v}p^{3} ,  (5.26)

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) + \frac{1}{6}c_{2}v ,  (5.27)

where the periods of ℘ are determined by the coefficients c_{1}, c_{2}, v. Using (5.26), the coefficients in (5.24) can be simplified. Further, we know from supersymmetry that this equation has a solution η_{i} = p^{2}. Substituting η_{i} = p^{2}q_{i} the equation becomes a second order equation for q′_{i}:

q′′′_{i} + \frac{3p′}{2p} q′′_{i} + (−\frac{3p}{2v} − \frac{3}{2p^{2}} + \frac{c_{2}}{2} ) q′_{i} = 0 ,  i = 1, 2 .  (5.28)

Supersymmetry (4.3) allows a reduction of the order of (5.25) as well. It implies that u_{1} = p′/p , η_{i}=p^{2} is a solution. So, writing u_{1}=rp′/p , η_{i}=p^{2}q_{i} in (5.25) yields a first order equation for r′:

r′′ + (c_{2}p − \frac{4p^{2}}{v} − \frac{1}{p}) \frac{r′}{p′} = \frac{p}{p′} (d_{1} + 4p(q_{1}q′_{2} − q_{2}q′_{1})) .  (5.29)

Multiplying by the integrating factor p^{′2}/p and integrating, we obtain

r′ = \frac{p}{p^{′2}} (d_{1}p + d_{2} + 4 ∫ (q_{1}q′_{2} − q_{2}q′_{1})pp′ dz) ,  (5.30)

where d_{2} is a further constant of integration.

Thus the Galilean-invariant reduction of the second deconstruction of superCH takes the form of the three equations (5.26),(5.28),(5.30), to which we now apply the Painlevé test. All substitutions hitherto have been ones which do not interfere with the test. Equation (5.26) 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}),

p(z) = −\frac{2v}{(z − z_{0})^{2}} + \frac{c_{2}v}{6} + \frac{12c_{1} − c_{2}^{2}v}{120} (z − z_{0})^{2} + \frac{\frac{54}{v} + c_{2}^{3}v − 18c_{1}c_{2}}{3024} (z − z_{0})^{4} + … (5.31)

and near a simple zero at z_{0} ,

p(z) = ±(z − z_{0}) − \frac{1}{2}c_{1}(z − z_{0})^{2} ± \frac{1}{6}c_{2}(z − z_{0})^{3} − \frac{1}{24}(\frac{6}{v} + c_{1}c_{2})(z − z_{0})^{4} + ….  (5.32)

At both the zeros and poles of p, equation (5.28), 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 (5.30) 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 (5.28), the substitution q′_{i} = p^{−3/4}h_{i} yields

h′_{i}′ + \frac{3}{8} (\frac{p}{v} − \frac{c_{2}}{6} + \frac{c_{1}}{p} − \frac{7}{2p^{2}} ) h_{i} = 0 ,  (5.33)

and similarly, on writing u_{1} = p^{−1/2}k , the homogeneous part of (5.25) takes the form,

k′′ + (\frac{3p}{v} − \frac{c_{2}}{4} − \frac{3}{4p^{2}} ) k = 0 .  (5.34)

By the arguments above, the latter is integrable by quadratures.

6 Outlook

In this paper we have examined fermionic extensions of the Camassa-Holm equation. In particular we have identified the superCH system (5.1), 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 of the choice of grassmann algebra, and especially for the field theoretically interesting case with infinitely many odd generators. We have not been able to find 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 first derivative, and the action of the supercharge on such functions gives objects with insufficient 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 flow 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 flow 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 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.

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