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ITP-UH-17/08 On the N=2 Supersymmetric Camassa-Holm and Hunter-Saxton Equations 9 J. Lenellsa and O. Lechtenfeldb 0 0 2 a Department of Applied Mathematics and Theoretical Physics, n University of Cambridge, Cambridge CB3 0WA, United Kingdom a J b Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover, 7 Appelstraße 2, 30167 Hannover, Germany 2 [email protected], [email protected] ] I S . n i l n [ 3 v Abstract 7 WeconsiderN=2supersymmetricextensionsofthe Camassa-HolmandHunter- 7 0 Saxton equations. We show that they admit geometric interpretations as Eu- 0 ler equations on the superconformal algebra of contact vector fields on the 12- . dimensional supercircle. We use the bi-Hamiltonian formulation to derive L|ax 9 0 pairs. Moreover,wepresentsomesimpleexamplesofexplicitsolutions. As aby- 8 product of our analysis we obtain a description of the bounded traveling-wave 0 solutions for the two-component Hunter-Saxton equation. : v i X r a PACS: 02.30.Ik, 11.30.Pb Keywords: Integrablesystem;supersymmetry;Camassa-Holmequation;bi-Hamiltonian structure. 1 Introduction The Camassa-Holm (CH) equation u u +3uu = 2u u +uu , x R, t > 0, (CH) t txx x x xx xxx − ∈ and the Hunter-Saxton (HS) equation u = 2u u uu , x R, t > 0, (HS) txx x xx xxx − − ∈ where u(x,t) is a real-valued function, are integrable models for the propagation of nonlinear waves in 1+1-dimension. Equation (CH) models the propagation of shal- low water waves over a flat bottom, u(x,t) representing the water’s free surface in non-dimensional variables. It was first obtained mathematically [20] as an abstract equation with two distinct, but compatible, Hamiltonian formulations, and was sub- sequently derived from physical principles [3, 10, 19, 27]. Among its most notable properties is the existence of peaked solitons [3]. Equation (HS) describes the evo- lution of nonlinear oritentation waves in liquid crystals, u(x,t) being related to the deviation of the average orientation of the molecules from an equilibrium position [24]. Both (CH) and (HS) are completely integrable systems with an infinite number of conservation laws (see e.g. [6, 9, 11, 21, 25, 28]). Moreover, both equations admit geometric interpretations as Euler equations for geodesic flow on the diffeomorphism group Diff(S1) of orientation-preserving diffeomorphisms of the unit circle S1. More precisely, the geodesic motion on Diff(S1) endowed with the right-invariant metric given at the identity by u,v = uv+u v dx, (1.1) h iH1 x x ZS1 (cid:0) (cid:1) is described by the Camassa-Holm equation [34] (see also [7, 8]), whereas (HS) describes the geodesic flow on the quotient space Diff(S1)/S1 equipped with the right- invariant metric given at the identity by [28] u,v = u v dx. (1.2) h iH˙1 ZS1 x x We will consider an N = 2 supersymmetric generalization of equations (CH) and (HS),whichwasfirstintroducedin[35]bymeansofbi-Hamiltonianconsiderations. In this paper we: (a) Show that this supersymmetric generalization admits a geometric interpretation as an Euler equation on the superconformal algebra of contact vector fields on the 12-dimensional supercircle;1 (b) Consider the bi-Hamiltonian structure; | (c) Use the bi-Hamiltonian formulation to derive a Lax pair; (d) Present some simple examples of explicit solutions. 1 Theinterpretation of theN =2 supersymmetricCamassa-Holm equation asan Euler equation related to thesuperconformal algebra was already described in [1]. 1 1.1 The supersymmetric equation In order to simultaneously consider supersymmetric generalizations of both the CH and the HS equation, it is convenient to introduce the following notation: γ 0,1 is a parameter which satisfies γ = 1 in the case of CH and γ = 0 in • ∈ { } the case of HS. Λ= γ ∂2. • − x m = Λu. • θ and θ are anticommuting variables. 1 2 • D = ∂ +θ ∂ for j = 1,2. • j θj j x U = u+θ ϕ +θ ϕ +θ θ v is a superfield. 1 1 2 2 2 1 • u(x,t) and v(x,t) are bosonic fields. • ϕ (x,t) and ϕ (x,t) are fermionic fields. 1 2 • A= iD D γ. 1 2 • − M = AU. • Equations (CH) and (HS) can then be combined into the single equation m = 2u m um , x R, t >0. (1.3) t x x − − ∈ Although there exist several N = 2 supersymmetric extensions of (1.3), some generalizations have the particular property that their bosonic sectors are equivalent to the most popular two-component generalizations of (CH) and (HS) given by (see e.g. [4, 16, 26]) m +2u m+um +σρρ = 0, σ = 1, t x x x ± (1.4) (ρt+(uρ)x = 0. The N = 2 supersymmetric generalization of equation (1.3) that we will consider in this paper shares this property and is given by 1 M = (MU) + [(D M)(D U)+(D M)(D U)]. (1.5) t x 1 1 2 2 − 2 Defining ρ by v+iγu if σ = 1, ρ= (1.6) (iv γu if σ = 1, − − the bosonic sector of (1.5) is exactly the two-component equation (1.4). If u and ρ are allowed to be complex-valued functions, the two versions of (1.4) corresponding to σ = 1 and σ = 1 are equivalent, because the substitution ρ iρ − → converts one into the other. However, in the usual context of real-valued fields, the equations are distinct. The discussion in [35] focused attention on the two- component generalization with σ = 1. The observation that the N = 2 super- − symmetric Camassa-Holm equation arises as an Euler equation was already made in [1]. 2 2 Geodesic flow Asnotedabove,equations(CH)and(HS)allowgeometricinterpretationsasequations forgeodesicflowrelatedtothegroupofdiffeomorphismsofthecircleS1 endowedwith a right-invariant metric.2 More precisely, using theright-invariance of the metric, the full geodesic equations can be reduced by symmetry to a so-called Euler equation in the Lie algebra of vector fields on the circle together with a reconstruction equation [33] (see also [29, 30]). This is how equations (CH) and (HS) arise as Euler equations related to the algebra Vect(S1). In this section we describe how equation (1.5) similarly arises as an Euler equation related to the superconformal algebra K(S12) of contact vector fields on the | 12-dimensional supercircle S12. Since K(S12) is related to the group of superdif- | | | feomorphisms of S12, this leads (at least formally) to a geometric interpretation of | equation (1.5) as an equation for geodesic flow. 2.1 The superconformal algebra K(S12) | The Lie superalgebra K(S12) is defined as follows cf. [15, 23]. The supercircle S12 | | admits local coordinates x,θ ,θ , where x is a local coordinate on S1 and θ ,θ are 1 2 1 2 odd coordinates. Let Vect(S12) denote the set of vector fields on S12. An element | | X Vect(S12) can be written as | ∈ ∂ ∂ ∂ X = f(x,θ ,θ ) +f1(x,θ ,θ ) +f2(x,θ ,θ ) , 1 2 1 2 1 2 ∂x ∂θ ∂θ 1 2 where f,f1,f2 are functions on S12. Let | α = dx+θ dθ +θ dθ , 1 1 2 2 be the contact form on S12. The superconformal algebra K(S12) consists of all | | contact vector fields on S12, i.e. | K(S12)= X Vect(S12) L α = f α for some function f on S12 , | | X X X | ∈ n (cid:12) o where LX denotes the Lie derivat(cid:12)ive in the direction of X. A convenient description of K(S12) is obtained by viewing its elements as Hamil- | tonian vector fields corresponding to functions on S12. Indeed, define the Hamilto- | nian vector field X associated with a function f(x,θ ,θ ) by f 1 2 ∂f ∂ ∂f ∂ X = ( 1)f +1 + , f | | − ∂θ ∂θ ∂θ ∂θ (cid:18) 1 1 2 2(cid:19) where f denotes the parity of f. The Euler vector field is defined by | | ∂ ∂ E = θ +θ , 1 2 ∂θ ∂θ 1 2 2Inthissectionweconsiderallequationswithinthespatiallyperiodicsetting—although formally the same arguments apply to the case on the line, further technical complications arise due to the need of imposing boundary conditions at infinity cf. [5]. 3 and, for each function f on S12, we let D(f)= 2f E(f). Then the map | − ∂ ∂f f K := D(f) X + E, f f 7→ ∂x − ∂x satisfies [K ,K ] =K where f g f,g { } ∂g ∂f ∂f ∂g ∂f ∂g f,g = D(f) D(g)+( 1)f + . (2.1) | | { } ∂x − ∂x − ∂θ ∂θ ∂θ ∂θ (cid:18) 1 1 2 2(cid:19) Thus, the map f K is a homomorphism from the Lie superalgebra of functions f f → on S12 endowed with the bracket (2.1) to K(S12). | | 2.2 Euler equation on K(S12) | For two even functionsU andV onS12 (weusuallyrefertoU andV as ‘superfields’), | we define a Lie bracket [, ] by · · 1 [U,V]= UV U V + [(D U)(D V)+(D U)(D V)]. (2.2) x x 1 1 2 2 − 2 It is easily verified that [U,V] = 1 U,V , where , is the bracket in (2.1). The 2{ } {· ·} Euler equation with respect to a metric , is given by [2] h· ·i U = B(U,U), (2.3) t where the bilinear map B(U,V) is defined by the relation B(U,V),W = U,[V,W] , h i h i for any three even superfields U,V,W. Recall that A = iD D γ. Letting 1 2 − U,V = i dxdθ dθ UAV, (2.4) 1 2 h i − Z a computation shows that 1 B(U,U) = A 1 (MU) + [(D M)(D U)+(D M)(D U)] . (2.5) − − x 2 1 1 2 2 (cid:20) (cid:21) It follows from (2.3) and (2.5) that (1.5) is the Euler equation corresponding to , h· ·i given by (2.4). Remark 2.1 We make the following observations: 1. The action of the inverse A 1 of the operator A= iD D γ in equation (2.5) − 1 2 − is well-defined. Indeed, the map A can be expressed as u iv γu − ϕ iϕ γϕ A:  1  2x − 1 , (2.6) ϕ2 7→ iϕ1x γϕ2 − −  v   iu γv     − xx−      4 where we identify a superfield with the column vector made up of its four component fields (e.g. U = u+θ ϕ +θ ϕ +θ θ v is identified with the column vector 1 1 2 2 2 1 (u,ϕ ,ϕ ,v)T). Let C (S1;C) denote the space of smooth periodic complex-valued 1 2 ∞ functions. Since the operator 1 ∂2 is an isomorphism from C (S1;C) to itself, − x ∞ we deduce from (2.6) that the operation of A 1 on smooth periodic complex-valued − superfields is well-defined when γ = 1. Considernowthecaseofγ = 0. Inthiscaseitfollowsfrom(2.6)thatA 1 involves − the inverses of the operators ∂2 and ∂ . In order to make sense of these inverses we x x restrict the domain of A to the set = U = u+θ ϕ +θ ϕ +θ θ v is smooth and periodicu(0) = ϕ (0) = ϕ (0) = 0 . 1 1 2 2 2 1 1 2 E { | } The operator A = iD D maps this restricted domain bijectively onto the set 1 2 E = M = i(n+θ ψ +θ ψ +θ θ m) is smooth and periodic 1 1 2 2 2 1 F (cid:26) (cid:12) (cid:12) (cid:12) mdx = ψ dx = ψ dx = 0 . 1 (cid:12) 2 ZS1 ZS1 ZS1 (cid:27) Hence the inverse A 1 is a well-defined map . Writing M = i(n + θ ψ + − 1 1 F → E θ ψ +θ θ m), A 1M is given explicitly by 2 2 2 1 − x y y m(z)dzdy +x m(z)dzdy − 0 0 x S1 0 ψ (y)dy A−1M =  R R − x0 2 R R . ψ (y)dy 0R 1  n(x)   R    Note that the expression 1 1 (MU) + [(D M)(D U)+(D M)(D U)] = i (D D U)U + (D U)(D U) − x 2 1 1 2 2 − 1 2 2 1 2 (cid:20) (cid:21)x acted on by A 1 in equation (2.5) belongs to for any even superfield U. Hence the − F operation of A 1 in equation (2.5) is well-defined also when γ = 0. The restriction − of the domain of A to is related to the fact that the two-component equation (1.4) E with γ = 0 is invariant under the symmetry u(x,t) u(x c(t),t)+c(t), ρ(x,t) ρ(x c(t),t), ′ → − → − for any sufficiently regular function c(t). Hence, by enforcing the condition u(0) = 0 we remove obvious non-uniqueness of the solutions to the equation. 2. If we restrict attention to the bosonic sector and let U = u +θ θ v , V = u +θ θ v , 1 2 1 1 2 2 1 2 theLiebracket(2.2)inducesontwopairsoffunctions(u ,v )and(u ,v )thebracket 1 1 2 2 [(u ,v ),(u ,v )] = (u u u u ,u v u v ). (2.7) 1 1 2 2 1 2x 1x 2 1 2x 2 1x − − 5 Werecognize(2.7)asthecommutation relation forthesemidirectproductLiealgebra Vect(S1)⋉C (S1), where Vect(S1) denotes the space of smooth vector fields on S1, ∞ see [22]. 3. The restriction of the metric (2.4) to the bosonic sector induces on pair of functions (u,v) the bilinear form (u ,v ),(u ,v ) = (iγ(u v +u v )+u u +v v )dx. (2.8) 1 1 2 2 1 2 2 1 1x 2x 1 2 h i Z Changing variables from (u ,v ) to (u ,ρ ), j = 1,2, according to (1.6), equation j j j j (2.8) becomes (u ,ρ ),(u ,ρ ) = (γu u +u u +σρ ρ )dx. (2.9) 1 1 2 2 1 2 1x 2x 1 2 h i Z Hence, as expected, when ρ = ρ = 0 the metric reduces to the H1 metric (1.1) in 1 2 the case of CH, while it reduces to the H˙1 metric (1.2) in the case of HS. 4. Our discussion freely used complex-valued expressions and took place only at the Lie algebra level. The extent to which there actually exists a corresponding geodesic flow on the superdiffeomorphism group of the supercircle S12 has to be | furtherinvestigated. Afirststeptowardsdevelopingsuchageometric pictureinvolves dealing with the presence of imaginary numbers in the definition of the metric (2.4) when γ = 1. Although these imaginary factors disappear in the bosonic sector when changingvariablesto(u,ρ)(see(2.9)),theyarestillpresentinthefermionicsector. It seems unavoidable to encounter complex-valued expressions at one point or another of the presentconstruction if one insists on thesystem beingan extension of equation (CH) (in the references [1] and [35] imaginary factors appear when the coefficients are chosen in such a way that the system is an extension of (CH)). 3 Bi-Hamiltonian formulation Equation (1.5) admits the bi-Hamiltonian structure3 δH δH 1 2 M =J = J , (3.1) t 1 2 δM δM where the Hamiltonian operators J and J are defined by 1 2 1 J = i ∂ M M∂ + (D MD +D MD ) , J = i∂ A, (3.2) 1 x x 1 1 2 2 2 x − − 2 − (cid:20) (cid:21) and the Hamiltonian functionals H and H are defined by 1 2 i i γ H = dxdθ dθ MU, H = dxdθ dθ MU2 U3 . (3.3) 1 1 2 2 1 2 −2 −4 − 3 Z Z (cid:16) (cid:17) 3By definition thevariational derivative δH/δM of a functional H[M] is required to satisfy d δH H[M+ǫδM] = dxdθ dθ δM, dǫ ˛ Z 1 2δM ˛ǫ=0 ˛ for any smooth variation δM of M. ˛ 6 δ i dxdθ dθ MU2 γU3 = δH2 ssggggggggδgMgJg2(cid:2)g−gg4ggRgggg 1 2(cid:0) − 3 (cid:1)(cid:3) δM Equation (1.5) kkWWWWWWWWWWWJW1WWWWWWWW δ i dxdθ dθ MU = δH1 ssggggggggggJg2ggδgMggg(cid:2)g−gg2R 1 2 (cid:3) δM M = M t − x kkWWWWWWWWWWWJW1WWWWWWWW δ i dxdθ dθ M = δH0 ssggggggggggggJg2ggggδgMgg(cid:2)g−g R 1 2 (cid:3) δM M = 0 t Figure 1 Recursion scheme for the operators J and J associated with the super- 1 2 symmetric equation (1.5). The bi-Hamiltonian formulation (3.1) is a particular case of a construction in [35], where it was verified that the operators J and J are compatible. The first few 1 2 conservation laws in the hierarchy generated by J and J are presented in Figure 1. 1 2 Restricting attention to the purely bosonic sector of the bi-Hamiltonian structure of (1.5), we recover bi-Hamiltonian formulations for the two-component generalizations of (CH) and (HS). More precisely, we find that equation (1.4) can be put in the bi-Hamiltonian form4 m = K gradG = K gradG , (3.4) ρ 1 1 2 2 (cid:18) (cid:19)t 4The gradient of a functional F[m,ρ] is defined by δF gradF = δm , „δF« δρ provided that thereexist functions δF and δF such that δm δρ d δF δF F[m+ǫδm,ρ+ǫδρ] = δm+ δρ dx, dǫ ˛ Z „δm δρ « ˛ǫ=0 ˛ for any smooth variations δm and δρ. ˛ 7 where the Hamiltonian operators are defined by m∂ ∂ m ρ∂ ∂Λ 0 K1 = − x∂−ρ x −0 , K2 = −0 σ∂ , (3.5) (cid:18) − (cid:19) (cid:18) − (cid:19) and the Hamiltonians G and G are given by 1 2 1 1 G = (um+σρ2)dx, G = (σuρ2 +γu3+uu2)dx. 1 2 2 2 x Z Z For j = 1,2, G are the restrictions to the purely bosonic sector of the functionals j H . j Thehierarchy of conserved quantities G for the two-component system (1.4) can n be obtained from the recursive relations K gradG = K gradG , n Z. 1 n 2 n+1 ∈ The associated commuting Hamiltonian flows are given by m = K gradG = K gradG , n Z. ρ 1 n 2 n+1 ∈ (cid:18) (cid:19)t Since K 1 is a nonlocal operator the expressions for members G with n 3 are 2− n ≥ nonlocal when written as functionals of (u,ρ). On the other hand, since the operator K can be explicitly inverted as 1 0 1∂ 1 K 1 = −ρ x− , 1− ∂ 11 ∂ 11(m∂ +∂ m)1∂ 1 − x− ρ x− ρ x x ρ x− ! it is possible to implement on a computer a recursive algorithm for finding the conservation laws G for n 2. The first few of these conserved quantities and their n ≤ associated Hamiltonian flows are presented in Figure 2. Note that G is a Casimir 0 for the positive hierarchy, G is a Casimir for the negative hierarchy, and G is a 2 1 − − Casimir for both the positive and negative hierarchies. Thisdiscussionillustratesthateventhoughmanypropertiesofthetwo-component system (1.4) carry over to its supersymmetric extension (1.5) (such as the property of being a geodesic equation as shown above, and the Lax pair formulation as shown below), the construction of a negative hierarchy does not appear to generalize. Cer- tainly, since the conservation laws G , G , ..., contain negative powers of ρ, these 1 2 − − functionals cannot be the restriction to the bosonic sector of functionals H , H , 1 2 − − ..., which involve only polynomials of the fields U and M and derivatives of these fields. 4 Lax pair Equation (1.5) is the condition of compatibility of the linear system iD D G= 1 M + γ G, 1 2 2λ 2 (4.1) (Gt = 12UxG(cid:0)− 21[(D1U(cid:1))(D1G)+(D2U)(D2G)]−(λ+U)Gx, 8 grad 1 σuρ2+γu3+uu2 dx = gradG ssggggggKgg2gggggg2ggRgg(cid:0)ggg x(cid:1) 2 m 2u m um σρρ = − x − x− x ρ (uρ) (cid:18) (cid:19)t (cid:18) − x kkW(cid:19)WWWWWWKWW1WWWWWWWWWWWW grad 1 um+σρ2 dx = gradG ssggggggggggggKg2ggggggggggg2ggRg(cid:0) (cid:1) 1 m m = x (cid:18)ρ(cid:19)t −(cid:18)ρx(cid:19) kkXXXXXXXXXXXXXKX1XXXXXXXXXXXXX grad mdx = gradG 0 m 0 ssffffffffffffffKf2ffffffffffffff R = (cid:18)ρ(cid:19)t (cid:18)0(cid:19) kkXXXXXXXXXXXXXXXKX1XXXXXXXXXXXXX grad ρdx = gradG 1 m 0 ssffffffffffffffKf2ffffffffffffff R − = (cid:18)ρ(cid:19)t (cid:18)0(cid:19) kkWWWWWWWWWWWWWWKWW1WWWWWWWWWWWW grad mdx = gradG ρ 2 sshhhhhhhKhh2hhhhhhhhhhhhhh R − γρx + 1 m ρ2 ρ = xxx ρ  σ (cid:16)m (cid:17)  (cid:18) (cid:19)t  (cid:16)ρ2(cid:17)x kkVVVVVVVVVKV1VVVVVVVVVVVV grad σm2 γ ρxx dx = gradG −2ρ3 − 2ρ − 4ρ2 −3 (cid:16) (cid:17) R Figure 2 Recursion scheme for two-component generalization (1.4) of the CH (corresponding to m = u u ) and the HS (corresponding to m = u ) equations. xx xx − − 9