On the N=2 Supersymmetric Camassa-Holm and Hunter-Saxton Equations

0.25 MB·English

by  Jonatan Lenells

ITP-UH-17/08
On the N=2 Supersymmetric Camassa-Holm and
Hunter-Saxton Equations
9 J. Lenellsa and O. Lechtenfeldb
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2 a Department of Applied Mathematics and Theoretical Physics,

n University of Cambridge, Cambridge CB3 0WA, United Kingdom
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J b Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover,

7 Appelstraße 2, 30167 Hannover, Germany
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[email protected], [email protected]

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v Abstract
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WeconsiderN=2supersymmetricextensionsofthe Camassa-HolmandHunter-
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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
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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.
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X
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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) de-
scribes 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;
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(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 gen-
eralizations 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 equa-
tion related to the superconformal algebra K(S12) of contact vector fields on the
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12-dimensional supercircle S12. Since K(S12) is related to the group of superdif-
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feomorphisms of S12, this leads (at least formally) to a geometric interpretation of
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equation (1.5) as an equation for geodesic flow.
2.1 The superconformal algebra K(S12)
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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
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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
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α = dx+θ dθ +θ dθ ,
1 1 2 2
be the contact form on S12. The superconformal algebra K(S12) consists of all
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contact vector fields on S12, i.e.
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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-
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tonian vector fields corresponding to functions on S12. Indeed, define the Hamilto-
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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)
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For two even functionsU andV onS12 (weusuallyrefertoU andV as ‘superfields’),
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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 compo-
nent 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
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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 generaliza-
tions 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 con-
servation 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 (cor-
responding to m = u u ) and the HS (corresponding to m = u ) equations.
xx xx
− −
9