Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

0.46 MB·English

by  Changzheng Qu

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 001, 19 pages
Multi-Component Integrable Systems
and Invariant Curve Flows in Certain Geometries(cid:63)
Changzheng QU † and Junfeng SONG ‡ and Ruoxia YAO §
† Center for Nonlinear Studies, Ningbo University, Ningbo, 315211, P.R. China
E-mail: [email protected]
‡ College of Mathematics and Information Science, Shaanxi Normal University,
Xi’an, 710062, P.R. China
E-mail: [email protected]
§ School of Computer Science, Shaanxi Normal University, Xi’an, 710062, P.R. China
3 E-mail: [email protected]
1
0 Received September 28, 2012, in final form December 27, 2012; Published online January 02, 2013
2 http://dx.doi.org/10.3842/SIGMA.2013.001

n
a
Abstract. In this paper, multi-component generalizations to the Camassa–Holm equation,
J
the modified Camassa–Holm equation with cubic nonlinearity are introduced. Geomet-
2
ric formulations to the dual version of the Schr¨odinger equation, the complex Camassa–

] Holm equation and the multi-component modified Camassa–Holm equation are provided.
I
It is shown that these equations arise from non-streching invariant curve flows respectively
S
. in the three-dimensional Euclidean geometry, the two-dimensional M¨obius sphere and n-
n dimensional sphere Sn(1). Integrability to these systems is also studied.
i
l
n Key words: invariant curve flow; integrable system; Euclidean geometry; M¨obius sphere;
[
dual Schr¨odinger equation; multi-component modified Camassa–Holm equation

1
2010 Mathematics Subject Classification: 37K10; 51M05; 51B10
v
0
8
1
1 Introduction
0
.
1
0 Integrable systems solved by the inverse scattering method usually arise from shallow water
3 wave, physics, optical communication and applied sciences etc. Integrable systems have many
1
interesting properties, such as Lax-pair, infinite number of conservation laws and Lie–B¨acklund
:
v symmetries, multi-solitons, B¨acklund transformations and bi-Hamiltonian structure etc. [1,44],
i
X which are helpful to explore other properties of integrable systems [1,44,56].
r It is of great interest to study geometric aspects of integrable systems. So far, very few inte-
a
grable systems were found to have geometric formulations. The relationship between completely
integrable systems and the finite-dimensional differential geometry of curves has been studied
extensively. It turns out that some integrable systems arise from invariant curve flows in certain
geometries [2–7,9–15,21,23,24,26,28–42,45,47–52,55,57,59,60]. The pioneering work on this
topic was done by Hasimoto [24]. He showed that the integrable nonlinear Schr¨odinger equation
(NLS)
iφ +φ +|φ|2φ = 0
t ss
is equivalent to the system for the curvature κ and τ of curves γ in R3
κ κ κ
sss s ss
κ = −2τκ −κτ , τ = − −2ττ +κκ (1)
t s s t κ κ2 s s
(cid:63)This paper is a contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants
and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html
2 C.Z. Qu, J.F. Song and R.X. Yao
(cid:82)s
via the so-called Hasimoto transformation φ = κexp(i τ(t,z)dz). Indeed, the system (1) is
equivalent to the vortex filament equation
γ = γ ×γ = κb, (2)
t s ss
where b is the binormal vector of γ. Mar´ı Beffa, Sanders and Wang [39,51] noticed that
Hasimoto transformation is a gauge transformation relating the Frenet frame {t,n,b} to the
parallel frame {t1,n1,b1}. It is also a Poisson map which takes Hamiltonian structure of the
NLS equation to that of the vertex filament flow [29]. The Hasimoto transformation has been
generalized in [51] to the Riemannian manifold with constant curvature, which is used to obtain
the corresponding integrable equations associated with the invariant non-stretching curve flows.
The parallel frames and other kinds of frames are also used to derive bi-Hamiltonian operators
and associated hierarchies of multi-component soliton equations from non-stretching curve flows
onLiegroupmanifolds[3,4,31,39]. TheKdVequation,themodifiedKdVequation,theSawada–
Kotera equation and the Kaup–Kuperschmidt equation were shown to arise from the invariant
curve flows respectively in centro-equiaffine geometry [7,9,48], Euclidean geometry [21], special
affine geometry [11,35] and projective geometries [11,30,41].
The integrable systems with non-smooth solitary waves have drawn much attention in the
last two decades because of their remarkable properties. The celebrated Camassa–Holm (CH)
equation
m +um +2u m+au = 0, m = u−u , (3)
t x x x xx
was proposed as a model for the unidirectional propagation of the shallow water waves over
a flat bottom, with u(x,t) representing the water’s free surface in non-dimensional variables [8].
It was also found using the method of recursion operators by Fokas and Fuchssteiner [19] as
a bi-Hamiltonian equation with an infinite number of conserved functionals. Geometrically,
the Camassa–Holm equation arises from a non-stretching invariant planar curve flow in the
centro-equiaffine geometry [9], and the periodic CH equation (3) describes geodesic flows on
diff(S1 ×R) with respect to right-invariant Sobolev H1 metric for a = 0 [13,14,28] and Bott–
Virasoroalgebrafora (cid:54)= 0[40]. AdualversionoftheItosystemisthetwo-componentCamassa–
Holm equation [46], the periodic two-component CH equation also describe geodesic flows on
an extended Bott–Virasoro algebra [22].
It is remarked that all nonlinear terms in the CH equation are quadratic. In contrast to
the integrable modified KdV equation with a cubic nonlinearity, it is of great interest to find
integrable CH-type equations with cubic or higher-order nonlinearity and non-smooth solitary
waves. To the best of our knowledge, two scalar integrable CH-type equations with cubic
nonlinearity have been discovered. The first equation is [18,46,49]
m + 1 (cid:0)(cid:0)u2+δu2(cid:1)m(cid:1) = 0, m = u+δu , (4)
t 2 x x xx
where δ = ±1, and the second one is the so-called Novikov equation [25,43]
m +3uu m+u2m = 0, m = u−u ,
t x x xx
which are completely integrable, and admit peaked solitons. Recently, systems of CH-type
equations with cubic nonlinearity were also obained [20,55].
The CH equation can also be derived by the tri-Hamiltonian duality approach basing on
bi-Hamiltonian structure of the KdV equation. Other examples of dual integrable systems
obtained using the method of tri-Hamiltonian duality can be found in [18,46]. Nonlinear dual
integrable systems, such as the CH equation and the modified CH equations, are endowed with
Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries 3
nonlinear dispersion, which in most cases, enables these systems to support non-smooth soliton-
like structures. It was remarked in [23] that the modified CH equation (4) can be regarded
as a Euclidean-invariant version of the CH equation (3), just as the modified KdV equation is
a Euclidean-invariant counterpart to the KdV equation from the viewpoint of curve flows in
Klein geometries [9,10,21,48].
The aim of this paper is to provide geometric formulations to multi-component integrable
systems admitting non-smooth solitons. We shall show that several multi-component integrable
systems with non-smooth solitons, such as a dual version of the Schr¨odinger equation [17,46],
the complex CH equation and multi-component modified CH equations arise from the invariant
curve flows respectively in three-dimensional Euclidean geometry, M¨obius sphere and the n-
dimensional unit sphere Sn(1). To obtain integrable systems relating to these geometric flows,
we shall use the scale limit technique. The outline of this paper is as follows. In Section 2,
a non-stretching invariant binormal curve flow in R3 is introduced and studied. Making use of
the system for curvature and torsion corresponding to this flow, we obtain a novel integrable
Schr¨odingerequationbyascalelimitapproach,whichiscompletelyintegrablesystemandcanbe
obtained by the so-called tri-Hamiltonian duality approach [18,46]. In Section 3, we give a brief
discussion on M¨obius 2-sphere PO(3,1)/H and the n-dimensional sphere SO(n+1)/SO(n), the
Cartan structure equations for curves in both geometries are reviewed, which will be used in
subsequent sections to study curve flows in both geometries. In Section 4, we consider the
non-stretching curve flows in M¨obius 2-sphere. It is shown that the complex Camassa–Holm
equationandcomplexHunter–Saxtonequationdescribethenon-stretchingcurveflowsinM¨obius
2-sphere. Thebi-HamiltonianstructureforthecomplexCamassa–Holmequationisobtained. In
Section 5, we study non-stretching curve flows in the n-dimensional sphere Sn(1). Interestingly,
we find that a multi-component modified CH equation (a multi-component generalization of
the modified Camassa–Holm equation) is equivalent to a non-stretching curve flow in Sn(1).
Integrability of the system is identified. Finally, Section 6 contains concluding remarks on this
work.
2 An integrable nonlinear Schr¨odinger equation
We consider the flows of space curves in R3, given by
γ = Un+Vb+Wt, (5)
t
where t, n and b denote the tangent, normal and binormal vectors of the curves, respectively.
The velocities U, V and W depend on the curvature and torsion as well as their derivatives with
respecttoarc-lengthparameters. Thearc-lengthparametersisdefinedimplicitlybyds = hdp,
h = |γ(cid:48)(p)|, where p is a free parameter and is independent of time. We denote by κ and τ the
curvature and torsion of the curves, respectively. Governed by the flow (5), time evolutions of
those geometric invariants are given by [24,42]
(cid:18) (cid:19) (cid:18) (cid:19)
∂U ∂V
t˙ = −τV +κW n+ +τU b,
∂s ∂s
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
∂U 1 ∂ ∂V τ ∂U
n˙ = − −τV +κW t+ +τU + −τV +κW b,
∂s κ∂s ∂s κ ∂s
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
∂V 1 ∂ ∂V τ ∂U
b˙ = − +τU t− +τU + −τV +κW n,
∂s κ∂s ∂s κ ∂s
(cid:18) (cid:19)
∂W
h˙ = 2h −κU (6)
∂s
4 C.Z. Qu, J.F. Song and R.X. Yao
and
∂τ ∂ (cid:20)1 ∂ (cid:18)∂V (cid:19) τ (cid:18)∂U (cid:19) (cid:90) s (cid:21) ∂V
= +τU + −τV +τ κUds(cid:48) +κτU +κ ,
∂t ∂s κ∂s ∂s κ ∂s ∂s
∂κ = ∂2U +(cid:0)κ2−τ2(cid:1)U + ∂κ (cid:90) sκUds(cid:48)−2τ∂V − ∂τV. (7)
∂t ∂s2 ∂s ∂s ∂s
Assumingthattheflowisintrinsic, namelythearc-lengthdoesnotdependontime, itimplies
from (6) that
W = κU. (8)
s
In terms of (7), one finds that the complex function
(cid:20) (cid:90) s (cid:21)
φ = κη, η = exp i τ(s(cid:48),t)ds(cid:48)
satisfies the equation [24,42]
(cid:18) ∂2 (cid:90) s ∂φ (cid:90) s (cid:19)
φ = +|φ|2+iφ ds(cid:48)τφ¯+ ds(cid:48)φ¯ (Uη)
t ∂s2 ∂s
(cid:18) ∂2 (cid:90) s (cid:90) s ∂φ¯(cid:19)
+ i +i|φ|2+φ ds(cid:48)τφ¯−iφ ds(cid:48) (Vη), (9)
∂s2 ∂s(cid:48)
where φ¯ denotes the complex conjugate of φ.
Let U = 0, V = κ, where κ is a real function, then (8) implies that W = C , where C is
1 1
a constant. Setting C = 0, we derive from (9) the celebrated Schr¨odinger equation
1
1
iφ +φ + |φ|2φ = 0. (10)
t ss
2
Let U = −κ , V = −κτ, then W = −1κ2 +C , where C is a constant. Letting C = 0, we
s 2 2 2 2
find that φ satisfies the mKdV system
3
φ +φ + |φ|2φ = 0.
t sss s
2
In the following, we shall consider the case U = W = 0. Denote θ(s,t) = (cid:82)sτ(s(cid:48),t)ds(cid:48),
g = Vη. It follows from (9) that φ satisfies the equation
(cid:90) s
iφ +g +|φ|2g−φ g(cosθ−isinθ)k ds(cid:48) = 0. (11)
t ss s(cid:48)
Set u˜ = κcosθ, v˜= κsinθ, g = g +ig , then the equation (11) is separated to two equations
1 2
u˜ = −g −v˜∂−1[κ(g cosθ+g sinθ) ],
t 2,ss s 1 2 s
v˜ = g +u˜∂−1[κ(g cosθ+g sinθ) ].
t 1,ss s 1 2 s
Furthermore, letting u˜ = u+v , v˜ = v −u , and choosing the binormal velocity V to be
s s
V = ∂−1[(cid:0)u2+v2(cid:1) /κ], we find that u and v satisfy the system
s s
(u+v ) = −g −(v−u )(cid:0)u2+v2(cid:1),
s t 2,ss s
(v−u ) = g +(cid:0)u2+v2(cid:1)(u+v ) (12)
s t 1,ss s
Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries 5
with
(u+v )(cid:0)u2+v2(cid:1) (v−u )(cid:0)u2+v2(cid:1)
g = s s , g = s s .
1 (u+v )2+(v−u )2 2 (u+v )2+(v−u )2
s s s s
Applying the scaling transformations
s (cid:55)−→ s, t (cid:55)−→ (cid:15)2t, u (cid:55)−→ (cid:15)−1u, v (cid:55)−→ (cid:15)−1v
to system (12) produces
(u+v ) = −(cid:15)2g −(v−u )(cid:0)u2+v2(cid:1),
s t 2,ss s
(v−u ) = (cid:15)2g +(cid:0)u2+v2(cid:1)(u+v ). (13)
s t 1,ss s
Expanding u and v in powers of the small parameter (cid:15)
u(t,s) = u (t,s)+(cid:15)u (t,s)+(cid:15)2u (t,s)+ ··· ,
0 1 2
v(t,s) = v (t,s)+(cid:15)v (t,s)+(cid:15)2v (t,s)+ ··· ,
0 1 2
and plugging them into system (13), we find that the leading order terms u (t,s) and v (t,s)
0 0
satisfy the system
(u +v ) +(cid:0)u2+v2(cid:1)(v −u ) = 0,
0 0,s t 0 0 0 0,s
(v −u ) −(cid:0)u2+v2(cid:1)(u +v ) = 0. (14)
0 0,s t 0 0 0 0,s
Again we use the notation φ(t,s) = u (t,s)+iv (t,s), then it is inferred from (14) that φ(t,s)
0 0
satisfies the equation
i(φ −iφ )+|φ|2(φ−iφ ) = 0, (15)
t ts s
whichisadualversionoftheSchr¨odingerequation(10), andcanbeobtainedbytheapproachof
tri-Hamiltonian duality [18,46]. Equation (15) is formally completely integrable since it admits
bi-Hamiltonian structure [46]
δH δH
2 1
ρ = E = E ,
t 1 2
δρ δρ
where ρ = φ−iφ , E and E defined by
s 1 2
E = ∂ +i and E (F) = ρ∂−1(ρ¯F −ρF¯)
1 s 2 s
are compatible Hamiltonian operators, while
(cid:90) (cid:90) (cid:90)
H = (cid:0)|φ|2−iφ¯φ (cid:1)ds = ρφ¯ds, H = 1 |φ|2(cid:0)|φ|2−iφ¯φ (cid:1)ds.
1 s 2 s
2
R R R
are the corresponding Hamiltonian functionals.
3 M¨obius sphere PO(3,1)/H and unit sphere SO(n + 1)/SO(n)
In this section, we give a brief account of M¨obius 2-sphere PO(3,1)/H and unit sphere Sn(1) =
SO(n+1)/SO(n). Please refer to the book [53] for the details of the two geometries.
6 C.Z. Qu, J.F. Song and R.X. Yao
3.1 M¨obius 2-sphere
Let (u ,u ,u ,u ) ∈ R4, we define the inner product on R4 by
0 1 2 3
(cid:104)x,y(cid:105) = xTΛ y,
3,1
where x,y ∈ R4, and the matrix Λ is
3,1
 
0 0 0 −1
 0 1 0 0 
Λ3,1 =  0 0 1 0 .
−1 0 0 0
A vector field x ∈ R4 is said to be light-like, if it satisfies (cid:104)x,x(cid:105) = 0. All the light-like vector
fields form a set L, which is called optical cone, defined by the equation
2x x −x2−x2 = 0.
0 3 1 2
Clearly it is homogeneous, namely for any λ ∈ R, if x ∈ L, then λx ∈ L. The projectivisation
of L is said to be M¨obius 2-sphere, which is isomorphic to S2. Recall that
O(3,1) = (cid:8)g ∈ GL(4,R) : gTΛ g = Λ (cid:9),
3,1 3,1
and the M¨obius group is defined to be PO(3,1) = O(3,1)/±I. We denote
H = {h ∈ PO(3,1) : h[e ] = [e ]}
3 3
a−1 0 0 
 
=  v A 0 ∈ O(3,1); A ∈ O(2), a ∈ R+, v ∈ R2 ,
 b ηT a 
where [e ] denotes the equivalent class of e in P(R4), [e ] = (0,0,0,∗). A straightforward
3 3 3
computation gives
a a
η = aATv, b = vTv = (cid:107)v(cid:107)2.
2 2
It is easy to verify that the group PO(3,1) acts on the M¨obius sphere transitively (the group
action is the usual conformal transformation).
Definition 1. The Klein pair (PO(3,1),H) is called the M¨obius 2-sphere.
For any g ∈ PO(3,1), there exists a unique decomposition around the identity of the group
 1 0 0a−1 0 01 uT 1(cid:107)u(cid:107)2
2
g = g1g0g−1 =  v I2 0 0 A 00 I2 u ,
1(cid:107)v(cid:107)2 vT 1 0 0 a 0 0 1
2
where a ∈ R+, A ∈ O(2), u,v ∈ R2, h = g g ∈ H, u in g part represent a local coordinate of
1 0 −1
the point in PO(3,1)/H. For the corresponding Lie algebra g, there exists also a decomposition
o(3,1) = g = g ⊕g ⊕g ,
1 0 −1
where
0 0 0 −(cid:15) 0 0 0 qT 0
p 0 0 ∈ g1,  0 S 0 ∈ g0, 0 0 q ∈ g−1,
0 pT 0 0 0 (cid:15) 0 0 0
p,q ∈ R2, (cid:15) ∈ R, S ∈ O(2). The Lie algebra of the isotropy group H is h = g ⊕ g while
1 0
g/h = g is identified to the tangent space of the conformal sphere PO(3,1)/H.
−1
Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries 7
3.2 n-dimensional sphere Sn(1) = SO(n + 1)/SO(n)
The n-dimensional unit-sphere is also a homogeneous space M = G/H = SO(n + 1)/SO(n).
The corresponding Lie algebra has the following Cartan–Killing decomposition
so(n+1) = h⊕m = so(n)⊕Rn,
with
(cid:18)0 −pT(cid:19) (cid:18)0 0(cid:19)
∈ m, ∈ h,
p 0 0 Θ
where p ∈ Rn, Θ ∈ so(n), and the decomposition satisfies
[h,h] ⊂ h, [h,m] ⊂ m, [m,m] ⊂ h,
wheremisidentifiedtothetangentspaceT M ∼= Rn ofM = SO(n+1)/SO(n). TheflatCartan
x
connection of principle SO(n) bundle SO(n+1) → Sn is given by the Maurer–Cartan form of
Lie group SO(n+1). The Cartan structure equation reads as
1
Ω = dω+ [ω,ω] = 0,
2
where so(n+1)-valued one-form ω is decomposed to
ω = ω +θ, ω ∈ Λ1(P,h), θ ∈ Λ1(P,g/h),
H H
whereg/h-valuedθrepresentsalinearcoframe,h-valuedω representsalinearconnectiononSn.
H
The corresponding Cartan structure equation is separated to
1 1
J ≡ dθ+ [ω ,θ]+ [θ,ω ] = 0
H H
2 2
and
1 1
R ≡ dω + [ω ,ω ] = − [θ,θ],
H H H
2 2
where J and R are called torsion and curvature forms, respectively.
4 Curve flows in PO(3,1)/H and the complex CH equation
For the M¨obius geometry PO(3,1)/H, its Cartan connection takes values on g = o(3,1), with
the form
−(cid:15) ξT 0
ω =  η Θ ξ,
0 ηT (cid:15)
and the Cartan structure equation reads as
1
Ω = dω+ [ω,ω] = 0.
2
Let’sconsidertheinvariantcurveflowsforcurvesγ(x,t) = (u (x,t),u (x,t))ontheconformal
1 2
sphere M2 = PO(3,1)/H, where x denotes the parameter of the curves, t is the time variable,
u and u denote the local coordinates on M2. Let γ = γ ∂ be the evolutionary vector field of
1 2 t ∗∂t
8 C.Z. Qu, J.F. Song and R.X. Yao
the curves, γ = γ ∂ denotes the tangent vector of the curves. Assuming that the curve flow
x ∗∂x
is intrinsic, namely the parameter x for the curve does not depend on time t, we have
[γ ,γ ] = 0.
x t
It was shown in [16,36] that there exists a conformally equivariant moving frame (the Frenet
frame) ρ = ρ(x,t) ∈ PO(3,1) along the curve γ(x,t) ⊂ M2. Let D and D denote respectively
x t
the vector field d and d along the curves ρ in PO(3,1), then the Frenet formulae for the
dx dt
conformally parametric curves is
ρ = ρωˆ(D ),
x x
with
 
0 1 0 0
ωˆ(Dx) = kk12 00 00 10, (16)
0 k k 0
1 2
where k and k are the conformally differential invariants for the curve γ(x,t). The time
1 2
evolution for the frame ρ(x,t) can be written as
ρ = ρωˆ(D ),
t t
where
 
−(cid:15) h h 0
1 2
ωˆ(Dt) = ff12 α0 −0α hh12, (17)
0 f f (cid:15)
1 2
and (cid:15), α, f , h (i = 1,2) are some conformal differential invariants related to k and k , to be
i i 1 2
determined. By the Cartan structure equation, one gets
d d
Ω(D ,D ) = ωˆ(D )− ωˆ(D )−[ωˆ(D ),ωˆ(D )] = 0. (18)
x t x t x t
dt dx
Plugging (16) and (17) into (18) results in the following equations
(cid:15) = −h , α = h , (19)
1,x 2,x
f = (cid:15) +k h +k h , (20)
1 x 1 1 2 2
f = α +k h −k h , (21)
2 x 2 1 1 2
k = f −k (cid:15)+αk , (22)
1,t 1,x 1 2
k = f −(cid:15)k −αk , (23)
2,t 2,x 2 1
where (19) is the torsion part (i.e.,the g part) of the Cartan structure equation (18), which
−1
can be written as
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
(cid:15) −∂ h h
= x 1 ≡ J 1 .
α ∂ h 1 h
x 2 2
Inserting (19) into (20) and (21) gives
f = −h +k h +k h , f = h +k h −k h . (24)
1 1,xx 1 1 2 2 2 2,xx 2 1 1 2
Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries 9
Substituting (19) and (24) into (22) and (23), we obtain the evolution equations for the curva-
tures k and k [32,36]
1 2
k = −h +2k h +k h +2k h +k h ,
1,t 1,xxx 1 1,x 1,x 1 2 2,x 2,x 2
k = h −2k h −k h +2k h +k h ,
2,t 2,xxx 1 2,x 1,x 2 2 1,x 2,x 1
which is equivalent to
(cid:18)k (cid:19) (cid:18)−∂3+k ∂ +∂k k ∂ +∂k (cid:19)(cid:18)h (cid:19) (cid:18)h (cid:19)
1 = 1 1 2 2 1 ≡ J 1 . (25)
k k ∂ +∂k ∂3−k ∂ −∂k h 2 h
2 t 2 2 1 1 2 2
The following cases are considered.
Case 1. Setting k = 1/2+m ≡ 1/2+u−u , k = n ≡ v −v , h = 1−u and h = v
1 xx 2 xx 1 2
in (25), we obtain the new two-component CH equation
m +2u m+um −2v n−vn = 0,
t x x x x
n +2u n+un +2v m+vm = 0. (26)
t x x x x
The above system admits the following Lax-pair
φ = Uφ, φ = Vφ, (27)
x t
with
0 1 +λm λn 0   −u f f 0 
2 x 1 2
U = 1 0 0 12 +λm and V = λ−1−u 0 vx f1,
0 0 0 λn   v −vx 0 f2
0 1 0 0 0 λ−1−u v u
x
where f = 1(λ−1+u)+λ(vn−um), f = 1v−λ(un+vm).
1 2 2 2
Case 2. Setting k = m = −u , k = n = −v , h = 1−u, h = v, we arrive at the
1 xx 2 xx 1 2
complex Hunter–Saxton equation
m +2u m+um −2v n−vn = 0,
t x x x x
n +2u n+un +2v m+vm = 0,
t x x x x
which admits the Lax-pair (27) with
 
0 −λu −λv 0
xx xx
U = 1 0 0 −λuxx
0 0 0 −λvxx
0 1 0 0
and
 
−u λ(vn−um) −λ(un+vm) 0
x
V = λ−1−u 0 vx λ(vn−um) .
 v −vx 0 −λ(un+vm)
0 λ−1−u v u
x
Case 3. In (25), letting h and h satisfy
1 2
1 h2−h2 h h
k = 1 2 , k = − 1 2 ,
1 2(cid:0)h2+h2(cid:1)2 2 (cid:0)h2+h2(cid:1)2
1 2 1 2
we get the two-component Harry–Dym equation
h = (cid:0)h3−3h h2(cid:1)h +(cid:0)h3−3h2h (cid:1)h ,
1,t 1 1 2 1,xxx 2 1 2 2,xxx
h = (cid:0)h3−3h2h (cid:1)h −(cid:0)h3−3h h2(cid:1)h .
2,t 2 1 2 1,xxx 1 1 2 2,xxx
10 C.Z. Qu, J.F. Song and R.X. Yao
Remark 1. Mar´ıBeffa[32,36]showedthatthecomplexKdVequationarisesfromtheinvariant
curve motion in M¨obius 2-sphere. Indeed, taking h = −k , h = −k in (25) yields the complex
1 1 2 2
KdV equation
k = k −3k k +3k k , k = k −3k k −3k k .
1,t 1,xxx 1 1,x 2 2,x 2,t 2,xxx 1 2,x 2 1,x
Its Hamiltonian structure J , see (25), was originally derived in [32]. One can see that the bi-
2
Hamiltonian structure J and J of the complex KdV equation comes from the Cartan structure
1 2
equation for the conformal invariant curve flow. According to the decomposition of the Lie
algebra
g = g ⊕g ⊕g ,
1 0 −1
the Cartan curvature form Ω is decomposed to Ω = Ω +Ω +Ω , where J comes from torsion
1 0 −1 1
part of the structure equation, i.e., the Ω part, and J arises from the Ω part. In the sequel,
−1 2 1
we will show that the complex CH equation admits a bi-Hamiltonian structure Jˆ and Jˆ. It
1 2
turns out that the complex CH equation is a dual version of the complex KdV equation (in the
sense of [46]).
It is well-known that the CH equation is a bi-Hamiltonian system [8]
δH δH
1 2
m = J = D ,
t
δm δm
where J = −(m∂ +∂m) and D = −(∂ −∂3) are Hamiltonian operators, H = −1 (cid:82) umdx and
1 2
H = −1 (cid:82) u(cid:0)u2+u2(cid:1)dxarethecorrespondingHamiltonianfunctionals. AsfortheCHequation,
2 2 x
the complex CH equation can be obtained by the approach of tri-Hamiltonian duality [46].
Indeed, we have the following result.
Theorem 1. The complex CH equation (26) is a bi-Hamiltonian system, which can be written
as
δHˆ  δHˆ 
2 1
(cid:18) (cid:19)
mn = Jˆ1δδHmˆ  = Jˆ2δδHmˆ 
t 2 1
δn δn
with
(cid:18) ∂3−∂ 0 (cid:19) (cid:18)m∂ +∂m n∂ +∂n (cid:19)
Jˆ = , Jˆ =
1 0 ∂ −∂3 2 n∂ +∂n −m∂ −∂m
and
(cid:90) (cid:90)
Hˆ = 1 (vn−um)dx, Hˆ = 1 (cid:2)u(cid:0)u2+u2(cid:1)−u(cid:0)3v2+v2(cid:1)−2vu v (cid:3)dx.
1 2 2 2 x x x x
Proof. Clearly, Jˆ and Jˆ are skew symmetric. To prove they are Hamiltonian operators, it
1 2
suffices to prove that the Poisson bracket defined by Jˆ satisfies the Jacobi identity.
2
The bi-vector associated with Jˆ is defined by [44]
2
(cid:90) (cid:90)
1
Θ = (ϑ∧Jˆϑ)dx = (mθ∧θ +nθ∧ζ +nζ ∧θ −mζ ∧ζ ),
Jˆ2 2 R 2 R x x x x
where ϑ = (θ,ζ), θ and ζ denote the basic unit vectors corresponding to m and n, respectively,
the notation ∧ denotes the usual inner product between ϑ and Jˆϑ. It suffices to show that the
2
Schouten bracket vanishes, namely [Jˆ,Jˆ] = 0. In terms of
2 2
prv (m) = 2mθ +m θ+2nζ +n ζ, prv (n) = 2nθ +n θ−2mζ −m ζ,
Jˆ2ϑ x x x x Jˆ2ϑ x x x x