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

In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric 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. It is shown that these equations arise from non-streching invariant curve flows respectively in the three-dimensional Euclidean geometry, the two-dimensional M\"obius sphere and $n$-dimensional sphere ${\mathbb S}^n(1)$. Integrability to these systems is also studied.


Introduction
Integrable systems solved by the inverse scattering method usually arise from shallow water wave, physics, optical communication and applied sciences etc. Integrable systems have many interesting properties, such as Lax-pair, infinite number of conservation laws and Lie-Bäcklund symmetries, multi-solitons, Bäcklund transformations and bi-Hamiltonian structure etc. [1,44], which are helpful to explore other properties of integrable systems [1,44,56].
It is of great interest to study geometric aspects of integrable systems. So far, very few integrable 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 Schrödinger equation (NLS) iφ t + φ ss + |φ| 2 φ = 0 is equivalent to the system for the curvature κ and τ of curves γ in R 3 κ t = −2τ κ s − κτ s , τ t = κ sss κ − κ s κ ss κ 2 − 2τ τ s + κκ s 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 via the so-called Hasimoto transformation φ = κ exp(i s τ (t, z)dz). Indeed, the system (1) is equivalent to the vortex filament equation where b is the binormal vector of γ. Marí 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 {t 1 , n 1 , b 1 }. 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 on Lie group manifolds [3,4,31,39]. The KdV equation, the modified KdV equation, the Sawada-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 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(S 1 × R) with respect to right-invariant Sobolev H 1 metric for a = 0 [13,14,28] and Bott-Virasoro algebra for a = 0 [40]. A dual version of the Ito system is the two-component Camassa-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] where δ = ±1, and the second one is the so-called Novikov equation [25,43] 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 nonlinear dispersion, which in most cases, enables these systems to support non-smooth solitonlike 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 Schrödinger 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, Möbius sphere and the ndimensional unit sphere S n (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 R 3 is introduced and studied. Making use of the system for curvature and torsion corresponding to this flow, we obtain a novel integrable Schrödinger equation by a scale limit approach, which is completely integrable system and can be obtained by the so-called tri-Hamiltonian duality approach [18,46]. In Section 3, we give a brief discussion on Möbius 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 Möbius 2-sphere. It is shown that the complex Camassa-Holm equation and complex Hunter-Saxton equation describe the non-stretching curve flows in Möbius 2-sphere. The bi-Hamiltonian structure for the complex Camassa-Holm equation is obtained. In Section 5, we study non-stretching curve flows in the n-dimensional sphere S n (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 S n (1). Integrability of the system is identified. Finally, Section 6 contains concluding remarks on this work.

An integrable nonlinear Schrödinger equation
We consider the flows of space curves in R 3 , given by 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 respect to arc-length parameter s. The arc-length parameter s is defined implicitly by ds = hdp, h = |γ (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] and Assuming that the flow is intrinsic, namely the arc-length does not depend on time, it implies from (6) that In terms of (7), one finds that the complex function satisfies the equation [24,42] whereφ denotes the complex conjugate of φ.
where C 1 is a constant. Setting C 1 = 0, we derive from (9) the celebrated Schrödinger equation where C 2 is a constant. Letting C 2 = 0, we find that φ satisfies the mKdV system In the following, we shall consider the case U = W = 0. Denote θ(s, t) = s τ (s , t)ds , g = V η. It follows from (9) that φ satisfies the equation Setũ = κ cos θ,ṽ = κ sin θ, g = g 1 + ig 2 , then the equation (11) is separated to two equations Furthermore, lettingũ = u + v s ,ṽ = v − u s , and choosing the binormal velocity V to be we find that u and v satisfy the system with Applying the scaling transformations Expanding u and v in powers of the small parameter and plugging them into system (13), we find that the leading order terms u 0 (t, s) and v 0 (t, s) satisfy the system Again we use the notation φ(t, s) = u 0 (t, s) + iv 0 (t, s), then it is inferred from (14) that φ(t, s) satisfies the equation which is a dual version of the Schrödinger equation (10), and can be obtained by the approach of tri-Hamiltonian duality [18,46]. Equation (15) is formally completely integrable since it admits bi-Hamiltonian structure [46] are compatible Hamiltonian operators, while are the corresponding Hamiltonian functionals.
3 Möbius sphere PO(3, 1)/H and unit sphere SO(n + 1)/SO(n) In this section, we give a brief account of Möbius 2-sphere PO(3, 1)/H and unit sphere S n (1) = SO(n + 1)/SO(n). Please refer to the book [53] for the details of the two geometries.

Möbius 2-sphere
Let (u 0 , u 1 , u 2 , u 3 ) ∈ R 4 , we define the inner product on R 4 by where x, y ∈ R 4 , and the matrix Λ 3,1 is A vector field x ∈ R 4 is said to be light-like, if it satisfies x, x = 0. All the light-like vector fields form a set L, which is called optical cone, defined by the equation Clearly it is homogeneous, namely for any λ ∈ R, if x ∈ L, then λx ∈ L. The projectivisation of L is said to be Möbius 2-sphere, which is isomorphic to S 2 . Recall that and the Möbius group is defined to be PO(3, 1) = O(3, 1)/±I. We denote where [e 3 ] denotes the equivalent class of e 3 in P (R 4 ), [e 3 ] = (0, 0, 0, * ). A straightforward computation gives It is easy to verify that the group PO(3, 1) acts on the Möbius sphere transitively (the group action is the usual conformal transformation). For any g ∈ PO(3, 1), there exists a unique decomposition around the identity of the group where p, q ∈ R 2 , ∈ R, S ∈ O(2). The Lie algebra of the isotropy group H is h = g 1 ⊕ g 0 while g/h = g −1 is identified to the tangent space of the conformal sphere PO(3, 1)/H.
3.2 n-dimensional sphere S n (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 where p ∈ R n , Θ ∈ so(n), and the decomposition satisfies where m is identified to the tangent space T x M ∼ = R n of M = SO(n+1)/SO(n). The flat Cartan connection of principle SO(n) bundle SO(n + 1) → S n is given by the Maurer-Cartan form of Lie group SO(n + 1). The Cartan structure equation reads as where so(n + 1)-valued one-form ω is decomposed to where g/h-valued θ represents a linear coframe, h-valued ω H represents a linear connection on S n . The corresponding Cartan structure equation is separated to where J and R are called torsion and curvature forms, respectively. 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) ⊂ M 2 . Let D x and D t denote respectively the vector field d dx and d dt along the curves ρ in PO(3, 1), then the Frenet formulae for the conformally parametric curves is

Curve f lows in PO(3, 1)/H and the complex CH equation
where k 1 and k 2 are the conformally differential invariants for the curve γ(x, t). The time evolution for the frame ρ(x, t) can be written as and , α, f i , h i (i = 1, 2) are some conformal differential invariants related to k 1 and k 2 , to be determined. By the Cartan structure equation, one gets Plugging (16) and (17) into (18) results in the following equations where (19) is the torsion part (i.e.,the g −1 part) of the Cartan structure equation (18), which can be written as Inserting (19) into (20) and (21) gives Substituting (19) and (24) into (22) and (23), we obtain the evolution equations for the curvatures k 1 and k 2 [32,36] which is equivalent to The following cases are considered. Case 1. (25), we obtain the new two-component CH equation The above system admits the following Lax-pair we get the two-component Harry-Dym equation Remark 1. Marí Beffa [32,36] showed that the complex KdV equation arises from the invariant curve motion in Möbius 2-sphere. Indeed, taking h 1 = −k 1 , h 2 = −k 2 in (25) yields the complex KdV equation Its Hamiltonian structure J 2 , see (25), was originally derived in [32]. One can see that the bi-Hamiltonian structure J 1 and J 2 of the complex KdV equation comes from the Cartan structure equation for the conformal invariant curve flow. According to the decomposition of the Lie algebra the Cartan curvature form Ω is decomposed to Ω = Ω 1 + Ω 0 + Ω −1 , where J 1 comes from torsion part of the structure equation, i.e., the Ω −1 part, and J 2 arises from the Ω 1 part. In the sequel, we will show that the complex CH equation admits a bi-Hamiltonian structureĴ 1 andĴ 2 . It 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] x dx are the corresponding Hamiltonian functionals. As for the CH equation, the complex CH equation can be obtained by the approach of tri-Hamiltonian duality [46]. Indeed, we have the following result. (26) is a bi-Hamiltonian system, which can be written as

Theorem 1. The complex CH equation
Proof . Clearly,Ĵ 1 andĴ 2 are skew symmetric. To prove they are Hamiltonian operators, it suffices to prove that the Poisson bracket defined byĴ 2 satisfies the Jacobi identity. The bi-vector associated withĴ 2 is defined by [44] 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 andĴ 2 ϑ. It suffices to show that the Schouten bracket vanishes, namely [Ĵ 2 ,Ĵ 2 ] = 0. In terms of a direct computation shows where the skew-symmetric property for the wedge product is used.
[Ĵ 1 , Through integration by parts, we get ThusĴ 1 andĴ 2 are a Hamiltonian-pair. Let's write the complex CH equation (26) as follows It is easy to get Thus Hence we deduce that To computeĤ 2 , we write (26) as .
It follows that Note that Hence we arrive at

Curve f lows on S n (1) and multi-component modif ied CH equations
Assume that γ(x, t) is a curve flow on unit sphere S n (1) = SO(n + 1)/SO(n), which satisfies γ = 1, where x is the invariant arc-length parameter, t is the time. The natural frame of the curve γ ∈ S n (1) is {e 1 = γ x , e 2 , . . . , e n }. Let ρ = (e 0 = γ, e 1 , . . . , e n ) ∈ SO(n + 1) be the lift from S n (1) to bundle space SO(n + 1), and D x and D t denote respectively the tangent and evolutionary vector field. It follows that whereω is the Cartan connection (the natural frame formulae (28) for curves on the sphere comes out from the Frenet formulae [15] through the Hasimoto transformation). Here k = (k 1 , k 2 , . . . , k n−1 ) is the natural curvature vector of γ. Assume that the curve flow is governed by where the tangent velocity f and normal velocities h i (i = 1, 2, . . . , n − 1) depend on the curvatures and their derivatives with respect to arc-length x.
The induced time evolution for the frame is where h, ξ ∈ R n−1 , ξ is a unknown vector, which will be determined later by the structure equations.
First, we assume that the flow is intrinsic, namely, the distribution {D x , D t } satisfies [D x , D t ] = 0 so that the integral submanifold is a smooth two-dimensional surface on Lie group SO(n + 1). Making use of the Cartan structure equation one gets the following equations where (29) is the arc-length preserving condition. For convenience, the following notations are used. For any a, b ∈ R n−1 , a, b denotes the usual Euclidean inner product, i.e., a, b = a T b, a ⊗ b denotes the tensor product, namely · · · · · · · · · a n−1 b 1 a n−1 b 2 · · · a n−1 b n−1 (29) and (30), it follows that In view of (31), we have Plugging (34) and (33) into (32) leads to the equation for the curvature vector where the identity for vectors was used. Analogous to the derivation for the CH equation [18] and the modified CH equation [46], we restrict our attention to the following cases. Case 1. h = u x , k =m = u + u xx . In this case, the tangent velocity f is determined by where c 0 is an integration constant. Substituting (36) with c 0 = −1 together with the expressions for k and h into (35), and noting that we arrive at the multi-component modified CH equatioñ Thus we have established the following result.

Remark 2.
In the case of n = 2, i.e., the case of S 2 (1), equation (37) reduces to the scaler modified CH equation (4) with δ = −1, which is completely integrable. In the case of n = 3, let u 1 = u, u 2 = v, m = u + u xx , n = v + v xx , then system (37) reduces to In general, the multi-component system (37) can be written as In this case, the tangent velocity f is given by where c 1 is an integration constant. Substituting (39) with c 1 = 1 into (35) and noting that we obtain the multi-component modified CH equation Thus we have proved the following result.
where p ∈ R n , Θ ∈ so(n). The Cartan connection matricesω(D x ) andω(D t ) of the natural frame are replaced witĥ for curves on H n = SO(n, 1)/SO(n). Similar results can be derived for the Hyperbolic space H n . Integrability of the equations (37) and (44) are guaranteed by the following results.

Concluding remarks
In this paper, geometrical formulations to several multi-component integrable systems are provided. These systems are regarded as multi-component generalizations of the CH equation and the modified CH equation, which can be obtained through the tri-Hamiltonian duality approach. We showed that an integrable generalization to the nonlinear Schrödinger equation arises from a non-stretching invariant curve flow in the three-dimensional Euclidean geometry. The integrable complex CH equation comes from an invariant curve flow on the Möbius 2-sphere. Furthermore, we verified that multi-component generalizations to the modified CH equation arise naturally from the curve flows in n-dimensional sphere S n (1) and the hyperbolic space H n = SO(n, 1)/SO(n).
In [27], Olver, Kamran and Tenenblat have established the theory for curves in affine symplectic geometry. The curve flows in four-dimensional affine symplectic geometry were studied in [58], and an integrable three-component equation with bi-Hamiltonian structure was obtained. The theory for curves in the centro-equiaffine symplectic geometry was established in [54]. It was shown that certain invariant curve flows in the centro-equiaffine symplectic geometry yield noncommutative KdV equations [47]. It is still not clear that what are the dual version of these integrable equations arising from curve flows in the affine and centro-equiaffine symplectic geometries.