Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

Higher KdV flows on spaces of closed equicentroaffine plane curves are studied and it is shown that the flows are described as certain multi-Hamiltonian systems on the spaces. Multi-Hamiltonian systems describing higher mKdV flows are also given on spaces of closed Euclidean plane curves via the geometric Miura transformation.


Introduction
A motion of a curve is a smooth one-parameter family of connected curves in a space. It is known that many differential equations related to integrable systems can be linked with special motions of curves [10,11,12,29]. For example, for a special motion of an inextensible curve in the Euclidean plane, the curvature evolves according to the modified Korteweg-de Vries (mKdV) equation [19] (cf. Section 4 below). There are a lot of preceding studies on motions of curves related to Euclidean geometry and the mKdV equation. See [24,30,32] and references therein. For special motions of a space curve, it is also known that the nonlinear Schrödinger equation appears [16]. In [13,14], the authors studied motions of a curve in the complex hyperbola under which the curvature evolves according to the Burgers equation.
In this paper, we shall study motions of an equicentroaffine plane curve. Under a special motion of an equicentroaffine plane curve, the equicentroaffine curvature evolves according to the Korteweg-de Vries (KdV) equation. In order to explain the above motion geometrically, Pinkall [28] introduced the natural presymplectic form on the space of closed equicentroaffine plane curves with fixed enclosing area, and showed that the equicentroaffine curvature evolves according to the KdV equation when the flow is generated by the total equicentroaffine curvature. Furthermore, the result has been generalized to the case of higher KdV flows (cf. [9,15]).
On the other hand, it is known that a lot of completely integrable systems are described as bi-Hamiltonian systems, from which the existence of many first integrals can be deduced (Magri's theorem [22,27]). In this context, many of motions of curves as above have been studied from the viewpoint of bi-Hamiltonian systems recently [1,2,3,4,5,6,7,8,21,23,24,31]. The purpose of this paper is to construct a multi-Hamiltonian structure associated to the higher KdV flows on each level set of Hamiltonian functions in a geometric way (Theorem 7). Moreover, we shall also introduce multi-Hamiltonian structures associated to the higher mKdV flows on the spaces of closed Euclidean plane curves via the geometric Miura transformation.
2 A bi-Hamiltonian structure on the space of closed equicentroaf f ine curves Throughout this paper all maps are assumed to be smooth. For a regular plane curve γ whose velocity vector is transversal to the position vector at each point, we can choose the parameter s of γ as det γ(s) γ s (s) ≡ 1 holds. A plane curve γ provided with such a parameter s is called an equicentroaffine plane curve. For an equicentroaffine plane curve γ, we can define a function κ, called the equicentroaffine curvature, by γ ss = −κγ.
We set the space M of closed equicentroaffine plane curves by where S 1 = R/2πZ. Let γ( · , t) ∈ M be a one-parameter family of closed equicentroaffine plane curves. As in [28], the motion vector field γ t is represented as and the equicentroaffine curvature κ evolves as where is the recursion operator of the KdV equation: Hence when we choose the one-parameter family γ( ·, t) as α = D −1 s Ω n−1 κ s , we obtain the nth KdV equation for κ: The tangent space of M at γ ∈ M is described as and we can define a presymplectic form ω 0 on M by When X and Y are given by a direct calculation shows that from which we see that the kernel of ω 0 at γ is R · γ s .
It is known that the higher KdV equation (3) as well as (2) has an infinite series of conserved quantities {H m } m∈N given in the form of where h m is a polynomial in κ and its derivatives up to order m, for example, (see [17,20,25,26]). Moreover, by using the conserved quantity, nth KdV equation (3) can be expressed as where δH n+2 /δκ is the variational derivative of H n+2 : The expression (5) played an important role in computation in [15], where we studied the higher KdV flows on the space of closed equicentroaffine curves as Hamiltonian systems; using the above presymplectic structure ω 0 , we gave the Hamiltonian flows associated with the higher KdV equations. The paper [15] deals also with the geometric Miura transformation as is mentioned in Section 5 below. For each n ∈ N, we define a vector field X n on M by Regarding {H m } m∈N as functions on M by substituting the equicentroaffine curvature of γ for κ, we have the following proposition, which is essentially due to Pinkall [28] in the case n = 1.

Proposition 1 ([15]
). For each n ∈ N, X n is a Hamiltonian vector field for H n with respect to ω 0 , i.e., dH n = ω 0 (X n , · ) holds. Hence H n is a Hamiltonian function for the nth KdV flow γ t = X n . Now, we define another form ω 1 on M by which is represented as for X, Y given by (4). The following shows that ω 0 and ω 1 with {H m } m∈N define a bi-Hamiltonian structure on M (cf. [22,27]).
Theorem 2. The form ω 1 is a presymplectic form on M. For each n ∈ N, X n is a Hamiltonian vector field for H n+1 with respect to ω 1 .
Proof . For two functions F and G on M of the form we set Then from [18,22], we see that { · , · } 1 provides a Poisson bracket with We and Hence ω 1 is skew-symmetric and its closedness follows from the Jacobi identity for { · , · } 1 since for functions F , G and H = S 1 h(κ, κ s , κ ss , . . . )ds on M we have Moreover, since we obtain X n = X H n+1 and hence Therefore X n is a Hamiltonian vector field for H n+1 with respect to ω 1 .
The special linear group of degree two SL(2; R) acts on M as M γ → Aγ ∈ M (A ∈ SL(2; R)). Two elements of M belong to the same orbit if and only if their equicentroaffine curvatures coincide. Hence ω 1 is invariant under the action of SL(2; R). Moreover, the kernel of ω 1 at γ is the tangent space of the orbit SL(2; R) · γ; indeed for a one-parameter family γ(·, t) ∈ M, it follows from (2) and (6) that the tangent vector (1) belongs to the kernel of ω 1 if and only if κ t = 0, that is, κ is independent of t and hence γ(·, t) is contained in an SL(2; R)-orbit. As a consequence, ω 1 defines a symplectic form on the quotient space M/SL(2; R).
We consider another action on M given by It is obvious that this S 1 -action is presymplectic, that is, it leaves ω 1 invariant. Moreover, the action is Hamiltonian as we see in the proof of the following theorem.
Theorem 3. The moment map µ 1 for the S 1 -action (8) with respect to ω 1 is given by Proof . The fundamental vector field A on M corresponding to ∂/∂σ ∈ Lie(S 1 ) is given by A γ = γ s (γ ∈ M). For any tangent vector γ t = −(1/2)α s γ + αγ s , we have which implies (9) by the definition of the moment map.
Remark 4. Let Φ τ n be the flow generated by X n , that is, Φ · n is a one-parameter transformation group of M such that As an R-action on M, Φ · n is Hamiltonian with respect to ω 0 and the corresponding moment map is given by H n . In the following, we assume that each M(C m ) is not an empty set.
Lemma 5. For functions α, β on S 1 , if D −1 s ΩD s α is determined as a function on S 1 , then we have Proof . Noting ΩD s = (1/2)D 3 s + κD s + D s κ, we can easily verify (10) by integration by parts.
We assume that the proposition holds for m = l for some l ≥ 1. Then, for X ∈ T γ M(C l+1 ) = T γ M(C l ) ∩ Ker(dH l+1 ) γ , by using (10) we get which implies that Ω l+2 α s is determined as a function on S 1 in the same way as in the case m = 1.
From Proposition 6, we can define a tensor field ω m+1 of type (0, 2) on M(C m ) by which is shown to be skew-symmetric by using (10). Furthermore, in a similar way to the proof of Theorem 2, we see that ω m+1 is a presymplectic form and X n is a Hamiltonian vector field for the Hamiltonian function H n+m+1 with respect to ω m+1 ; indeed, for functions F , G given by (7) and for an integer k, putting we have a family of Poisson brackets { · , · } k with which implies that ω m+1 is presymplectic. Moreover, since Hence X n is a Hamiltonian vector field of H n+m+1 with respect to ω m+1 .
Besides ω m+1 , we have m + 1 more presymplectic forms on M(C m ) by restricting ω 0 , ω 1 on M and ω k+1 's on M(C k )'s for k = 1, 2, . . . , m − 1 to M(C m ); we denote them by the same symbols. By the discussion so far, we obtain the following theorem.

Theorem 7.
On M(C m ), for each n ∈ N and k = 0, 1, . . . , m + 1, X n is a Hamiltonian vector field for H n+k with respect to ω k , that is, the set {H n } n∈N , {ω k } m+1 k=0 is a multi-Hamiltonian system on M(C m ) describing the higher KdV flows.
As on M, we have the following theorem for a Hamiltonian S 1 -action on M(C m ): Theorem 8. The moment map µ m+1 for the S 1 -action on M(C m ) with respect to ω m+1 is given by Remark 9. We can define ω m+1 in a manner similar to the definitions of ω 0 and ω 1 . We put a map φ from T γ M to the space of all vector fields along γ as For any tangent vector X of M, (D 2 s + κ)X has no γ s -component and it belongs to the image of φ if X is tangent to M(C 1 ). Then for X ∈ T γ M(C 1 ) we have holds. More generally, [φ −1 (D 2 s + κ)] m X can be defined for any tangent vector X of M(C m ) and we obtain on M(C m ). We note that this formula is valid in the case ω 1 (m = 0) and even in the case ω 0 (m = −1) since

A bi-Hamiltonian structure on the space of closed curves in the Euclidean plane
We denote by E 2 the Euclidean plane equipped with the standard inner product · , · , and we set the spaceM of closed curves in the Euclidean plane E 2 bŷ Forγ ∈M, the curvatureκ is defined by T s =κN , where T =γ s is the velocity vector field and N is the left-oriented unit normal vector field alongγ. Letγ( · , t) ∈M be a one-parameter family of closed curves in E 2 . Thenγ t is represented aŝ γ t = λT + µN, λ, µ : S 1 → R, λ s =κµ, and the curvatureκ evolves aŝ κ t = µ ss +κλ s +κ s λ =Ω(2µ), is the recursion operator of the mKdV equation: Hence when we choose µ = (1/2)Ω n−1κ s , we have the nth mKdV equation forκ: The tangent space ofM atγ ∈M is described as and we can define a presymplectic formω 0 onM bŷ When X and Y are given by we havê and we see that the kernel ofω 0 atγ is R ·γ s . As in the case of the higher KdV equation (3), the nth mKdV equation (11)  whereĥ m is a polynomial inκ and its derivatives up to order m, for example, For each n ∈ N, we define a vector fieldX n onM by then we have the following.
Proposition 10 ( [15]). For each n ∈ N,X n is a Hamiltonian vector field forĤ n with respect toω 0 . HenceĤ n is a Hamiltonian function for the nth mKdV flowγ t =X n .
In addition, we define another formω 1 onM bŷ which is represented aŝ (κλ + µ s )Ωμds for X, Y given by (12). The following theorem is proved in a similar way to the proof of Theorem 2.
Theorem 11. The formω 1 is a presymplectic form onM. For each n ∈ N,X n is a Hamiltonian vector field forĤ n+1 with respect toω 1 .
Note that the Euclidean motion group E(2) = O(2) R 2 of E 2 acts onM. It is easily verified thatω 1 is invariant under the E(2)-action and the kernel ofω 1 at TγM contains the tangent space of the orbit. Hence ω 1 determines a presymplectic form onM/E (2).
As well as on (M, ω 1 ), S 1 acts onM leavingω 1 invariant and the following theorem holds.
Theorem 12. The moment mapμ 1 for the S 1 -action onM with respect toω 1 is given bŷ

The geometric Miura transformation and multi-Hamiltonian structures on spaces of closed curves in the Euclidean plane
First, we briefly review the geometric Miura transformation which relates the Hamiltonian structures on M and onM (see [15] for more details). We consider the complexification of M: We determine the curvature of γ ∈ M C , (complex) presymplectic forms on M C , etc. by the same formulas as in the case of M, hence we use the same symbols κ, ω 0 , ω 1 , . . . to denote them. By identifying the range E 2 ofγ ∈M with a complex plane C, we define the geometric Miura transformation Φ :M → M C by Φ(γ) = (−γ s ) − 1 2 (γ, 1) ,γ ∈M.
The curvature κ of Φ(γ) is related with the curvatureκ ofγ by the Miura transformation: Moreover, we have the following.
Therefore, Φ gives a map fromM(C m ) to M C (C m ) and we have a presymplectic formω m+1 = Φ * ω m+1 onM(C m ). Under these settings the following theorems are directly deduced from Theorems 7 and 8.
Theorem 14. OnM(C m ), for each n ∈ N and k = 0, 1, . . . , m + 1,X n is a Hamiltonian vector field forĤ n+k with respect toω k , that is, the set {Ĥ n } n∈N , {ω k } m+1 k=0 is a multi-Hamiltonian system onM(C m ) describing the higher modified KdV flows.
We note that (15) impliesω m+1 is a real form, though ω m+1 on M C (C m ) is complex.