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From Euler's elastica to the mKdV hierarchy, through the Faber polynomials

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.


Introduction
When the modified KdV (mKdV) equation was added to the unsystematically growing number of "completely integrable PDEs", the ad hoc explanation given was Miura's powerful idea: starting with the celebrated KdV equation "the simplest modification of the nonlinear term" becomes mKdV; what is now called a "Miura transformation" [14] u = v x − v 2 factors, with the mKdV equation as the righthand factor, (−2v + ∂ x )(v t − 6v 2 v x + v xxx ) = 0. Solutions to mKdV thus transform into solutions of KdV, though the converse does not always hold.
However, there appeared to be no direct motivation for mKdV, as opposed to the wave motion in a shallow canal that was modeled by KdV in the nineteenth century, even though Miura (loc. cit.) does refer to anharmonic lattices. Goldstein and Petrich [4,5] showed that going from the real (x, y) to the complex z = x + √ −1y representation of a number on the plane, takes KdV to mKdV, and they obtained the latter equation by the description of a plane region which moves in time with conserved area and perimeter: the curvature satisfies the mKdV equation (up to rescaling).
The first-named author et al. in a series of papers [8,9,10,11,12], defined "quantized elastica", based on Euler's Elastica theory, and derived mKdV. Euler's Elastica have a long history, spanning from Pappus of Alexandria around 300 A.D., to splines and computer vision today, through the discovery of the addition theorem for elliptic functions, which give the closed-form solution, as thoroughly surveyed in a recent Ph.D. thesis [7,6], as well as Truesdell's classic works [17] on the history of mechanics, particularly Euler's works. Put in modern terms, the elastica are curves that solve an isoperimetric variational problem, by minimizing a potential energy, but there are many equivalent formulations; in particular, by writing the equivalent dynamics of the simple swinging pendulum, the general solution can be parametrized by one number, a Lagrange multiplier, representing the force of gravity in a suitable normalization [6]. "Quantized elastica" replace the Euler-Bernoulli energy functional [17] (cf. Section 2 for the meaning of the symbols), the partition function [15] of the temperature β. The mKdV hierarchy then appears by deforming the coefficients of the expansion of the integral at its extrema [8].
In this paper, we derive the mKdV hierarchy by introducing time parameters over a suitable moduli space of plane loops; we use the setting of Brylinski [2] to construct differential flows on several related moduli spaces, and we partition the appropriate moduli space by the mKdV orbits under a commuting hierarchy of isometric and isoenergy flows. We present the invariant that defines the elastica as the integral of a coefficient in the expansion of a holomorphic function used by Tjurin [16] to represent projective connections on a Riemann surface, so that the setting is that of the moduli space of curves; both the isometric and isoenergy conditions are therefore constraints on the coefficients of said expansion, which points out a relationship with modular forms and Faber polynomials (Subsection 6.5).
Acknowledgments: The first-named author expresses his thanks to Y. Kodama, V. Enolskii, M. Guest and A. Fujioka for their invitations to their seminars and discussions on this subject. Both authors are very grateful to John McKay for supporting their visit in Concordia University, where they had the opportunity to hear his talks and questions about replicable functions and Faber polynomials.

Setting and review
We work in the free loop space LM of smooth (i.e., real analytic) maps S 1 → M (cf. Section 3.1 in Ref. [2]), where the (smooth, paracompact, oriented) manifold M is the real plane. In other words, our space LM is the set of real-analytic immersions of S 1 in C, Z : S 1 ֒→ C, Z(s) ∈ C ω (S 1 , C), which we parametrize by arclength; moreover, we fix the length and take the quotient space under Euclidean motion: The condition |dZ/ds| = 1, namely dZ/ds ∈ U(1), allows us to define an action g s 0 Z(s) = Z(s−s 0 ) (where we identified S 1 with R modulo the integers); we call "Euclidean motion" a transformation with this property. We set: We where ϕ is a real-analytic function. The curvature of the loop is given by: so that, by equating separately the real and imaginary part [4], the Frénet-Serret relation can be expressed as, Moreover, the integral of k with respect to ds is the winding number of the loop. The integral where the subscript SD denotes the Schwarzian derivative (2.3), is the Euler-Bernoulli energy functional-here we return to the representation of s as a complex number of norm 1.
To address applications to biology and other sciences, the first-named author [12] replaced the Euler-Bernoulli functional with: We note that these invariants appear in an expansion that was used by Tjurin [16] to encode the projective structure of a Riemann surface, but we point out that in our case it is a formal expansion because, while s is viewed as a complex number by Tjurin (and in the expression of the Schwarzian derivative below), it must be viewed as a real number when taking differences of the type s − s 0 . (2.2) Using the representation the first term of the expansion is √ −1ϕ(s 0 )/2, the coefficient of the second term is the curvature k(s 0 ), the coefficients of the third and fourth term are expressed by: We write them together: The fourth coefficient is the Schwarzian derivative, namely the operator, The integrals of the coefficients of both the third and fourth terms are therefore proportional to the Euler-Bernoulli energy functional (2.1).
For an element Z ∈ M, Z(0) vanishes up to Euclidean motion; assuming this to be the case, we introduce the expansion, where q = e √ −1s and will use the coefficients a i as moduli of the loops; we introduce a dependence of the a i 's on parameters t = (t 2 , t 3 , . . .) ∈ [0, ε), where ε is a small positive number, to be viewed as deformations independent of each other, and we require that they also satisfy the condition which we call "isometric"; we think of s as the stationary flow t 1 , cf. Section 5. We will derive the mKdV hierarchy by imposing an extra condition on the loops We denote by A p S 1 (R) the real-analytic p-forms on S 1 with R-valued coefficients, and by A p S 1 (C) the real-analytic p-forms on S 1 with C-valued coefficients.
For a given Z ∈ M, the tangential angle ϕ ∈ A 0 S 1 (R) and the curvature k ∈ A 0 S 1 (R) are defined as consistent with the terminology at the beginning of this Section. Since |∂ s Z(s)| = 1, for fixed parameters t i , we view the curvature k as an element of A 0 S 1 (R) and kds as an element of A 1 S 1 (R). 4

Isometric Deformations
We consider the tangent space of the isometric deformations in M, namely a fiber space Regarding the parameter s in Z as fixed, the fiber is given by the isometric deformations of M. We let IDiff(M) be the group acting on M that gives isometric deformations, and we consider the action of its Lie algebra idiff(M) at a point Z in M. We identify the infinitesimal orbit of idiff(M), or the infinitesimal deformations, and the elements of T Z M at Z ∈ M. We do not need to prove the existence of this, say, analytic Lie group, because it will consists of the flows that we construct concretely. We take the limit of the flow parametrized by t ∈ [0, ε) at Z ∈ M as ε approaches zero, in order to investigate the fiber T Z M. As a result, this limit gives us tangent vectors and differential operators belonging to the fiber, without a need to introduce a connection: for our purposes, we only need local calculations for given coordinates.
A deformation is defined by: where v belongs to A 0 where v (r) and v (i) denote real and imaginary parts respectively, so that v (r) , v (i) ∈ A 0 S 1 (R) := C ω (S 1 , R). Recalling that the "isometric" condition means that s, t are independent variables, namely ∂ t ∂ s Z = ∂ s ∂ t Z, we check that: Proof. The statement follows from the calculations, An isometric deformation of ∂ s Z = e √ −1ϕ belongs to C ω (S 1 , U(1)), and its derivative along t ∈ [0, ε) is given by By applying ∂ s to both sides of (3.1), we have Since ∂ t ϕ is real-valued, we obtain two conditions by equating the real and imaginary parts: For a point Z ∈ M, the isometric condition [∂ s , ∂ t ]Z = 0 is reduced to two conditions [4,5]: In the limit as ε approaches zero, these conditions hold on the fiber of the tangent space. Therefore we can define the following length-preserving linear transformation on the tangent space, which we identified with idiff(M), We derive some consequences from the second condition, (3.3). Let us consider the map, We define ℓ r : We will need (cf. Lemma 5.1) only deformations whose real part v (r) is unique up to multiplicative constant, so we fix a representative for the image in the quotient ℓ 0 r : We have an induced map, which is a Euclidean motion, By introducing a loop of constant derivative, we can arrange for similarly, we can add a constant to ∂ t φ(s) by using rotation as the Euclidean motion. We will use this choice of representative in M for an element Z ∈ M.
The following proposition is essentially the same as Proposition 3.3.4 (i) in Ref. [2], following from (3.3) for the map ℓ. For v ∈ A 0 S 1 (R) andZ ∈ M, ℓ induces the bijection ℓ ♯ and the surjection ℓ ♭ , in the sense that, for every element v in the vector space T Z (M), where Z has the repre-sentativeZ in M, we have an element v (i) ∈ A 0 S 1 (R) such that v = ℓ(v (i) ). Given this result, it will be more convenient and sufficient for our purposes to deal with elements of M rather than M.
For the first condition, (3.2), let us introduce the differential operators, where Ω (II) is the recursion operator for the mKdV hierarchy.

Lemma 3.4. The infinitesimal isometric deformation (3.1) is reduced to
as an element in the fiber of T Z M.

Isoenergy deformations
We will now impose, in addition to the isometric condition, that the Euler-Bernoulli energy functional E is preserved by the deformation. We call the Lie group of isometric and isoenergy motions acting on M IIDiff(M) ⊂ IDiff(M), and its Lie algebra acting at a point Z in M, iidiff(M). We identify the infinitesimal orbit of an element of iidiff(M) with the infinitesimal deformation itself and with the corresponding element of T Z M of Z ∈ M. From Lemma 2.1, we have the following result: there exists a function f ∈ A 0 S 1 (R) satisfying k∂ t k = ∂ s f . Note that (3.3), one of the isometric conditions, says that k∂ t k is an exact differential, akin to the isoenergy condition.
We now construct an isometric deformation: let v ′(i) := ∂ t k and v ′(r) := ℓ r (v ′(i) ), which satisfies k∂ t k = kv ′(i) = ∂ s v ′(r) . This yields the isometric condition, The following proposition and corollary use this construction: The converse also holds: If the isometric deformation, is isoenergy, then there is another isometric flow given by ℓ(v ′(i) ) satisfying ∂ t k = v ′(i) .

The stationary deformation
Since Z(s) and Z(s − s 0 ) have the same image loop, we call this the "stationary" deformation, in accordance with the terminology for the mKdV equation, when the solution u(x, t 2 , . . . , t n , . . .) is viewed as a function of x only (N.B. In previous work by the first-named author (Section 3 in Ref. [8]), this is called the "trivial" deformation).
Lemma 5.1. For any given real number c ∈ R and Z ∈ M, Z(s + ct), the stationary deformation, is a solution of ∂ t Z = cZ.
each element of the sequence {Ω (II) n ∂ s k} n=0,1,2,... belongs to A 0 S 1 (R), yielding a sequence of isometric and isoenergy relations, This recursive sequence is known as the modified KdV hierarchy.
Proof. We let v (i) in Proposition 4.6 correspond to the stationary deformation v (i) = ∂ s k, andt i = t i−1 for i = 2, 3, . . .. The equalities hold because the sequence of t i are the higher mKdV flows, since Ω (II) is the mKdV recursion operator. Then we can take v (i) in Proposition 4.6 equal to 0. Then, with v (i) = 0,t i = t i for i = 1, 2, 3, . . ..
By letting w j = ∂ t j ϕ, we also have a matrix format: .
It is well-known that the mKdV hierarchy is integrable and its orbits are well-defined, thus the Lie-group and Lie-algebra actions we used, for isometric and isoenergy diffeomerphism IIDiff(M) on M, make sense.
If the curvature k is given, ϕ is determined modulo a constant of integration. If ϕ is given, Z is determined modulo Euclidean motion. Therefore, the mKdV hierarchy acts on the space M rather than M: Proposition 5.6. There exists an action IIDiff(M) on M given by the mKdV hierarchy.

Related constructions, finite orbits, Comments
In this section, we review previous results (Prop. 3.11 in Ref. [10]), to relate the KdV flow with the deformations we described; we give equations for a class of finite mKdV orbits; and conclude with comments.
6.1. Connection between KdV flow and mKdV flow. In this subsection, we show the connection between the KdV flow (Prop. 3.11 in Ref. [10]) and the mKdV flow described above.
For a solution ψ of the deformation ∂ t Z = v∂ s Z induces the deformation, Since this deformation gives the mKdV hierarchy, v induces the KdV hierarchy, via the Miura map.
We have the natural symplectic structure on T M, under which the KdV flow is Hamiltonian. Moreover, the KdV hierarchy has a bi-Hamiltonian structure, one of the two compatible structures being associated to the Hamiltonian flow which preserves the Euler-Bernoulli energy of the elastica.
6.2. Symplectic structures. Besides the symplectic structure for the mKdV flow associated with (6.1) through the correspondence between KdV and mKdV, we have a sequence of natural symplectic structures, as follows.
where ℓ is a natural number. Clearly, u, v ℓ = − v, u ℓ . Moreover, for u and v giving isometric deformations,  The orbits are finite-dimensional when the hierarchy defined recursively contains a finite number of linearly-independent flows. This happens, in particular, when the deformations ∂ t i are stationary for i > g + 1, although this is not the only case: as an analogy, the flows of a finite-dimensional orbit of the KdV equation span the Jacobian of a spectral curve, of given genus g; they are coordinatized by the coefficients of the expansion of a basis of holomorphic differentials in (the inverse of) a local parameter t on the curve, ω i = ∞ j=0 c i j t j , i = 1, ..., g, and of course all vectors (c 1 j , ..., c g j ) could be non zero. We call the union of the particular finite-dimension orbits for which the flows become stationary exactly at the g stage M g , although the orbit in fact could have any smaller dimension, equal to the dimension of the span of the first g − 1 flows: , Ω (II) g k = 0}, , Ω (II) g k = 0} = pr 2 (M g ), giving a filtration: M g ⊂ M g+1 , and M g ⊂ M g+1 .
where the multiplicative-constant ambiguity of each Ω (II) is adjusted by rescaling the corresponding t j .
6.4. Action on cohomology. As in Ref. [10], we can interpret the hierarchy of flows on the filtration by orbits, by viewing M, the loops in the Euclidean plane, as the topological space obtained by compactifying C to P. We recall the result (Section III.16 in Ref. [1]): Theorem 6.2. The cohomology of the loop space ΩS n over S n is given by H p (ΩS n , R) = Rδ [p mod(n−1)],0 .
On the other hand, in Prop. 7.1 of Ref. [10], the following is proved,