Lagrangian Curve Flows on Symplectic Spaces

: A smooth map γ in the symplectic space R 2 n is Lagrangian if γ , γ x , . . ., γ ( 2 n − 1 ) x are linearly independent and the span of γ , γ x , . . . , γ ( n − 1 ) x is a Lagrangian subspace of R 2 n . In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in R 2 n with respect to the symplectic group Sp ( 2 n ) , (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve ﬂows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve ﬂows of C-type and A-type are solutions of the Drinfeld-Sokolov’s ˆ C ( 1 ) n -KdV ﬂows and ˆ A ( 2 ) 2 n − 1 -KdV ﬂows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve ﬂows.


Introduction
The modern theory of soliton equations dates from the famous numerical computation of the interaction of solitary waves of the Korteweg-de Vries (KdV) equation by Zabusky and Kruskal [1] in 1965. In 1967, Gardner, Green, Kruskal, and Miura [2] applied the Gelfand-Levitan's inverse scattering transform of the one-dimensional linear Schrödinger operator to solve the Cauchy problem for rapidly decaying initial data for the KdV equation. In 1968, Lax [3] introduced the Lax-pair for KdV. Zakharov and Faddeev [4] gave a Hamiltonian formulation of KdV, and proved that KdV is completely integrable by finding action-angle variables. Zakharov and Shabat [5] found a Lax pair of 2 × 2 first order differential operators for the non-linear Schrödinger equation (NLS), Adler-Kostant-Symes gave a method to construct completely integrable Hamiltonian systems using splitting of Lie algebras (cf. [6][7][8][9]), Kupershmidt-Wilson [10] constructed n × n modified KdV (mKdV) using a loop algebra, and finally Drinfeld-Sokolov [11] gave a general method to construct soliton hierarchies from affine Kac-Moody algebras. In particular, soliton equations have many remarkable properties including: a Lax pair, infinite families of explicit soliton solutions, Bäcklund and Darboux transformations that generate new solutions from a given one by solving a first order system, a permutability formula to superpose solutions, a rational loop group action, a scattering theory and an inverse scattering transform to solve the Cauchy problem, a bi-Hamiltonian structure, and infinitely many commuting Hamiltonians. For more detail and references, we refer readers to the following books and survey articles: [11][12][13][14][15][16][17][18].
Soliton equations are also found in classical differential geometry: the sine-Gordon equation (SGE) arose first through the theory of surfaces of negative constant Gauss curvature in R 3 , and the reduced 3-wave equation can be found in Darboux's work [19] on triply orthogonal coordinate systems of R 3 . These equations were rediscovered later independently of their geometric history. The main contribution of the classical geometers lies in their methods for constructing explicit solutions of these equations from geometric transformations.
Next we discuss how curve flows appeared in soliton theory. In 1906, da Rios, a student of Levi-Civita, wrote a master's thesis, in which he modeled the movement of a thin vortex by the motion of a curve propagating in R 3 along its binormal with curvature as speed, i.e., γ t = kb.
This is the vortex filament equation (VFE). It was much later, in 1971, that Hasimoto showed in [30] the equivalence of VFE with the NLS, q t = i(q xx + 2|q| 2 q).
In fact, if γ(x, t) is a solution of VFE, then there exists a function θ(t) such that is a solution of the NLS, where k, τ are the curvature and torsion of the curve. This correspondence between the VFE and NLS given above uses the Frenet frame. If we use the parallel normal frame, then the correspondence can be stated as follows: If γ is a solution of the VFE, then there exists an orthonormal moving frame g = (e 1 , e 2 , e 3 ) : R 2 → SO(3) such that and q = k 1 + ik 2 is a solution of the NLS, where e 1 (·, t) is tangent to the curve γ(·, t), e 2 (·, t) and e 3 (·, t) are parallel normal fields along γ(·, t), and k 1 (·, t) and k 2 (·, t) are the principal curvatures along e 2 (·, t) and e 3 (·, t) respectively. Since the NLS is a soliton equation, we can use techniques in soliton theory to study geometric and Hamiltonian aspects of the VFE. The NLS admits an so(3) valued Lax pair with phase space C ∞ (R, V), where Please note that the differential invariants constructed from the parallel frames for curves in R 3 lie in C ∞ (R, V). Hence a good way to construct integrable curve flows on a homogeneous space M = G · p 0 = G/H is to find a class of curves in G/H, which has a moving frame g : R → G so that γ = g · p 0 , g −1 g x gives a complete set of differential invariants, and g −1 g x lies in the phase space of a soliton equation. A more detailed discussion of how to use this scheme to construct integrable curve flows can be found in [31].
There are many recent works on integrable geometric curve flows in homogeneous spaces. For example, Langer-Perline studied Poisson structures and local geometric invariants of the VFE in [32,33], and constructed curve flows that relate to Fordy-Kulish NLS type hierarchies associated with Hermitian symmetric spaces in [34]. Doliwa-Santini constructed curve flows in R 2 and R 3 that give the mKdV and NLS respectively in [35]. Ferapontov gave hydro-dynamic type curve flows on homogeneous isoparametric hypersurfaces in sphere in [36]. Yasui-Sasaki studied the integrability of the VFE in [37]. Chou-Qu constructed integrable curve flows in affine plane in [38] and integrable curve flows in the plane for all Klein geometries in [39]. Anco constructed integrable curve flows on the symmetric space U K in [40]. Sanders-Wang studied curve flows in R n whose curvatures are solutions of the vector mKdV in [41]. Terng-Thorbergsson constructed curve flows on Adjoint orbits of a compact Lie group G that relate to the n-wave equation associated with G in [42], Terng-Uhlenbeck explained the relation between the Schrödinger flow on compact Hermitian symmetric space and the Fordy-Kulish NLS system and wrote down a bi-Hamiltonian structure, geometric conservation laws, and commuting curve flows in [43] for the Schrödinger flows. Terng constructed Darboux transforms and explicit soliton solutions of the Airy curve flow in R n in [44]. Mari Beffa gave natural Poisson structures on semi-simple homogeneous spaces and discussed their relations to integrable curve flows in [45,46]. Readers are referred to these papers for more references. Drinfeld and Sokolov in [11] associated with each affine Kac-Moody algebraĜ a hierarchy of soliton equations of KdV type, which will be called theĜ-KdV hierarchy. It was proved in [11] that the KdV hierarchy is theÂ (1) 1 -KdV hierarchy and the Gelfand-Dickey hierarchy is theÂ (1) n−1 -KdV hierarchy. There are recent works on integrable curve flows on flat spaces whose differential invariants satisfy theĜ-KdV hierarchies. The first example was given by Pinkall, who in [47] constructed a hierarchy of central affine curve flows on R 2 invariant under the group SL(2, R) and showed that their differential invariant (the central affine curvature) satisfies the KdV hierarchy. Calini-Ivey-Mari Beffa in [48] (for n = 3) and Terng and Wu in [49] (for general n) constructed a hierarchy of curve flows on the affine space R n invariant under SL(n, R) whose differential invariants satisfy theÂ (1) n−1 -KdV hierarchies. Terng and Wu also constructed in [50] two hierarchies of curve flows on R n+1,n , whose differential invariants under the group O(n + 1, n) are solutions of theB (1) n -KdV andÂ (2) 2n -KdV hierarchies respectively. In this paper, we construct two hierarchies of curve flows on the symplectic space R 2n whose differential invariants under the symplectic group are solutions of thê C (1) n -KdV and theÂ (2) 2n−1 -KdV hierarchies respectively. We need to set up some more notations before we explain our results. Let R 2n be the symplectic space with the symplectic form Sp(2n) = {g ∈ GL(2n, R) | g t S n g = S n } the group of linear isomorphisms of R 2n that preserves w, and sp(2n) = {A ∈ sl(2n) | A t S n + S n A = 0} the Lie algebra of Sp(2n). A linear subspace V of R 2n is isotropic if ω(x, y) = 0 for all x, y ∈ V. A maximal isotropic subspace has dimension n, and is called Lagrangian. The action of Sp(2n) on the space of Lagrangian subspaces of R 2n defined by g · V = gV is transitive.
We show that if γ : R → R 2n is Lagrangian then there exists a unique orientation preserving parameter x = x(s) such that ω(γ (n) x , γ (n−1) x ) = (−1) n . We call such parameter the Lagrangian parameter for γ. Let We prove that given γ ∈ M 2n , there exists a unique g = (g 1 , . . . , g 2n ) : R → Sp(2n) such that g i = γ (i−1) x for 1 ≤ i ≤ n + 1 and We call this g the Lagrangian moving frame and u = ∑ n i=1 u i e n+1−i,n+i the Lagrangian curvature along γ.
It is easy to see that is in M 2n with Lagrangian frame g(x) = exp(bx) and zero Lagrangian curvature.

Definition 2. The Lagrangian curvature map
It follows from the theory of existence and uniqueness of solutions of ordinary differential equations that the Lagrangian curvatures form a complete set of differential invariants for curves in M 2n .
A Lagrangian curve flow is an evolution equation on M 2n , i.e., the flow preserves the Lagrangian parameter. Such flow can be written in the form γ t = gξ(u) so that gξ(u) is tangent to M 2n at γ, where g(·, t) and u(·, t) are the Lagrangian moving frame and Lagrangian curvature along γ(·, t) and ξ(u) is a R 2n×1 valued differential polynomial of u in x variable.
Please note that when n = 1, we have sp(2) = sl(2, R), ω(X, Y) = det(X, Y), the Lagrangian parameter, frame, curvature are the central affine parameter, frame, central affine curvature on R 2 under the group SL(2, R), and the Lagrangian curve flows on R 2 are the central affine curve flows studied in [47] (see also in [51,52]). For example, is a Lagrangian flow on R 2 and its Lagrangian curvature u satisfies the KdV, In this paper, we construct two hierarchies of Lagrangian curve flows on R 2n whose Lagrangian curvatures are solutions of theĈ (1) n -KdV andÂ (2) 2n−1 -KdV hierarchies respectively. In particular, we obtain the following results: (1) We construct a sequence of commuting Lagrangian curve flows of C-type and A-type respectively on M 2n such that the third flows are respectively, where u 1 is the first Lagrangian curvature. (2) The Lagrangian curvature map Ψ maps the space of solutions of Lagrangian curve flows of C-type (A-type resp.) modulo Sp(2n) bijectively onto the space of solutions ofĈ (1) n -KdV (Â (2) 2n−1 -KdV resp.) flows. For example, the Lagrangian curvatures u 1 , u 2 of a solution γ of (3) and (4) satisfy the thirdĈ and the thirdÂ respectively. (3) A bi-Hamiltonian structure and commuting conservation laws for Lagrangian curve flows of C-and A-types are given. For example, the curve flows (3) and (4) are Hamiltonian flows for functionalŝ respectively on M 2n with respect to the second Hamiltonian structure, where u is the Lagrangian curvature of γ. (4) We construct Darboux transforms (DTs), Permutability formulas, scaling transforms, and give an algorithm to compute explicit soliton solutions of these flows.
This paper is organized as follows: We construct Lagrangian moving frames in Section 2, and review the constructions of theĈ (1) n -KdV andÂ (2) 2n−1 -KdV hierarchies in Section 3. Lagrangian curve flows of Cand Atypes and the evolutions of their Lagrangian curvatures are given in Section 4. In Section 5, we construct Darboux transforms (DTs) and a Permutability formula for theĈ (1) n -KdV and for the Lagrangian curve flows of C-type. DTs for the A case and its Permutability formula are given in Section 6. The scaling transforms are given in Section 7. Bi-Hamiltonian structures and commuting conserved functionals are given in Section 8. We give an outline of a method for constructing integrable curve flows whose differential invariants satisfy theĜ (1) -KdV hierarchy for general simple real non-compact Lie algebra G and give some open problems in the last section.

Lagrangian Moving Frame
In this section, we prove the existence of Lagrangian parameter and construct the Lagrangian moving frame and curvatures for Lagrangian curves (cf. Definition 1).
We derive g i 's and u i 's by the recursive formula: Then g = (γ, . . . , γ (n) x , g n+2 , . . . , g 2n ) satisfies g −1 g x = b + u, i.e., g is a Lagrangian moving frame along γ. Example 1. For n = 1, we have ω(X, Y) = det(X, Y), thus γ ∈ M 2 if and only if det(γ, γ x ) = 1. So the Lagrangian parameter is the central affine parameter, the Lagrangian frame along γ is g = (γ, γ x ) is the central affine moving frame along γ, and the Lagrangian curvature is the central affine curvature. Moreover, It follows from the Existence and Uniqueness of ordinary differential equations that {u 1 , · · · , u n } forms a complete set of local differential invariants for γ ∈ M 2n under the Sp(2n)-action. So we have the following: Proposition 2. The Lagrangian curvature map Ψ : M 2n → C ∞ (R, V n ) defined by Definition 2 is onto and Ψ −1 (u) is a Sp(2n)-orbit.

Example 3.
A Lagrangian curve in R 2n with zero Lagrangian curvature is of the form:
A meromorphic map f : C → SL(2n, C) is said to satisfy the Sp(2n)-reality condition if For Let B + n and N + n denote the subgroups of upper, strictly upper triangular matrices in Sp(2n) respectively, and B + n , N + n the corresponding Lie subalgebras of sp(2n). Set It is easy to check that J 2j−1 is in (Ĉ (1) Next we use the general method given in [53] to construct theĈ (1) n -hierarchy generated by the vacuum sequence {J 2j−1 | j ≥ 1}. First a direct computation gives the following known results: 49,53]). Given q ∈ C ∞ (R, B + n ), then there exists a unique n satisfying [∂ x + J + q, P(q, λ)] = 0, P 2n (q, λ) = λI 2n .
Moreover, P 1,i (q) can be computed recursively by equating the coefficients of λ i in (10) and they are polynomials in u and x-derivatives of u (i.e., a differential polynomial in u).
Please note that if operators A, B commute, then A and B j also commute. Hence it follows from the first equation of (10) that we have Write the power series We compare coefficient of λ i of (11) to obtain which implies that the left hand side lies in B + n . So defines a flow on C ∞ (R, B + n ). We call (14) the (2j − 1)-thĈ (1) n -flow. We need the following well-known elementary result to explain the Lax pair: Proposition 3. Let G be the Lie algebra of G, and A, B : R 2 → G smooth maps. Then the following statements are equivalent: (1) the linear system g x = gA, g t = gB is solvable for g : R 2 → G, Proposition 4. The following statements are equivalent for smooth q : R 2 → B + n : (1) q is a solution of (14), (2) the following linear system is solvable for h : R 2 → Sp(2n), (3) the following linear system is solvable for F(x, t, λ) ∈ SL(2n, C), The last equation says that F(x, t, λ) satisfies the Sp(2n)-reality condition (7) in λ.
The group C ∞ (R, N + n ) acts on C ∞ (R, B + n ) by gauge transformation, for The following Proposition shows that C ∞ (R, V n ) is a cross-section of this gauge action.
Proposition is proved by equating components of G j of (20) for |j| ≤ 2n − 1.
It can be checked by the same method for theÂ (1) n -hierarchy (cf. [53]) that flow (14) is invariant under the C ∞ (R, N + n )-action. So given u ∈ C ∞ (R, V n ) and j ≥ 1, there exists a unique N + n -valued differential polynomial η j (u) satisfying The induced quotient flow of (14) on the cross-section C ∞ (R, V n ) is obtained by projecting (14) down along gauge orbits. So the induced quotient flow on The above equation is the (2j − 1)-thĈ (1) n -KdV flow. As a consequence of the construction, we have the following. Proposition 6. The following statements are equivalent for smooth u : (iii) The following linear system is solvable for g : R 2 → Sp(2n), (iv) The following linear system is solvable for E(x, t, λ) ∈ SL(2n, C) for all parameter λ ∈ C,

Proof. It was proved in [11] that given any
Let θ ∈ S 2n be the cyclic permutation defined by θ(1) = 2n, and θ(i Please note that where q = (q ij ) and k i are diagonal matrices defined by We compare the coefficients of J j 's of both sides of each equation in (26) and use (27) to solve h j uniquely as differential polynomial of q. This gives the formula for Q(q, λ). We plug in Formulas (8) and (9) to obtain Q 1,j (q)'s.
The first equation of (26) implies that Write Q 2j−1 (q, λ) as a power series in λ, We compare the coefficient of λ i of (28) to obtain where β is defined by (25). So the left hand side of (30) is B + n -valued and is a flow on C ∞ (R, B + n ). This is the (2j − 1)-th flow in theÂ (2) 2n−1 -hierarchy. We use the same proof of Proposition 4 to obtain the following: Proposition 7. The following statements are equivalent for smooth q : R 2 → B + n : (i) q is a solution of (31). (ii) The following linear system is solvable for smooth g : R 2 → Sp(2n), (iii) The following linear system is solvable for F(x, t, λ) ∈ SL(2n, C) for all parameter λ ∈ C, It follows from Proposition 5 that there exist a unique The (2j − 1)-thÂ (2) 2n−1 -KdV flow is the following flow on C ∞ (R, V n ): Proposition 8. The following statements are equivalent for smooth u : R 2 → V n : (i) u is a solution of (34).

Lagrangian Curve Flows on R 2n
In this section, we (i) give a description of the tangent space of M 2n at γ and show that it is isomorphic to C ∞ (R, R n ), (ii) construct two hierarchies of Lagrangian curve flows whose curvatures satisfy thê n -KdV and theÂ (2) 2n−1 -KdV flows respectively. Henceforth in this paper we set Theorem 4. Let g and u denote the Lagrangian frame and Lagrangian curvature along γ ∈ M 2n , and Ψ : M 2n → C ∞ (R, V n ) the Lagrangian curvature map. Then if and only if gCe 1 is tangent to M 2n at γ, (2) if ξ is tangent to M 2n at γ then there exists a unique smooth C : R → sp(2n) satisfying (37) such that ξ = gCe 1 .
Suppose C satisfies (37). Let η i denote the i-th column of gC. Please note that ξ is tangent to M 2n at γ if and only if To prove η 1 satisfies (38) (gC) t S n g + g t S n gC = C t g t S n g + g t S n gC = C t S n + S n C = 0, so η 1 satisfies (38).

Example 7. Lagrangian curve flows of C-type
γ ∈ M 2 if and only if γ satisfies det(γ, γ x ) = 1, and the Lagrangian parameter, moving frame, and curvature for γ ∈ M 2 are the central affine parameter, moving frame and curvature respectively. The thirdĈ The third Lagrangian curve flow of C-type on M 2 is which is the third central affine curve flow on the affine plane (cf. [47]). Moreover, if γ is a solution of (42), then its Lagrangian curvature is a solution of the KdV (41). (ii) Let g = (γ, γ x , γ xx , g 4 ) be the Lagrangian moving frame of γ ∈ M 4 , and u 1 , u 2 the Lagrangian curvatures as in Example 2. From Example 5, we see that the first column of So the third Lagrangian curve flow of C-type on M 4 is where g 4 is the fourth column of the Lagrangian frame of γ. This is the curve flow (3) for n = 2 because g 4 = γ xxx − u 1 γ x (given in Example 2). Similar computation implies that the first column of P 5,0 (u) is Hence the fifth Lagrangian curve flow of C-type on M 4 is (iii) We use Equation (10) to compute P 1,i (u) and the first column of P 3,0 (u) for general n.
Then we see that the third Lagrangian curve flow of C-type on M 2n for n ≥ 3 is (3).

Example 8. Lagrangian curve flows of A-type
We use the algorithm given in Theorem 3 to compute Q 1,i (u). Then we use these Q 1,i (u)'s to compute Q i,0 (u). Then we obtain the following: The fifth Lagrangian curve flow of A-type on M 4 is Theorem 4 (1) states that gξ is tangent to M 2n at γ if and only if there is a C satisfying (37) and ξ = Ce 1 . So to get a better description of the tangent space of M 2n at γ, we need to understand properties of C that satisfies (37).
n be the linear projection onto V t n defined by then we have the following: (iv) C i,j 's are differential polynomials of u, C 21 , · · · , C 2n,1 .
We prove (i) by induction.
Then by (46) and induction, C j (j < 0) are differential polynomials in u, v i and the linear system (46) implies (ii).
Please note that ad(b) : G 0 → G −1 is bijection, and [u, C] G −1 depends only in u, v 1 , · · · , v n . Hence C 0 can be solved uniquely from C i , i < 0. This proves (iii).

Corollary 2.
Given C 1 , C 2 : R → sp(2n) satisfying (37), then we have the following: (1) If the first columns of C 1 and C 2 are the same, then C 1 = C 2 .
It follows from Proposition 5 (i) that we have the following: Corollary 3. Given smooth u : R → V n and v : R → V t n , there exists a unique C : R → sp(2n) satisfying (45) and entries of C are polynomial differentials of u, v and linear in v.
The above Corollary leads us to define a natural linear differential operator P u defined below.
It follows from the definition of P u and Theorem 5 that we have the following: (37), then C = P u (π 0 (C)).

Remark 1.
We use the same proof as in [49] for the n-dimensional central affine curve flow to show that solutions of the Cauchy problem of (22) give solutions of the Cauchy problem for Lagrangian curve flow (39) with both rapidly decaying and periodic initial data. Similar results hold for the Lagrangian curve flows (34) and (40). (1) n -Hierarchy In this section, we use the loop group factorization method given in [54] to construct Darboux transformations for theĈ (1) n -,Ĉ

Darboux Transforms for theĈ
n -KdV, and the Lagrangian curve flows of Ctype. We also give a Permutability formula for these Darboux transforms. To use this method, we need to identify the loop groups, find simple rational elements, and write down formulas for the factorizations.
The following result was proved in [54] for soliton hierarchies constructed from a splitting of loop algebras. So it works for both theĈ (1) n -andÂ (2) 2n−1 -hierarchies given in Section 3.
Sincef (x, t, ·) andg(x, t, ·) are in RĈ (1) n ,gf is in RĈ (1) n . Please note that the coefficient of λ −1 ofgf isf −1 +g −1 . Hence it follows from Theorem 7 that we have Given a linear subspace V of R 2n , let Lemma 1. Let R 2n = V 1 ⊕ V 2 be a direct sum of linear subspaces, and π the projection of R 2n onto V 1 along V 2 . Then we have ω(πX, Y) = ω(X, π s Y), where Proof. Please note that where ω is the symplectic form defined by (1).
We use Lemma 1 and a direct computation to get:

Lemma 2.
(1) A linear subspace of R 2n is Lagrangian if and only if V ⊥ = V.
(2) Let π be a projection of R 2n . Then Imπ and Kerπ are Lagrangian subspaces, if and only if π s = I 2n − π.

Theorem 9 (Darboux transform for theĈ
is a new solution of (14) andF is a frame forq.
The following DTs for (22) is a consequence of Proposition 9 and Theorem 9.
As a consequence of Theorems 9 and 6 (iii), we have Theorem 12 (DT for Lagrangian curve flow of C-type).

Example 11. [1-soliton solutions of C-type]
First, we apply Theorem 9 to the trivial solution q = 0 of the thirdĈ is a frame of the solution q = 0 of the thirdĈ (1) 2 -flow. We use λ = z 4 to write down F(x, t, λ) in terms of known functions, (Although the entries of F(x, t, z 3 ) involves z i in the denominators, use power series expansion and a simple computation to see that they are holomorphic at z = 0).
Next we apply DTs for the thirdĈ (1) 2 -flow to the trivial solution q = 0 and z = 1. Let π be the projection onto V 1 along V 2 , where Thenπ is the projection ontoṼ 1 alongṼ 2 , wherẽ From a direct computation, we havẽ π(x, t) = p 1 ,p 2 , 0, 0 p 1 ,p 2 ,p 3 ,p 4 Applying (54), we can get a solution of the thirdĈ Using the algorithm in the proof of Proposition 5, we get a new solution of (5), u = * q =ũ 1 e 23 + u 2 e 14 , We use Theorem 12, and the formula forπ, and a direct computation to see that is a solution of the third Lagrangian curve flow of C-type on M 4 , where Next we give a Permutability formula for DTs of theĈ (1) n flows. The following Lemma follows from Lemma 4.
Then we have In particular, q 12 can be obtained algebraically fromπ 1 andπ 2 .
The Permutability Theorem 13 gives an algebraic formula for constructing k-solitons and their frames from k 1-solitons for theĈ (1) n -flow. IfF is a frame of the k-soliton solutioñ q ofĈ (1) n -flow, thenγ =F(x, t, 0)e 1 is a k-soliton solution of the Lagrangian curve flow of C-type and its Lagrangian curvatureũ is a k-soliton of theĈ (1) n -KdV flow. (2) 2n−1 -Hierarchy In this section, we construct Darboux transformations for theÂ (2) 2n−1 ,Â (2) 2n−1 -KdV, and the Lagrangian curve flows of A type. We also give a Permutability formula for these Darboux transforms.
Please note that the second condition of (59) is equivalent to where A s = S −1 n A t S n . Please note that the restriction of the symplectic form w to a linear subspace V of R 2n is non-degenerate if and only if R 2n = V ⊕ V ⊥ . Lemma 6. Let π be a projection. Then Ker(π) = (Im(π)) ⊥ if and only if π = π s .
Theorems 14 and 6 (iii) give the following:

Theorem 17 (DT for Lagrangian curve flows of A-type).
Let γ be a solution of the Lagrangian curve flow (40) of A-type, and g(·, t), u(·, t) the Lagrangian frame and Lagrangian curvature along γ(·, t). Let E be the frame of the solution u of (31) satisfying E(0, 0, λ) = g(0, 0). Let , α, π,π be as in Theorem 16. Theñ is a new solution of (40) and its Lagrangian curvatureũ is a solution of (31).

Example 12. 1-soliton solutions of A-type
Please note that u = 0 is the trivial solution of the thirdÂ (2) 2n−1 -flow with frame F(x, t, λ) = exp(xJ B (λ) + tJ 3 B (λ)). By Theorem 6 (iii), is the Lagrangian curve flow (39) with zero Lagrangian curvature and as its Lagrangian frame. Please note that the linear system B q,λ given by (63) for q = 0 is , the solution of B 0,λ for any given initial data can be written down explicitly. Hence Theorem 15 gives an algorithm to compute explicit formula for 1-solitonsq and its frame for the thirdÂ (2) 2n−1flow. Theorem 17 gives the corresponding 1-soliton solutionγ of the third Lagrangian curve flow of A-type and the Lagrangian curvatureũ ofγ is a 1-soliton solution of the thirdÂ (2) 2n−1 -KdV flow.
We use (65) and a direct computation to see that P x + [J + r q,P] = 0.
This shows thatP = P(r q, λ). A direct computation implies that It follows from Proposition 4 that r q is a solution of (14) andF is a frame of r q.
It follows from Theorem 19 (2) and Theorem 6 (iii) that we have the following: In particular, letγ be the solution of the third Lagrangian curve flow on M 4 constructed in Example 11. Then c γ is also a solution for all c ∈ R \ 0.
(2) If E(x, t, λ) is a frame of the solution u of (22), then is a frame of r u.
(3) If E(x, t, λ) is a frame of the solution u of (34), then is a frame of r u. Corollary 6. r u defines an action of the multiplicative group R + on the space of solutions of (22) ((34) resp.).

Bi-Hamiltonian Structure
The existence of a bi-Hamiltonian structure and using it to generate the hierarchy are two of the well-known properties for soliton hierarchies (cf. [11,55,56]). In this section, we use the linear operator P u defined in Definition 4 to write down the bi-Hamiltonian structure for theĈ (1) n -KdV andÂ (2) 2n−1 -KdV. The pull back of this bi-Hamiltonian structure to M 2n via the Lagrangian curvature map Ψ gives the bi-Hamiltonian structure for the Lagrangian curve flows of C and A-type. Let ξ, η = tr(ξη)dx denote the standard L 2 inner product on C ∞ (S 1 , sl(2n, R)).

Example 15. Conservation laws for theÂ
(1) For n = 2, we have (2) For general n, the first two densities of conservation laws are

Example 16. Hamiltonian flows for F 3 and H 3
A simple computation implies that ∇F 3 (u) = 1 4 u 1 e 32 + e 41 , where u = u 1 e 23 + u 2 e 14 . We use notations and formulas as in Example 9 to compute P u (∇F 3 (u)) and obtain x + u 2 , The Hamiltonian flow of F 3 with respect to { , } 2 is We use the formula for P u (∇F 3 (u)) to compute directly and see that (69) is the following system for u 1 , u 2 , Substitute C ij into the above equation to see that it is (5). Similarly, we use the same notations and formulas as in Example 9 to compute P u (∇H 3 (u)). Here ∇H 3 (u) = e 32 . We see thatbe So the Hamiltonian flow for H 3 with respect to { , } 2 written in terms of u 1 , u 2 is (6).

Review and Open Problems
In this section, we give an outline of the construction ofĜ (1) -KdV hierarchy (cf. [11,53]), explain the key steps needed in constructing curve flows whose differential invariants satisfy theĜ (1) -KdV, and give some open problems.
Let G be a non-compact, real simple Lie group, G its Lie algebra, and Then (Ĝ − ) is a splitting ofĜ (1) . Let {α 1 , . . . , α n } be a simple root system of G, and B + , B − , N + the Borel subalgebras of G of non-negative roots, non-positive roots, and positive roots respectively. Let B + , B − , N + be connected subgroups of G with Lie algebras B + , B − , N + respectively. Let where b = − ∑ n i=1 α i and β is the highest root.
The construction ofĈ (1) n -hierarchy in Section 3 works forĜ (1) except that the generating function P(q, λ) in Proposition 2 should satisfy [∂ x + b + q, S(q, λ)] = 0, m(S(q, λ)) = 0, where m is the minimal polynomial of J defined by (70). Assume that there is a sequence of increasing positive integers {n j | j ≥ 1} such that J n j lies inĜ (1) + for all j ≥ 1. Write S n j (q, λ) = ∑ i S n j ,i (q)λ i .
Then the n j -th flow in theĜ (1) -hierarchy is for q : R 2 → B + .
(iii) If we find a linear subspace V of G such that C ∞ (R, V) is a cross-section of the gauge action of C ∞ (R, N + ) on C ∞ (R, B + ). Then we can push down theĜ (1) -flows to the cross-section C ∞ (R, V) along gauge orbits and obtain aĜ (1) -KdV hierarchy on C ∞ (R, V). Moreover, there exists a polynomial differentials ξ j (u) such that the n j -th flow in theĜ (1) -KdV hierarchy is u t = [∂ x + b + u, S n j ,0 (u) − ξ j (u)]. (vi) F n j (q) = − (S n j ,−1 (q)β)dx is the Hamiltonian for the n j -th flow with respect to { , } ∧ 2 . Although properties (i)-(vi) can be proved in a unified way for anyĜ (1) , the following results need to be proved case by case depending on G: (1) Find a linear subspace V such that C ∞ (R, V) is a cross-section of the gauge action of C ∞ (R, N + ) on C ∞ (R, B + ). (2) Suppose G is a subalgebra of gl(n) and C ∞ (R, V) is a cross-section of the gauge action.
We consider the following class of curves in R n : for some u ∈ C ∞ (R, V)}.
Find geometric properties of curves in M that characterize γ ∈ M (so g is the moving frame and u is the differential invariant of γ under the group G). For example, for theĈ (1) n case, it is easy to see that if γ ∈ M, then γ is Lagrangian (see Definition 1). Conversely, if γ is Lagrangian then g ∈ M.
(3) Identify the tangent space of M at γ.
(4) Show that γ t = gS n j ,0 (u)e 1 (74) is a flow on M, i.e., the right hand side is tangent to M. (5) Show that if γ(x, t) is a solution of (74), then the differential invariants u(·, t) satisfies theĜ (1) -KdV flow (73). This also gives a natural interpretation of theĜ (1) -KdV. (6) Write down the formula for the induced bi-Hamiltonian structure for theĜ (1) -KdV hierarchy. (7) We pull back the bi-Hamiltonian structure on C ∞ (S 1 , V) to M via the curvature map Ψ : M → C ∞ (S 1 , V) defined by Ψ(γ) = u the differential invariant of γ. Then soliton properties ofĜ (1) -KdV can be also pulled back to the curve flows (74) on M. (8) Prove an analogue of Theorem 5, i.e., if C : R → G satisfies [∂ x + b + u, C] ∈ C ∞ (R, V), then (a) C is determined by Ce 1 , (b) C is determined by the projection of C onto V t , where u ∈ C ∞ (R, V). We need this result to give a precise description of the tangent space of M at γ and to write down the formula for the induced bi-Hamiltonian structure on C ∞ (R, V) for theĜ (1) -KdV hierarchy. (9) To construct Darboux transforms, we need to find rational maps g : R → G C satisfies g(λ) = g(λ) with minimal number of poles and work out the factorization formula explicitly.
Finally we give a list of open problems: Find integrable curve flows on R 2n,1 whose differential invariants satisfy thê B n -KdV flows. Find integrable curve flows on R k,2n−k whose differential invariants satisfy thê D (1) n -KdV flows.
Find integrable curve flows on R 2n whose differential invariants satisfy thê D (2) n -KdV flows. Find integrable curve flows on R 8 whose differential invariants satisfy thê D (3) 4 -KdV flows. Find integrable curve flows on R 7 whose differential invariants satisfy thê G (1) 2 -KdV flows. Calini and Ivey constructed finite gap solutions for the VFE in [57]. It would be interesting to construct finite-gap solutions for central affine curve flows, isotropic curve flows, and Lagrangian curve flows. The Gauss-Codazzi equations of submanifolds occurring in soliton theory are often given by the first level flows of the soliton hierarchy, i.e., the commuting flows generated by degree one (in λ) elements in the vacuum sequence. It would be interesting to see whether the flows of theĜ (1) -KdV hierarchy generated by degree one elements in the vacuum sequence also arise as the Gauss-Codazzi equations for some class of submanifolds.
Author Contributions: Both authors are equally responsible for all results in this paper. All authors have read and agreed to the published version of the manuscript.