数理解析研究所講究録
第 1929 巻 2014 年 100-111

100

TWO TYPICAL EXAMPLES ON INTEGRABLEGEOMETRY FLOWSZHIWEI WU

1. INTRODUCTION. An
acts onLet
be a manifold of dimension , and a groupis called a geometric curve
evolution equation for curves in a manifold
flow, if the equation can be written as$\mathcal{M}$

$G$

$n$

$\mathcal{M}$$\mathcal{M}$$\gamma_{t}=\xi_{0}\gamma+\sum_{i=1}^{n}\xi_{i}e_{i}.$Here $(e_{1}, e_{2}, \cdots, e_{n})\in G$ is a moving frame along under the group action
on . And all ’s are smooth functions of the invariants (or curvatures)
under the action of on . A specific case is the Frcnet frame, it is corre. Under the SO(3) action,
sponding to the transitive action of $SO(3)$ on
a generic curve can be determined up to a constant element in $SO(3)$ by
two functions and , which are called the curvature and torsion of respectively. Let
denote the Erenet frame of , then a geometriccurve flow in this case can be written as$\gamma$$G$$\mathcal{M}$$\xi_{i}$$G$$\gamma$$\mathbb{R}^{3}$$k$$\tau$$\gamma$$(e_{1}, e_{2}, e_{3})$$\gamma$$\gamma_{t}=\xi_{0}(k, \tau)\gamma+\sum_{i=1}^{3}\xi_{i}(k, \tau)e_{i}.$Diﬀerent group actions usually leads to diﬀerent type of moving frames.
be a parameterized curve such thatLet :, then $g=(\gamma, e_{2})\in SL(2, \mathbb{R})$ is a moving frame
never vanishes, set
for under the aﬃne action. Moreover, the structure equation of is$\triangle=\det(\gamma, \gamma_{x})$$\mathbb{R}arrow \mathbb{R}^{2}\backslash \{O\}$$\gamma$$e_{2}=2_{\frac{x}{\triangle}}$$g$$\gamma$$g_{x}=g(\begin{array}{ll}0 qr 0\end{array})$

form a set of diﬀerential invariants for
It can be proved that
underaction. Here there exist two invariants because we do not
specify the parameter. Consider a third ordered diﬀerential equation for :$\gamma\in \mathbb{R}^{2}\backslash \{O\}$$q,$ $r$$SL(2, \mathbb{R})$$\gamma$$\gamma_{t}=-\frac{1}{4}(q_{x}r-qr_{x})\gamma-\frac{1}{4}(2qr^{2}-r_{xx})\gamma_{x}.$From the previous setting, this is a geometric curve flow. Its solvability
condition is equivalent to a couple PDE system in aﬃne curvatures:$\{\begin{array}{l}q_{t}=\frac{1}{4}(q_{xxx}-6qrq_{x}) ,r_{t}=\frac{1}{4}(r_{xxx}-6qrr_{x}) .\end{array}$101
This is the third flow in a soliton hierarchy-the AKNS hierarchy [1].
Fixing the parameter such that, then there is a natural
moving framewith only one invariant such that
. Pinkallconsider the following geometric curve flow [15]:$\det(\gamma, \gamma_{x})\equiv 1$$(\gamma, \gamma_{x})$$\gamma_{xx}=q\gamma$$q$$\gamma_{t}=\frac{1}{4}q_{x}\gamma-\frac{1}{2}q\gamma_{x}$It is solvable if and only if

$q$

(1.1)

.

is a solution of the $KdV$ equation:$q_{t}= \frac{1}{4}(q_{xxx}-6qq_{x})$

,

(1.2)

This article is based on the talk I gave in RIMS workshop on “Development of group actions and submanifold theory” (06/25/2014-06/27/2014).
In this article, we will focus on two examples connecting integrable systems
and geometric curve flows. One of the equation is the nonlinear Schr\"odinger
equation (NLS), and the other one is the $KdV$ equation. They are twoclassical equations in the soliton theory literature.

2. NONLINEAR SCHR\"oDINGER EQUATION AND VORTEX FILAMENTEQUATION:Let
be a family of curves, and $\{e_{1}(\cdot, t), e_{2} t), e_{3} t)\}$the Frenet frame of). Lett) andt) be the corresponding curvature and torsion. We say an evolution equation for
is a geometricflow if it can be written as$\mathbb{R}^{2}arrow \mathbb{R}^{3}$$\gamma(x, t)$

$k$

$t$

$\gamma$$\tau$$\gamma(x, t)$$\gamma_{t}=F_{0}(k, \tau)\gamma+\sum_{i=1}^{3}F_{i}(k, \tau)e_{i}$

(2.1)

,

where $\{F_{i}|0\leq i\leq 3\}$ are diﬀerential polynomials in
to arc-length parameter.

$k$

and

$\tau$

with respect

Remark 2.1. Given any curve :such thatfor
The Frenet frame andare rational functions ofisalso called a diﬀerential invariants for the curve. Therefore, (2.1) is a partial
diﬀerential equation for curves in$\mathbb{R}arrow \mathbb{R}^{3}$$||\gamma_{x}||>0$$\gamma$$k,$$\tau$$\gamma,$$\partial_{x}\gamma,$$\partial_{xx}\gamma.$$\forall x\in \mathbb{R}.$$\{k, \tau\}$$\mathbb{R}^{3}.$Da Rios modeled the movement of a thin tube in viscous fluid as anevolution equation for curve in:$\mathbb{R}^{3}$$\gamma_{t}=\gamma_{x}\cross\gamma_{xx}$

.

(2.2)

This equation is called Vortex Filament equation (VFE).
One of the important properties of VFE is that it is arc-length preserving.
That is, if
is a solution of VFE and) is parameterized by arclength, so ist) for all . Therefore, (2.2) can be rewritten in terms of
the Frenet frame,),$\gamma(x, t)$$0$$\gamma$$t$$\gamma$$\{e_{1}(\cdot, t)$$e_{2}$$t$$e_{3}$$t$$\gamma_{t}=\gamma_{x}\cross\gamma_{xx}=e_{1}\cross ke_{2}=ke_{3}.$

102
with
In other words, the curve evolves along the bi-normal direction(2.2)
problemfortheinitialvalue
curvature as the speed. If we consider
is a circle, then under the flow it will evolve as a smoke ring.
such thatThat is why sometime this equation is also called the smoke ring equation.
In 1972, Hasimoto pointed out the connection between the VFE and theNLS:$e_{3}$$k$$\gamma(x, 0)$Theorem 2.2. ([11]) If$k(x, t)$and torsion$\tau(x, t)$$\gamma(x, t)$

is a solution to $VFE(2.2)$ with curvature, then there exists a function $c(t)$ of such that$t$$q(x, t)=k(x, t)e^{i(\int_{0}^{x}\tau(s,t)ds+c(t))}$is a solution of the nonlinear Schr\"odinger equation:$q_{t}=i(q_{xx}+ \frac{1}{2}|q|^{2}q)$.

Remark 2.3. The formula $q(x, t)=k(x, t)e^{i(\int_{0}^{x}\tau(s,t)ds+c(t))}$ is called theHasimoto transformation.

Hasimoto transformation provides a explicit formula from solutions of
VFE to NLS. But on the other hand, given a solution of the NLS, it is notclear from the formula that how to construct a curve flow solution for the
VFE. Therefore, we need to explore more on the NLS.
From integrable system point of view, NLS is equivalent to the following$su(2)$ -value flat connection 1-from (Lax pair):$\theta_{\lambda}=(a\lambda+u)dx+Q(u, \lambda)dt$

where $a=diag(i, -i)$ ,$u=(\begin{array}{ll}0 q-\overline{q}0 \end{array}),$$q\in C^{\infty}(\mathbb{R}, \mathbb{C})$

,

(2.3)

, and

$Q(u, \lambda)=(_{-i\overline{q}\lambda+\frac{iq}{2}}^{i\lambda^{2}-\frac{i}{2}|_{\frac{1}{q}}^{2}}x -i\lambda^{2}+\frac{i}{2}|q|^{2)}q\lambda+\frac{i}{2}q_{x}$Note that if we carry out the computation, the NLS is of the form:$q_{t}=i(q_{xx}+|q|^{2}q)$

.

(2.4)

We will see the reason using this equation in the following part of this section.
We call $E(x, t, \lambda)\in SU(2)$ an extended frame of if $E(x, t, \lambda)$ is a parallel
frame of . That is, $E(x, t, \lambda)$ satisfies the following equations:$q$$\theta_{\lambda}$$\{\begin{array}{l}E(x, t, \lambda)_{x}=E(x, t, \lambda)(a\lambda+u) ,E(x, t, \lambda)_{t}=E(x, t, \lambda)Q(u, \lambda) .\end{array}$

Associate $su(2)$ with the following inner product: X, $Y \rangle=-\frac{1}{2}Tr(XY)$ ,and consider the isomorphism from $su(2)$ to : let$\langle$$\mathbb{R}^{3}$$a=(\begin{array}{ll}i 00 -i\end{array}), b=(\begin{array}{ll}0 -i-i 0\end{array}), c=(\begin{array}{ll}0 1-1 1\end{array})$103
Then under the inner product, $(a, b, c)$ forms an ordered orthonormal basis
for $su(2)$ . Moreover, we note that$[a, b]=2c, [b, c]=2a, [c, a]=2b.$$H_{(^{s}I1}c^{\backslash },ew(^{\Delta}$

can nlake the following identification between$\mathbb{R}^{3}$

and $su(2)$ :$e_{1}=(1,0,0)^{t}\mapsto a, , e_{2}=(0,1,0)^{t}\mapsto b, e_{3}=(0,0,1)^{t}\mapsto c,$

and$x \cross yarrow\frac{1}{2}[X, Y], x, y\in \mathbb{R}^{3}, X, Y\in su(2)$.

Pohlmeyer-Sym construction (cf. [14], [17]) presents a converse version of
the Hasimoto transformation:
Theorem 2.4. [Sym’s formula]
Let be a solution of the $NLS$ equation (2.4) and $E(x, t, \lambda)$ is an extended
is a solution of $VFE$ of theframe for . Then$q$$\gamma(x, t)=\frac{\partial E}{\partial\lambda}E^{-1}(x, t, \lambda)|_{\lambda=0}$

$q$

form

$\gamma_{t}=\frac{1}{2}\gamma_{x}\cross\gamma_{xx}$

.

(2.5)

Moreover,$ifFisa$ nother$rameforq,then \tilde{\gamma}=\frac{\partialfF}{0\partial\lambda}F^{-1}(x_{\mathfrak{e}},t,\lambda)|_{\lambda=0}isalsoas$
olution o$f(2.5).AndthereexistsA_{0}\in O(3),$constant, $suchthat$$b\in \mathbb{R}^{3}$$\tilde{\gamma}=A_{0}\gamma+b_{0}.$Proof. This proof is by direct computation. Let $g(x, t)=E(x, t, 0)$ , then$\gamma_{t}=(\frac{\partial E}{\partial\lambda}E^{-1}(x, t, \lambda)|_{\lambda=0})_{t}=(\frac{\partial}{\partial_{t}}(\frac{\partial E}{\partial\lambda})E^{-1})|_{\lambda=0-}\frac{\partial E}{\partial\lambda}E^{-1}E_{t}E^{-1}|_{\lambda=0}$$=gug^{-1}$

Similarly,$\gamma_{x}=gag^{-1}, \gamma_{xx}=g[u, a]g^{-1}$

Therefore, from the isometric between $su(2)$ and$\mathbb{R}^{3}$, we have$\gamma_{x}\cross\gamma_{xx}=\frac{1}{2}9[a, [u, a]]_{9^{-1}}=2gug^{-1}=2\gamma_{t}.$This proves the theorem.

$\square$$-1$
Note tthenwhere $g=$$=gag$$E(x,t,0)isthev$alue
extended frame a $t\lambda=0Itisa$ unit vector in. Therefore, is the arc-length parameter. From direct computation,$hatif\gamma=\frac{\partial E}{\partial\lambda,of}E^{-1}(x,t, \lambda)|_{\lambda=0},$$\gamma_{x}.$$\mathbb{R}^{3}$$x$$(\gamma_{x})_{x}=g[g^{-1}g_{x}, a]g^{-1}=2{\rm Re}(q)b+2{\rm Im}(q)c$

In other words, $(gag^{-1}, gbg^{-1}, gc9^{-1})$ is a parallel frame along with principle curvatures $k_{1}=2Re(q)$ and $k_{2}=2{\rm Im}(q)$ . Therefore, if is a solutionof, and
its principle curvatures, thenis a solution of the NLS equation:$\gamma$$\gamma$$\gamma_{t}=\frac{1}{2}\gamma_{x}\cross\gamma_{xx}$$k_{1},$$k_{2}$$q_{t}=i(q_{xx}+|q|^{2}q)$$q$.

$:= \frac{1}{2}(k_{1}+k_{2}i)$

(2.6)

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3. CENTRAL AFFINE PLANE CURVES AND THE $KDV$ EQUATIONIn this section, we discuss the geometric explanation of the $KdV$ equation:$q_{t}= \frac{1}{4}(q_{xxx}-6qq_{x})$

(3.1)

.

In this part, we will give a brief scheme of constructing commutative geometric curve flows from the $KdV$ hierarchy. More details can be found in[15], [20].
As we mentioned before, the moving frame depends on the group action
on the space. On the plane, instead of the rigid motion, we consider thesuch that $A\cdot y=Ay$ for $A\in SL(2, \mathbb{R})$onaﬃne action of
belongs. Then the moving frame of a curveand.
, and the invariant set
to
, if
in
never vanishes, we canGiven a smooth curve
such that $\det(\gamma, \gamma_{x})=1.$
change the parameter toSuch parameter is called the central aﬃne arc-length parameter for . TakeO.
the derivative of $\det(\gamma, \gamma_{x})=1$ with respect to , thenHence there exists a unique smooth function such that
$SL(2, \mathbb{R})$$\mathbb{R}^{2}\backslash \{O\}$$y\in \mathbb{R}^{2}\backslash \{0\}$$\gamma\in \mathbb{R}^{2}\backslash \{O\}$$g$$g^{-1}g_{x}\in sl(2, \mathbb{R})$$SL(2, \mathbb{R})$

$x$

$s$

$\det(\gamma, \gamma_{S})$$\mathbb{R}^{2}\backslash \{0\}$$\gamma(s)$$( \frac{ds}{dx}=\det(\gamma, \gamma_{s})^{-1})$$\gamma$$\det(\gamma, \gamma_{xx})=$

$x$

$q$

$\gamma_{xx}=q\gamma.$

This is called the central aﬃne curvature of . ltom the uniqueness of
is a complete set of local invariants ofordinary diﬀerential equations,curves inunder th aﬃne action.
$I=S^{1}$Letor , denote$q$$\gamma$$\{q\}$$\mathbb{R}^{2}\backslash \{O\}$$(s$$\mathbb{R}$$\mathcal{M}_{2}(I)=\{\gamma :Then the tangent space of

Iarrow \mathbb{R}^{2}|\det(\gamma, \gamma_{x})=1\}.$$\mathcal{M}_{2}(I)$at$\gamma$

is of the form:$T_{\gamma} \mathcal{M}_{2}(I)=\{\tilde{\xi}=-\frac{\xi_{x}}{2}\gamma+\xi\gamma_{x}|\xi\in C^{\infty}(I, \mathbb{R}$An equation on$\mathcal{M}_{2}(I)$is called a central afine curve flow if$\gamma_{x}=-\frac{\xi_{x}}{2}\gamma+\xi\gamma_{x},$

where

$\xi$

$\mathcal{M}_{2}(\mathbb{R})$is a diﬀerential polynomial of . In [15], Pinkall show that if
is a solution of the following equation:$q$$\gamma_{t}=\frac{1}{4}q_{x}\gamma-\frac{1}{2}q\gamma_{x}$then its central aﬃne curvature$q$$\gamma\in$

(3.2)

,

is a solution of the $KdV$ equation:$q_{t}= \frac{1}{4}(q_{xxx}-6qq_{x})$

.

(3.3)

.
such that
action on
There is a natural$H(q)=\frac{1}{2}\ointqdx$
with respect toThen this equation is the Hamiltonian for$S^{1}$$\mathcal{M}_{2}(S^{1})$$e^{i\theta}\cdot\gamma(x)=\gamma(x+\theta)$

105
the following symplectic form on$\mathcal{M}_{2}(S^{1})/S^{1}$:$\omega_{\gamma}(\tilde{\xi},\tilde{\eta})=\oint\det(\tilde{\xi},\tilde{\eta})dx=-\oint\xi_{x}\eta dx.$

Periodic solutions in -part of (3.2) with finite-gap central aﬃne curvature
is studied in [3]. Higher-order central aﬃne curve flows, conservation law
and bi-Hamiltonian structure are given in [4, 5, 8, 9].
As in the previous section, it is natural to ask whether we can constructcentral aﬀne curve flows from the knowledge of $KdV$ equation. In thesoliton theory literature, Lie algebra splitting theory is a powerful method
to generate soliton equations (cf. [1], [7], [12], [16], [19] and etc.)$x$

I. There are several ways to derive the $KdV$ equation. First we take a lookof the construction from certain constrain on the $SL(2)$ -hierarchy (or the
ANKS hierarchy [1]).
Let $L(SL(2))$ be the group of smooth loops on $SL(2)$ andits Lie
algebra. Consider the following splitting of:$2\cross 2$$\mathcal{L}(sl(2))$$\mathcal{L}(sl(2))$(3.4)$\{\begin{array}{l}\mathcal{L}(sl(2))_{+}=\{\sum_{i\geq 0}A_{i}\lambda^{i}|A_{i}\in sl(2)\},\mathcal{L}(sl(2))_{-}=\{\sum_{i<0}A_{i}\lambda^{i}|A_{i}\in sl(2)\}.\end{array}$

Let $a=diag(1, -1)$ , given$u=(\begin{array}{ll}0 qr 0\end{array})$, set $Q(u, z)=z\lambda+Q_{0}+Q_{-1}z^{-1}+\cdots,$then we can solve $Q(u, z)$ uniquely from the following condition:(3.5)$\{\begin{array}{l}[\partial_{x}+az+u, Q(u, z)]=0,Q^{2}=z^{2}.\end{array}$

The j-th flow in the $SL(2)$ -hierarchy is a coupled system for

and :
$r$

$q$

$u_{t}=[\partial_{x}+az+u, (Q(u, z)z^{j-1})_{+}]=[\partial_{x}+u, Q_{1-j}]$

.

(3.6)

For example, the first several terms of $Q(u, z)$ is$Q_{0}=u= (\begin{array}{ll}0 qr 0\end{array}), Q_{-1}=\frac{1}{2}(\begin{array}{ll}-qr -q_{x}r_{x} qr\end{array}),$$Q_{-2}= \frac{1}{4}(\begin{array}{ll}q_{x}r-qr_{x} -2q^{2}rq_{xx}-r_{xx}2qr^{2} qr_{x}-q_{x}r\end{array})$

We write down the first three flows:$q_{t_{1}}=q_{x}, r_{t_{1}}=r_{x},$$q_{t_{2}}= \frac{1}{2}(q_{xx}-2q^{2}r) , r_{t_{2}}=-\frac{1}{2}(r_{xx}-2q^{2}r)$,$q_{t_{3}}= \frac{1}{4}(q_{xxx}-6qq_{x}r) , r_{t_{3}}=\frac{1}{4}(r_{xxx}-6qrr_{x})$.

Proposition 3.1. The third flow admits the constraints $r=1$ , which gives
the $KdV$ equation.

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Lax pair of the $KdV$ equation is the following $sl(2)$ -value connection 1from:$\theta_{z}=(\begin{array}{ll}z q1 -z\end{array})dx+$$(^{z_{Z-\frac{+1}{2}q}^{3_{-\frac{1}{@}qz\frac{1}{4}q_{x^{Z}}}}}$$qz^{2}- \frac{1}{\S}q_{x}z+\frac{1}{4}q_{xx}-\frac{1}{2}q^{2}-z+\frac{1}{2}qz-\frac{1}{4}q_{x}z)dt.$

II. To construct the B\"acklund transformation, it is more eﬃcient to consider the following splitting which is gauge equivalent to the $SL(2)$ -hierarchy.
Families of both pure soliton and rational solutions for central aﬃne curvatures and solution of (3.2) are constructed from this splitting [18], [20].
Define$\phi(z)=(\begin{array}{ll}1 z0 1\end{array})$. Let$\mathcal{L}=\{A(z)\in \mathcal{L}(sl(2))|\phi(z)^{-1}A(z)\phi(z)=\phi(-z)^{-1}A(-z)\phi(-z)\}.$

Consider the following splitting of :$\mathcal{L}$$\mathcal{L}_{+}=\{\sum_{i\geq 0}A_{i}z^{i}\}, \mathcal{L}_{-}=\{\sum_{i<0}A_{i}z^{i}\}.$

Then the Lax pair is gauge equivalent to$\hat{\theta}_{z}=\phi^{-1}(z)\theta_{z}\phi(z)$$= (\begin{array}{ll}0 z^{2}+q1 0\end{array})dx+ (_{z^{2^{4}}-q}L^{x}2 z^{4q}+_{2}z^{2}+\frac{q_{xx}}{4}-L^{2}-L^{x}42)dt.$

III. In the end, we consider another algebraic structure of the $KdV$ equationfrom the Drifeld-Sokolov construction [7]. From this construction, the curve
flows will rise naturally.
Let :defined as$sl(2, \mathbb{R})arrow \mathbb{R}e_{12}$$\mathcal{B}$$\mathcal{B}((\begin{array}{ll}a bc -a\end{array}))=(\begin{array}{ll}0 0c 0\end{array})$

Consider the splitting$\mathcal{L}(sl(2))$$(\tilde{\mathcal{L}}_{+},\tilde{\mathcal{L}}_{-})$

of$\mathcal{L}(sl(2))$

such that for$A( \lambda)=\sum_{i}A_{i}\lambda^{i}\in$,$A( \lambda)_{+}=\sum_{i>0}A_{i}\lambda^{i}+A_{0}-\mathcal{B}(A_{-1})$

Let $J=e_{12}\lambda+e_{21}$ , given a smooth function
there exists unique

$q$

:

.

(3.7)

$\mathbb{R}arrow \mathbb{R}$$Q(q, \lambda)=e_{12}\lambda+\sum_{i\geq 0}Q_{i}(q)\lambda^{-i}$, let$u=(\begin{array}{ll}0 q0 0\end{array}),$such that(3.8)$\{\begin{array}{l}[\partial_{x}+J+u, Q(q, \lambda)]=0,Q(q, \lambda)^{2}=\lambda I_{2}.\end{array}$

The $(2j+1)$ -th flow in the $KdV$ hierarchy is$u_{t_{2j+1}}=[\partial_{x}+J+u, (Q(q, \lambda))_{+}^{2j+1}]$

.

(3.9)

107
In fact, set$Q_{j}(q)=(_{C_{j}}^{A_{j}}$$-A_{j}B_{j)}$, then the previous recursive formula in-duces:$C_{j+1}(q)=-((A_{j}(q))_{x}+qC_{j}(q)-B_{j}(q))$ ,$A_{j+1}(q)= \frac{1}{2}((B_{j}(q))_{x}-qA_{j}(q)$$A_{j}(q)=- \frac{1}{2}(C_{j}(q))_{x}$,.

(3.10)

Then written in , the $2j+1$ -th flow (3.9) is$q$$q_{t_{2j+1}}=(B_{j}(q)-C_{j+1}(q))_{x}-2qA_{j}(q)$.

In particular, the first, third, and fifth flows are:$q_{t_{1}}=q_{x}$(3.11),$q_{t_{3}}= \frac{1}{4}(q_{xxx}-6qq_{x})$(3.12),$q_{t_{5}}= \frac{1}{16}(\partial_{x}^{5}q-10q\partial_{x}^{3}q-20(\partial_{x}q)\partial_{xx}q+30q^{2}\partial_{x}q)$.

(3.13)

Note that (3.12) is the $KdV$ equation.
Now we can write down Commuting higher order central aﬃne curve flowsfor (3.2). Let $E(x, t, \lambda)\in SL(2, \mathbb{R})$ be an extended frame for the $2j+1$ -thflow in the $KdV$ hierarchy, that is(3.14)$\{\begin{array}{l}E_{x}(x, t, \lambda)=E(x, t, \lambda)[Matrix],E_{t}(x, t, \lambda)=E(x, t, \lambda)(e_{12}\lambda^{j+1}+\cdots+Q_{j}(q)) .\end{array}$

Let $g(x, t)=E(x, t, 0)$ , then from the first equation of (3.14), $g(x, t)=$, and.
if andis a tangent vector field on
Recall thatisonly if. So it follows from (3.10) thattangent toat and$(\gamma, \gamma_{x})$$\gamma\in \mathcal{M}_{2}(I)$$\tilde{\xi}(\gamma)=\xi_{1}\gamma+\xi_{2}\gamma_{x}$$\mathcal{M}_{2}(I)$$A_{j}(q)\gamma+C_{j}(q)\gamma_{x}$$\xi_{1}=-\frac{1}{2}(\xi_{2})_{x}$$\mathcal{M}_{2}(I)$$\gamma$(3.15)$\gamma_{t_{2j+1}}=A_{j}(q)\gamma+C_{j}(q)\gamma_{x}=-\frac{1}{2}(C_{j}(q))_{x}\gamma+C_{j}(q)\gamma_{x}$is a central aﬃne curve flow on$(_{C_{j}(q)}^{A_{j}(q)}$$-A_{j}(q)B_{j}(q))$$\mathcal{M}_{2}(I)$is the coeﬃcient ofof order $2j+1$ , where $Q_{j}(q)=$$\lambda^{-j}$of the solution

We call this the $(2j+1)$ -th central aﬃne curve flow on$Q(q, \lambda)$$\mathcal{M}_{2}(I)$.

of (3.8).

108
For example, the first, third, and fifth central aﬃne curve flow on$\mathcal{M}_{2}(I)$is$\gamma_{t_{1}}=\gamma_{x},$$\gamma_{t_{3}}=\frac{1}{4}q_{x}\gamma-\frac{1}{2}q\gamma_{x},$$\gamma_{t_{5}}=\frac{1}{16}(q_{xxx}-6qq_{x})\gamma+\frac{1}{8}(3q^{2}-q_{xx})\gamma_{x}.$Note that the third central aﬃne curve flow is the curve flow (3.2).

The concept of central aﬃne curve can be generalized to higher dimen:
sional case naturally. If
is a smooth curve such that, then there exists a parameter $x=x(s)$ unique up
to translation such that$\gamma$$\mathbb{R}arrow \mathbb{R}^{n}\backslash \{O\}$$\det(\gamma, \gamma_{s}, \ldots, \gamma_{S}^{(n-1)})>0$$\det(\gamma, \gamma_{x}, \ldots, \gamma_{x}^{(n-1)})\equiv 1.$

Take -derivative of the above equation to see that$x$$\det(\gamma, \gamma_{x}, \ldots, \gamma_{x}^{(n-2)}, \gamma_{x}^{(n)})\equiv 0.$

Therefore, $\gamma_{x}^{(n)}=u_{1}\gamma+u_{2}\gamma_{x}+\cdots+u_{n-1}\gamma_{x}^{(n-2)}$ , for some smooth func. Such parameter is called the central aﬃne arc-lengthtions
parameter, and , . . ., $u_{n-1}$ are the central afine curvatures. The frame
is called the central aﬃne moving frame along (cf.
[4], [21]). Moreover,$u_{1},$$x$$\cdots,$ $u_{n-1}$$u_{1}$$g=(\gamma, \gamma_{x}, \ldots, \gamma_{x}^{(n-1)})$$\gamma$$9^{-1}g_{x}=b+u, b= \sum_{i=1}^{n-1}e_{i+1,i}, u=\sum_{i=1}^{n-1}u_{i}e_{in}.$Let$\mathcal{M}_{n}(\mathbb{R})=\{\gamma\in C^{\infty}(\mathbb{R}, \mathbb{R}^{n}\backslash \{0\})|\det(\gamma, \gamma_{x}, \ldots, \gamma_{x}^{(n-1)})=1\}.$

Define :curvature map by$\Psi$$\mathcal{M}_{n}(\mathbb{R})arrow C^{\infty}(\mathbb{R}, V_{n})$,$V_{n}=\oplus_{i=1}^{n-1}\mathbb{R}e_{in}$, to be the central aﬃne$\Psi(\gamma)=u=\sum_{i=1}^{n-1}u_{i}e_{in},$where , . . . ,are the central aﬃne curvatures ofIn [21], we construct a sequence of commuting higher order central aﬃnecurves on
such that the second flow is$u_{1}$$u_{n-1}$$\gamma.$$\mathcal{M}_{n}(\mathbb{R})$$\gamma_{t}=-\frac{2}{n}u_{n-1}\gamma+\gamma_{xx}.$

Thc,sc equations are all integrable, in the sense that under the central aﬃne
$u= \sum_{i=1}^{n-1}u_{i}e_{in}$ satisfies equations belonging
curvature map
to theGelfand-Dickey $(GD_{n})$ hierarchy (or the $A_{n-1}^{(1)}-KdV$ hierarchy) [6]. From
the Cauchy problems for the $GD_{n}$ -hierarchy with rapidly decaying initial
data and periodic initial data(cf. [2], [10], [13]), we can solve the Cauchy$\Psi,$

109
problem for the central aﬀne curve flows with periodic initial data and with
rapidly decaying initial data (i.e., the central aﬃne curvatures are rapidly
decaying).
Ftlrthermore, these flows are all integrable, we obtain a bi-Hamiltonian
for theon
structure and a sequence of Poisson structures
central aﬃne curve flow hierarchy. We prove that these Poisson structuresarise naturally from the Poisson structures of certain co-adjoint orbits.
, where ’s
Letbe the algebra of pseudo-diﬀerential operators:ofsplittingfollowingthe.
Considerare smooth functions on$\{,$$\mathcal{M}_{n}(S^{1})$$\}_{j}^{\wedge}$$\sum_{i}f_{i}\partial_{x}^{i}$$\mathcal{D}$$f_{i}$$\mathcal{D}$$\mathbb{R}$$\mathcal{D}_{+}=\{\sum_{i\geq 0}f_{i}\partial_{x}^{i}\}, \mathcal{D}_{-}=\{\sum_{i<0}f_{i}\partial_{x}^{i}\}.$

The $GD_{n}$ -hierarchy is generated by n-th order diﬀerential operator:$L= \partial^{n}-\sum_{i=1}^{n-1}u_{i}\partial_{x}^{i-1},$

and the j-th $(j\neq 0mod (n))$ flow is a PDE system for$u_{i}’ s$:$L_{t_{j}}=[L_{+}^{i_{\vee}}n, L].$For example, the second flow in the $GD_{3^{-}}KdV$ hierarchy is(3.16)$\{\begin{array}{l}(u_{1})_{t}=(u_{1})_{xx}-\frac{2}{3}(u_{2})_{xxx}+\frac{2}{3}u2(u_{2})_{x},(u2)_{t}=-(u_{2})_{xx}+2(u_{1})_{x}.\end{array}$

The second flow in the $GD_{4^{-}}KdV$ hierarchy is the following system for$u=u_{1}e_{14}+u_{2}e_{24}+u_{3}e_{34}$ :(3.17)$\{\begin{array}{l}(u_{1})_{t}=(u_{1})_{xx}-\frac{1}{2}u_{3}^{(4)}+\frac{1}{2}u_{3}(u3)_{xx}+\frac{1}{2}u2(u_{3})_{x},(u_{2})_{t}=2(u_{1})_{x}+(u_{2})_{xx}-2(u_{3})_{xxx}+u_{3}(u_{3})_{x},(u_{3})_{t}=u_{1}+2(u_{2})_{x}-2(u_{3})_{xx}.\end{array}$

And the third flow is$(u_{1})_{t}= \frac{3}{8}(u_{3}^{(5)}-2u_{2}^{(4)}-u_{3}^{(3)}u_{3}+2u_{2}^{(2)}u_{3}-u_{3}^{(2)}u_{2}+2u_{2}’u_{2}-2u_{1}’u_{3})+u_{1}^{(3)},$$(u_{2})_{t}= \frac{3}{4}u_{3}^{(4)}-2u_{2}^{(3)}+2u_{1}^{(2)}+u_{1}u3+\frac{3}{4}(u_{2}u_{3})_{x},$1(3)$(u_{3})_{t}=\overline{4}^{u_{3}}$3(2) $+u_{1}’+ \frac{3}{4}u_{3}’u_{3}.$$–u_{2}2$

In [21], we give a systematic method to construct higher ordered com. Here we give some examples.
muting central aﬃne curve flows on$\mathcal{M}_{n}(I)$

Example 3.2. [Higher order central aﬃne curve flows]is the flow:(1) For $n\neq 3$ , the third central aﬃne curve flow on$\mathcal{M}_{n}(\mathbb{R})$$\gamma_{t}=(-\frac{3}{n}u_{n-2}+\frac{3(n-3)}{2n}(u_{n-1})_{x})\gamma-\frac{3}{n}u_{n-1}\gamma_{x}+\gamma_{xxx}$.

(3.18)

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When $n=2$ , this is the Pinkall’s central aﬃne curve flow on$\mathbb{R}^{2}\backslash \{0\},$$\gamma_{t}=\frac{1}{4}q_{x}\gamma-\frac{1}{2}q\gamma_{x}.$So (3.18) is a natural analogue of Pinkall’s flow in -dimension $(n\neq 3)$ .
When $n=4$ , (3.18) is$n$$\gamma_{t}=(\frac{3}{8}(u_{3})_{x}-\frac{3}{4}u_{2})\gamma-\frac{3}{4}u_{3}\gamma_{x}+\gamma_{xxx}.$(2) The fourth and the fifth central aﬃne curve flows on$\mathcal{M}_{3}(\mathbb{R})$are$\gamma_{t}=-\frac{1}{9}(2u_{2}"-3u_{1}’-2u_{2}^{2})\gamma+\frac{1}{3}(u_{2}’-u_{1})\gamma_{x}-\frac{u_{2}}{3}\gamma_{xx},$$\gamma_{t}=\frac{1}{9}(-u_{1}"+u_{1}u_{2})\gamma-\frac{1}{9}(u_{2}"-3u_{1}’+u_{2}^{2})\gamma_{x}+\frac{1}{3}(u_{2}’-2u_{1})\gamma_{xx}.$

Acknowledgment.
The author would like to thank Professor Takashi SAKAI from Tokyo Metropolitan University for the kind invitation of RIMS workshop and the supports during his visit last summer.

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DEPARTMENT OF MATHEMATICS, NINGBO UNIVERSITY, ZHEJIANG, 315211, CHINA.,EMAIL: WUZHIWEI@NBU.EDU.CN