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首都大学東京大学院

理 工学研 究科 数 理 情報 斜 学 専攻

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m1t12* A-7-7> '1

X A46 : Deformations of Plane Curves and Soliton Equations ( fs ) : , , ~:~g)o.;V 9 1- /--fi=C ()

The purpose of this thesis is to study and summarize the main results in deriving soliton equations through the deformation of plane curves. Namely, we study how, given certain integrability conditions on the Frenet frame of a plane curve p in R2 and the conformal Frenet frame of a projective curve p = {pi : p2]

in RP', a smooth evolution of either curve with respect to a new variable, t, gives rise to two different classes of PDEs. Among these PDEs, we find that the mKdV equation is a special case of deformations of planar curves, and the KdV equation is a special case of deformations of projective curves. We then compare this geometric approach with an approach using D-modules to show how the mKdV and KdV equations arise, again, as special cases of a general method of constructing PDEs. Finally, we compare the two different approaches and discuss the correspondence between plane curves and D-modules.

In the Euclidean plane case, we can express the evolution of a smooth, regular curve p, parameterized by its arc length s, as:

pt(s, t) = g(s, t)T(s, t) + f (s, t)N(s, t),

where T and N are the unit tangent and normal vector fields to p, and f, g are smooth functions. We show that whenever the integrability condition:

gs=fic

is satisfied, then we obtain the class of PDEs:

Kt, = (fs + g/c)s,

where ic is the curvature of p. The choice f = —Ks and the integrability condi- tion yield the mKdV equation:

3 2 'c+/c

8+2 ~tcs = 0.

In the projective line case, we can express the evolution of a smooth, regular curve p as:

pt(u, t) = a(u, t)p(u, t) + b(u, t)p. (u, t),

1

where a, b are smooth functions. We show that whenever the integrability condition:

bu = —2a is satisfied, then we obtain the class of PDEs:

Ct = buuu 2bucd- bcu,

where c is the Schwarzian derivative of p. The choice a = Ac „ and the inte-

grability condition yield the KdV equation:

Ct = cau. + 3ccu•

We next consider the ring of differential operators Ds , a t-family of oper- ators {L(t)} t c Ds where L(t) := u(x , t), u is smooth in both variables, and a t-family of operators {P(t)} t c Ds where P(t) := Enk=O Pk (X, t)(9k and the coefficients Pk are all smooth in x and t. It is known that the D-module Ds ,t1(L, at - P) has rank 2 if and only the following "mod" commutator relation holds:

[at - P, = 0 mod L.

Making the simple choice of P = (-12-us h) — ?Las, where h = h(t) is smooth , this relation is then shown to be equivalent to u satisfying the KdV equation:

Ut = 22--ttxxs 3uux•

By making the additional assumption that u = q — q2 for some smooth function q(x , t), we show that Dx,t/(ax +q, at - P) has rank 1 if and only a similar "mod"

commutator relation holds:

[at - P, + = 0 mod (Ox + q), which is equivalent to q satisfying the mKdV equation:

qt = v‘i(Ix• Q,2,

Having presented the above results, we then discuss the parallels between plane curves, in both R2 and RP', and the aforementioned D-modules, with particular focus on the occurence of the KdV and mKdV equations in both

approaches.

References

1. Guest, Martin A. From quantum cohomology to integrable systems. Ox- ford Graduate Texts in Mathematics, 15. Oxford University Press, Oxford,

2008.

2. F olgsk-s, 2010.

2

Deformations of Plane Curves and

Soliton Equations

Stefan Horocholyn

1

Contents 1 Introduction

2 Curves in 2 and the Frenet Equation

3 Deformations of Curves in R2 and the mKdV Equation 4 Curves in RP1 and 2nd-order Linear ODEs

5 Deformations of Curves in 11P1 and the KdV Equation 6 D-Modules and the KdV and mKdV Equations

7 Plane Curves and D-Modules

A The Special Euclidean Group SE(2) B The Schwarzian Derivative and GL(2, IR)

3 5 11 18 24 28 35 40 44

2

1 Introduction

The purpose of this thesis is to study and summarize the main results in deriving soliton equations through the deformation of plane curves. In the Euclidean plane, we can express the evolution of a smooth, regular curve p, parameterized by its arc length s, as:

pt(s, t) = g(s, t)T(s, t) + f(s,t)N(s,t),

where T and N are the unit tangent and normal vector fields to p, and f, g are smooth functions. We show that whenever the isoperimetric condition:

gs=

is satisfied, then we obtain the class of PDEs:

Kt (h

where IC, is the curvature of p. The choice f = and the isoperimetric condition yield the mKdV equation:

Kt ±nsss= O.

In the projective line case, we can express the evolution of a "lift" p in R2, of a smooth, regular curve p E RP' as:

Pt (u, t) = a(u, t)p(u, t) + b(u, t)pu(u, t),

where a, b are smooth functions. We show that whenever the conformal condi- tion:

= —2a is satisfied, then we obtain the class of PDEs:

Ct buuu + 2buc + bcu,

where c is the Schwarzian derivative of p. The choice a = Acu and the confor-

mal condition yield the KdV equation:

Ct Cuuu 3CCu •

We next consider the ring of differential operators El:, with coefficients smooth functions, a t-family of operators {L(t)} t C Dx, where L(t) := t9, + u(x, t) and u is smooth in both variables, and another t-family of operators {P(t)} t c Dx, where P(t) := a=0 pk(x, t)8 and the coefficients Pk are all smooth in x and t. It is known that the D-module Dx,t/(L, at - P) has rank 2 if and only if the following "modulo" commutator relation holds:

— P, = 0 mod L.

3

Making the simple choice of P = (2 ux + h) — uax, where h = h(t) is smooth,

this relation is then shown to be equivalent to u satisfying the KdV equation:

ut = ZUxxx + 3uux.

By making the additional assumption that u = qx — q2 for some smooth function q(x, t), we show that DG,t/(ax + q, at - P) has rank 1 if and only if a similar

"modulo" commutator relation holds:

[at — P, ax + q] = 0 mod (ax + q), which is equivalent to q satisfying the mKdV equation:

qt = — 2 gxxx + 3g2gx. .

Having presented the above results, we then discuss the parallels between plane curves, in both R2 and RP', and the aforementioned D-modules, with particular focus on the occurrence of the mKdV and KdV equations in both

approaches.

4

2 Curves in I".2 and the Frenet Equation

We will be using the treatment given in Inoguchi [61; for a complete treatment of the theory of curves (in English), we refer to Pressley [8].

Definition 2.1 A parameterized curve (in R2) is a vector-valued function p : [a, b] - R2 such that p(u) = (x(u), y(u)). Furthermore, we say that p is a smooth parameterized curve whenever it is smooth on any open subinterval

of [a, b]. We will say that p is regular whenever a 0 0 for all u E [a, b].

In this exposition, the term "parameterized curve" is to be interpreted as referring to a smooth, regular parameterized curve, unless otherwise qualified.

Definition 2.2 For a parameterized curve p : [a, b] Il 2, the arc length of p from p(a) to p(u) is defined to be the function s : [a, b] --} [0, s(b)] given by the following integral equation:

s(u) := Ju—dpdt.(2.1.)

Remark 2.3 By regularity of p, s(u) is a strictly increasing function on [a, b], so it is injective, and thus, it has an inverse u = u(s) defined on the interval I := [0, s(b)]. Using this inverse function, we can re-parameterize our curve by s instead of u, giving us the arc length parameterization of p. One immediate advantage to using the arc length parameter s over u is the following identity:

dp dsdu MS)) =dp(u)du=dp(u)dp(u) ds dudu = I.

This also shows that regularity of p is unaffected by re-parameterizing to arc length. -

Remark 2.4 More than just being convenient to work with, the most impor- tant thing about the arc length s is that it is invariant if the curve is rotated or translated. Suppose A E SO(2, R) and b E R2 are arbitrary but fixed; then the arc length s' of the curve Ap + b is:

Ias'(91)

= J(Z

fu •a

au a

= s(u).

d(Ap + b)

dt dAp d

t dt

Adpat

dP dt dt

dt

by Lemma A.5(ii)

5

(See Appendix A for the relevant discussion.) Of course, the above is true if A E 0(2, R), but we will see shortly that SO(2, R) has a particularly special role in our exposition.

We introduce three quantities in the following definition, which will be im- portant in our analysis later. We will also make much use of the following special orthogonal matrix:

0 —1 J := 1 0

Definition 2.5 Let p be a curve parameterized by its arc length, s. Then the unit tangent vector field, T, of p is the vector field:

T(s) Os),(2.2)

and the unit normal vector field, N, of p is defined to be:

N(s) := JT(s).(2.3)

The matrix consisting of these two vector ,fields is called the Frenet frame, F(s), of p:

F(s) [T(s) N(s)].(2.4)

Remark 2.6 By Remark 2.3, 11Th = JINN = 1 and T • N = 0, so {T, N} forms an orthonormal basis at p; furthermore, det(F) = 1, so F(s) E SO(2, R) for all

s E I.

" Differentiating IITII2 = 1 by s shows that T • T' = 0

, so r N, which means that there must be a smooth function n(s) such that T'(s) K(s)N(s).

Differentiating N then shows that:

N1(s) JT'(s) ic(s)JN(s)).

Consequently, we obtain the Frenet equation:

F'(s) = [T'(s) N1(s)]

= n(s)JF(s).(2.5)

Owing to the relatively simple nature of this equation, we can in fact solve this immediately: first, using the fact that F(s) E SO(2, R) for each s, there is a 0 : R such that:

[cos 0(s) — sin 0(s) F(s) sin 0(s) cos 9(s)

6

(See Appendix A for a derivation of this fact.) Second, if we interpret F(s) as the matrix representation of a complex function f : I -* C, then:

f(s) = eZs(s)• •

Finally, if we re-write the Frenet equation as:

f(s) = ii (s)f(s), which we know has solution:

f (s) f (0)ei f0 .(-) dr,

then comparing the two expressions for f(s) implies that a Oo E lR such that f(0) = eie0. (Strictly speaking, 00 + 2irn, for any n E Z, is also a solution.) Therefore, we conclude that:

fS0(s)=nr) dr + 00.(2.6)

This function is called the tangential angle, or less often the turning angle, of p; we will see it again in Theorem 2.7.

With the above background on plane curves, we are now in a position to state and prove the fundamental theorem of plane curves, in two parts:

Theorem 2.7 (Existence) Let > 0. For every smooth, real-valued function n(s) on [0, £1, there is a parameterized curve p : [0, Q] —* lR2 with s as its arc length parameter and ic(s) as its curvature.

Proof: We first define the function 0 : [0, £] —> I': by:

rS

0(s) := J i(t) dt + 00,

0 where 00 E R is arbitrary but fixed, and then define the curve p : [0,r.] R2

by:

p(s) := f(cos O(t), sin 0(t)) dt,+ po,

0 where po E R2 is also arbitrary but fixed. We then see that:

p'(s) = (cos 0(s), sin 0(s)) lip'(s) II = 1,

which tells us that s is the arc length parameter of the curve p(s). Then by Definition 2.5, the Frenet frame of p is:

F(s) = [T(s) N(s)], and differentiating T yields:

T'(s) = 0'(s)(—sin0(s),cos0(s)) = O'(s)JT(s),

so we see that we have obtained the ODE:

F'(s) _ [T'(s) N'(s)]

= H'(s)J[T'(s) M(s)]

= H'(s)JF(s),

which is precisely the Frenet equation (2.5). Thus, p must have curvature:

h(s) = H'(s), as was to be shown. •

Theorem 2.8 (Uniqueness) Let pi, P2 : [a, b] -4 R2 be two parameterized curves with arc lengths si, s2, and curvatures /£i,' 2i respectively. Then s := si = s2 and 'ci(s) _ 1c2(s) for all s E I := [0, s(b)] if there is a spe- cial Euclidean motion (A, b) E SE(2) := SO(2, IR) v R2 such that:

P2=(A,b) *pi =APi+b.

Proof: () Suppose that pi and p2 share the same arc length s, and ni(s) _ n2(s) for all s E I. To find an element (A, b) E SE(2) such that p2 (s) = (A, b) * pi (s) for all s, we will first want to show that there is a fixed 0 E [0, 2ir) such that F2(s) = R0Fi(s) V s, which is equivalent to showing that F2 F1 = R_0. To that end, we differentiate FZ Fi:

(F2Fi)' = 1"2Fi + F2Fi

= (,c2JF2)TF1 + F2('c2JF1) by the Frenet equation

= ic2(JTF'Fi +F2JF1) by 'c = r,,2

= 0 since JT = —J and by commutativity of SO(2, R).

Hence, F2 F1 is constant, so for all s:

F2(s)F1(s) = F2(0)F1(0).

Since F2(0), F1(0) E SO(2, R), they "differ" by a rotation R0, where H E [0, 27r), such that:

F2(0)F1(0) = R_©

F2(s)F1(s) = R_0

F2(s) = R0Fi(s) V s E I, as we had claimed. In particular:

T2 (s) = RBTi (s) 44. p'2(s) = Rop (s)

8

Therefore, integrating this equation gives us:

P2(s) = RePi (s) + (Ropi (0) — p2 (0))

= (Re, Repi(0) — p2(0)) * pi (s)

by inspection. Thus, we have shown that two curves pi and 132 with identical arc length and curvature differ at most by a special Euclidean motion (Re, Ropi (0) — P2 (0)).

() Let p1 be parameterized by its arc length si, and suppose:

P2(81) := Api(si) + b,

so pi (si) and p2(81) differ by only a special Euclidean motion. (Note that p2 is still otherwise parameterized by its own arc length, s2.) Then:

dp2 T

2 (S2) = 3 (Si (S2))

us2

dp2 ds1 ds1 ds2

= ATi (si) (cit—Z and since A E SO(2, R):

1 = 111'211 =- V:, 1,

so either s s2 or si = —s2. However, if u is a local coordinate of [a, b]

for both curves, then by Definition 2.2, si(u) and s2(u) are strictly increasing functions on [a, b], which excludes the possibility that si(u) + s2(u) = 0 for all u E [a, hi, Hence, s 2 = s1 =: s is the arc length parameter for both pi and p2.

The Frenet frame F2 = [T2 N2] of p2 is found as follows:

T2 = 132 = AT1,

and (using commutativity of SO(2, R)):

N2 = JT2 = JAT1 = AJT1 ANi ,

so F2(s) = AFi (s).

To find the curvature n2 of 132, we use the definition given in Remark 2.6:

= IC2N2 , and comparing it with T2 = ATi , we find that:

= AT = A(niNi) = k1AN1

It then follows that:

so two curves differing at arc length and curvature.

ni(s) n2(s) VsE I, most by a special Euclidean U.

motion share the same

9

In summary, the fundamental theorem gives an isomorphism between smooth functions n(s) on closed intervals I, and equivalence classes of smooth, regular curves p, parameterized by arc length s E I, in the plane, where the equivalence is up to special Euclidean motion. That equivalence is up to special Euclidean motion, and not general Euclidean motion, is due to the fact that the curvature function ic(s) accounts for the curve's orientation, so reflections will cause a change in the sign of n and are thus ruled out.

10

3 Deformations of Curves in R2 and the mKdV Equation

Having completed our preliminary discussion on parameterized curves in the plane, we wish to investigate how plane curves evolve with respect to a time variable, t. This will lead us to a partial differential equation that describes the time evolution of the curvature function, and will allow us to provide a geometric derivation of the modified Korteweg-de Vries equation (mKdV equation).

To formalize this appropriately, we make the following definition:

Definition 3.1 The function p : [a, b] x I' -3 R2 is said to be the time evolu- tion of a parameterized curve if p(u; t) is a parameterized curve for each t E R.

In addition, we will demand that the time evolution preserve the arc length, s, of p(u; t) in the following way:

Definition 3.2 We say that p, as in Definition, 3.1, satisfies the isoperimet- vie condition whenever:

°=at- fjpv dv.(3.1)

Notice that, if the original curve was closed, then preservation of the arc length would mean that the curve would neither expand nor contract - the curve would be isoperimetric.

With respect to the Frenet frame F(s; t) = [T(s; t) N(s; t)], we may write:

pt(s; t) = g(s; t)T(s; t) + f (s; t)N(s; t),

for some smooth functions f, g. As such, applying the isoperimetric condition yields:

0 =I~tIIPv1Idv=fup1I P Ilutdv=`1.0T' Pvt dv.

Now, using our expression for pt and the Frenet equation (2.5), we find:

as

Put = (Pt)u = (Pt)s au

= IJPull(gT + f N)s

= IIPuII ((gs — f tt)T + (j. + gc)N),

so the isoperimetric condition reduces to:

0 = IIPzj II (gs - f dv.

11

This holds for all u E [a, bi, so by Remark 2.3, we can re-write it as:

0 f lip,11(g-§ - f

rs

jo iiPv Egg ftc) liPv11-1

rs

=j

o(g§— f

gs f(3.2)

Let us now consider the time evolution of the Frenet frame:

Tt (Ps)t (Pt)s = (gT + fN),

= (gs - tc)T (f, gn)N

= (f8 gic)N by Equation (3.1), and thus:

Nt = JTt —(f + g tc)T Combining our results, we obtain:

Ft = [T Nt]

[( f s + gic)N — (h + g tc,)T]

(fs + gic)J[T N]

= (fs + g/c)JF.(3.3)

In fact, by Equations (2.5) and (3.3), we have derived the following simulta- neous partial differential equations for F(s; t):

aFaF

=FU,=FV at where:

U = KJ, V = (fs + gn)J.

By smoothness of F, we can take the difference of its mixed second derivatives to find:

0 = (F5)t - (Ft)„ = (FU) t (FV)„

= FU + FU t - F8V FV,

= FVU + FUt - FUV - FV, F(VU + Ut - UV - V8)

= -F(V, - Ut + [U, 1/1)

<#. 0= V, - Ut + [U,(3.4)

12

This equation is called the integrability condition for F. Observe that, by definition of U and V:

V, - Ut + = (( + gn)s - Kt)

since [U, = 0, so the integrability condition is equivalent to the PDE:

= (fs + gic)s.(3.5)

Remark 3.3 Let us suppose that we did not initially demand g = 0; then

carrying through the calculations of the time evolution of the Frenet frame would have yielded:

Ft= g5-ftc(fsgic) f F, s+ gicg5-fic

-

and so:

F-1Ft e .so(2) <4> gs(s, t) - f(s,t)K,(s,t) = 0, where we recall that:

.so(2) { aJ E M2 '^ a e

By then taking the definite integral L9 and working backwards up to Equation (3.1), we see that gs - f = 0 implies that 0; hence, "isoperimetricity"

is an underlying Lie algebraic property of the Frenet frame, and we will indeed see this again in Section 5, when we consider deformations of a curve in the projective line IRP'.

We supplement this discussion with one last comment, slightly generalizing the integrability condition:

Theorem 3.4 Let a,,8 E C" (R2 ; R), and define U, V : R2 (2) by:

U(s, t) := a(s, t)J, V(s, 1) := 13(s,l)J.

Then there exists a smooth solution F : R2 -+ SO(2, R) to the simultaneous PDEs: aFaF

—=FU ,—= asat FV iff:

V, Ut [U, = 0 <4> /3, - at O.

Proof: Given a smooth F E SO(2, R) that simultaneously satisfies the above system of PDEs, the integrability condition follows by repeating the above calculation for Fst = Fts, mutatis mutandis.

(=) Let A, = at, which by direct calculation is equivalent to the integrability condition V, - Ut [U, = 0. Then the 1-form w := a ds /3 dt is closed,

since:

dw = at dt A ds + /3, ds A dt = 0.

13

By the Poincare Lemma (see [2]), there is a smooth function f : R2 —÷ R such that:

.1s ds + ft dt = df w = ds + /3 dt

<=> a, ft= 0.

Hence, if we let 11(s, := eif(s,t) , then:

Ps= ifseif = iaP, and:

Pt = ifteif OE.

We now identify the complex-valued function F with the special orthogonal matrix F given by:

F(cos( )s,t f (s, t)) — sin( f (s, t)) sin( f (s,t)) cos( f (s,t)) • It then follows that F simultaneously satisfies:

aF aF

—= FU andFV asat ,

where U(s, t) a(s, t)J and V(s, := 13(s, t)J, as was to be shown. •

Remark 3.5 An important result of the derivation of the family of PDEs (3.5) can be seen if we choose f = —ns. Then gs = —KK,s, so for simplicity, we take g n2 . In this case, the time evolution of p(s; t) is:

Pt = —

and the curvature ic(s, t) satisfies:

nt + n,,,,,, + :4n2n, = 0.(3.6)

Equation (3.6) is the modified KdV (mKdV) equation, and is an im- portant and well-known equation in the area of non-linear wave motion. Inter- estingly, if we define the integro-differential operator C2 by:

h„ + n2h + ns f kit ds,

where h is any sufficiently smooth function, then the mKdV equation can be

simplified to:

—= -ICSSS2— ic,, f 1cK,,, ds

=

14

Re-writing the isoperimetric condition (3.2)

and defining for

we see that

gs = kf <=>

all T1 E N:

fn :=

f(fit,g71)171EN

and

similarly:

g f kJ' ds,

gr, := ficf„ ds,

all satisfy the isoperimetric condition:

grt,s icfn,

and for each n, K, satisfies the n-th mKdV equation:

it = frt,ss gn,sn 9nKs

_a82 (fin- 1 no n2fr,-1 ns f scr-lns ds

= - (8.,20 + n2()f no ds)s-r- iss

The collection of all n-th mKdV equations is called the mKdV hierarchy, and C2 is the recursion operator for the mKdV hierarchy. For example, the 2nd mKdV equation is:

-4-121cs

= _0 820 + n2 I.^ --r K,s f (•) ds)(nss, is,^)

= —nsssss — 3tt — 6nicsnss — pc2nsss — — f K,K ,s,„dS - .1Ks f K3tcsds

— — ns(ksicss(.14n4)

= —K sssss3K3s6nicsicss4K2Ksss

=— —icsssssrbsss 5,2, — 7ickskss — 8 -

Unfortunately, we will not take this opportunity to delve into hierarchies of integrable systems; however, the theory has deep connections with Hamiltonian systems, and we refer to [4J for a recent account of this area of work.

Before giving some example solutions, we make one other remark on the form of the mKdV equation: notice that making the substitution ic 1-> -ic leaves the equation unchanged. This means that the curves p, q with curvatures n, -it, respectively, are both solutions to the same mKdV equation. We will explain this point some more at the end of Section 7.

Let us give a couple of examples of (equivalence curvature K. satisfies the niKdV equation.

Example 3.6 (The Solitary Wave) Let k(s,t) be verified that n satisfies the mKdV equation (3.6).

then:

classes of) curves whose 2 sech(s - t); then it can

The tangential angle is 0(s, t) 4 arctan(es-t) - 4 arctan(e-t)

= 2 gd(s - t) - 2 gd(-t),

15

where gd(x) is the "Gudermannian" function, defined by:

x dy

gd(x) J

o cosh y

= 2 arctan(ex) — 2~.

(See [10] for a definition of this function.) The Gudermannian is a way of relating trigonometric and hyperbolic functions without reference to the complex numbers, so we have the following useful identities:

sin gd(x) = tanh(x) cos gd(x) = sech(x).

Then:

cos 0 = 1 — 2 tanh2 t, — 2 tanh2 (s — t) + 4 tanh2 (s — t.) tanh2 t + 4 tanh(s — t) sech(s — t) tanh t sech t

and sin 0 = 2 (1 — 2 tanh2 t) tanh(s — t) sech(s — t)

+ 2tanhtsecht(1 — 2tanh2(s — t)),

so the equivalence class of curves (up to special Euclidean motion) with curva- ture n(s, t) = 2 sech(s — t) is:

fSp(s, t) =(cos0, sin 0) ds,

which can be integrated to give:

r 5 cosFlds= (1—2 sech2t,) s — (2 — 4 sech2t,) (tanh(s — t) — tanht,)

0

+ 4 tanh t sech t (sech(s — t) — sech t)

and:

.108 sin 0 ds = (2 — 4 sech2 t) (sech(s — t) — sech t) — (2 tanht,secht)s

+ 2 tanhtsech t(tanh(s — t) + tanh t). A

Example 3.7 (The Algebraic Soliton.) Let us now consider the curvature function:

^c(s,t)=c— e2(s4t)2+1'where c=f 3.

With a bit of work, it can be shown that this satisfies the mKdV equation (3.6).

(Incidentally, the value of c depends on the coefficients in the mKdV equation.) This can be integrated to give the tangential angle function:

0(s, t) = cs — 4 arctan(c(s — t)) + 4 arctan(—ct) = cs — 4 arctan x, x cs

1 — (cs)(ct) I (ct)2 •

16

cos 0

and Sift)

These can be fur

In addition to using the standard trigonometric identities, if we make use of the following identities:

cos 0 = 1 — 8 sin2 (1) cos2 (1),

sin 0 = 4(2 cos2 () — 1) sin (&) cos (1), cos(arctan x) 1

x2 + 1

and sin(arctan x) = --- 1/x2+1 then we obtain:

x4 cos cs — 4x3 sin cs — 6x2 cos cs + 4x sin cs + cos cs cos 0 = --- (

x2 + 1)2

x4 sin es + 4x3 cos cs — 6x2 sin es — 4x cos cs + sin cs ---(

x2 + 1)2 These can be further simplified to:

(x + .!-,88)4 cos 0 = ( (cos cs) x2 + 1)2

= COS4 (arctan X) (:i; as) 4 (COS cs)

and:

(x + as)4(sin cs) sin 0 =

(x2 + 1)2

= cos4(arctan x) (x + icas) 4 (sin cs),

where we have used the binomial expansion:

(x + u)\4= x4 + x38_+6x2 32,+4.xa3sJt34, c'

Hence, the equivalence class of curves with curvature s(s, t) c -

p(s , t) f ( cos 0, sin 0) ds. *

4c c2(s-02-1-1is:

17

4 Curves in RP1 and 2nd-order Linear ODEs In this section, we give a brief theory of curves in I[8P1, starting by considering such curves and a projective differential invariant, the Schwarzian derivative, and ending with a way of viewing them in terms of certain 2nd-order linear ODEs. In doing so, we follow the treatment given in both Inoguchi [6] and Sasaki [9].

Definition 4.1 Let M be an open interval with coordinate u.

(a) A smooth map p : M -4 RP1 taking u p(u) is called a projective curve.

(b) A projective curve p(u) is said to be regular whenever dp.0 0 for all 'a E M .

In homogeneous coordinates, the curve p has expression:

p(u) = [X1 (U): x2 (u)],

so if p is regular, then:

dpu 0:41:2 — X2x1 O.

Writing p in inhomogeneous coordinates as p(u) = [f (u) : 1], regularity is then equivalent to f'(u) 0 0 for all u.

For convenience, we will refer to regular, projective curves as simply "projective curves" from now on, and will alternate between homogeneous and inhomoge- neous notation when convenient.

Let us make the following definition:

Definition 4.2 Let f : M —* R be a smooth function. Then the Schwarzian derivative (or more commonly, the Schwarzian) of f , with respect to u, is defined to be the quantity:

f~~(u) 1 fli (u)2(4

.1) Su( f)=f'(

u)2f,(u)

f (u) 3 f."(,u) 2

=

f'(u) 2 f'(u)

Using the discussion from Appendix B, we have the following theorem:

Theorem 4.3 (Uniqueness) Let M be an open interval with coordinate 'a, and let p, q : M —4 ll P1 be two curves, expressed in inhomogeneous coordinates

as p(u) = [ f (u) : 1] and q(u) = [g(u) : 1]. Then 3 A = (a 1,)i) E GL(2,R) such

that:

ef (u) + d' if and only if S,L(f) = S.u(g).

18

Proof: The (=) 9 = g' and gni (

cf + d)2(cf +(cf +d)`1

and since g' 0 by regularity, we can take the following ratios:

g,,f" 2c f'

9' f'cf + d

9"2_"2 4cf" 4c2 f'2

g'f' cf+d+(cf+d)2

andg"'—f„"—6cf~~+6c2r2

9' f' cf + d (cf + d)2 The last equality can be simplified, using the two above it, into:

direction follows by direct calculation:

(ad — bc) f' (cf+d)2

(ad — bc) f" 2c(ad — bc)ft2 (c f + d)2 (cf + d)3

(ad— hAfin Rc(nd — be)f'f" Rf•2(ad— bc)f-''s

9//I 3 f111 32

2(92 ^_ .i2f'

Therefore, if two functions f and g are related by a linear fractional trans- formation, then Su(f) = Su(g).

For the () direction, Lemma B.4 tells us that:

Sf(g) df2 = S,L(g) du2 — Su(f) du2

= 0,

by assumption that Su(f) = Su(g). Therefore, by Lemma B.3, 3 A = (( i) E GL(2, R) such that:

g(u) =7'A(f (u))=a f (u) + b.^ c .f(u)+d

Just as we showed that the Euclidean curvature is invariant under the action of SF'(2) in Section 2, what this theorem shows is that the Schwarzian of a projective curve p (or rather, its inhomogeneous coordinate f) is invariant under the action of GL(2, IR). (Additionally, Corollary B.5 shows that the "parameter"

u is unique up to linear fractional transformations.) As Proposition B.1.3 shows, the transformation TA is not unique, but if we instead consider the projective general linear group:

PGL(2, IR) := GL(2, lR)/{cI2 r. E IR*},

19

then the above result can be refined by replacing "a A E GL(2, R)" with

"3! [A] E PGL(2 , IR)", and similarly, the parameter u would be unique up to the action of PGL(2, I' ).

To give an "existence" theorem (i.e. given a Schwarzian, produce a pro- jective curve), it turns out that curves in RP1 and their Schwarzian have an interpretation with respect to certain 2nd-order linear ODEs. For p = [xi : x2}

a projective curve, consider the following 2nd-order linear ODE:

i„ I

X' x x

0 = xl xC x1

x2 x2 x2

_ (xcx2 – x2x1)x + (x2x1 – x1x2)x + (x1x2 – x2x1)x.

By regularity of p, we can divide through to obtain:

"x2x1— x1x2 ,x1x2—

xx2x1

x1x2 – x2x1 x1x2 – x2x1

By Equation (4.2), it is clear that both x1(u) and x2(u) are solutions, so it seems natural to associate Equation (4.3) with p(u). However, if µ : M R*

is any smooth function, then p = [x1 : x2] = [px1 : / x2], so making the substitutions x1 F Ax1 and x2 i ,ux2 would yield an ODE that also includes A. Therefore, our goal is to find an ODE that is somehow "naturally" associated with the projective curve p, independent of any smooth, non-vanishing scaling function µ.

Let us now re-write p in inhomogeneous coordinates as:

p(u) = [f(u) : 1] = [µ(u)f(u) : µ(u)]-

where A : M –> R* is any smooth function. Substituting this into Equation (4.3) allows us to simplify to:

2`i2

N f'N .t ` la 11,

for all such p. In particular, we may restrict p. to satisfy:

21-=-`+d=0—d u Only') = --2du(ln'f')•

Integrating and then exponentiating gives us the solution:

µ(u) = ^'f`(

u)', Po E R*.

Therefore, for any projective curve p(u), when we take its homogeneous com- ponents to be:

x1(u) = Poi (u) ,x2(u) = V µo IP (u)I ,/'f'(u)'

20

where f is a smooth function, the associated ODE (4.3) becomes:

_ _I f)

((f/)2(JI)2) 1li--I-2(

=x"+ (—

\' p-x li

(1 f,,, 3 f/, 2

= x" + 2Su(f)x•(4.4)

As such, if we are given a smooth function f : M -- 1R, we can then consider the 2nd-order linear ODE (4.4) and construct a projective curve p(u) with ho- mogeneous coordinates xi and x2 given above. We formalize this conclusion in the following theorem:

Theorem 4.4 Let f : M —+ R be a smooth function, and S„ (f) be the Sch,warzian of f . Then the end-order linear ODE:

x" + 2S.u(f)x = 0

has solutions:

xi (u) = µ/o f (u)

VIf'(u)I

and x2 (u) = ---/µ0---

Vil(u•)i

where µo E lR*, which represent the projective curve p : 1V1 —* RP1:

p(u) := [xi (u) : 52(u)] = [.f (u) : 11.

When it is convenient, we will also write this ODE as:

x" + -Su(2)x = 0.(4.5)

Having said that, let us take another look at Equation (4.5), in terms of its two solutions xi and x2: if we let x(u) = (xi(u),x2(u)) represent a trajectory in R2 \ {O}, then:

xi,uu+ S.u

((~)xi = 0 xuu+ 2S.u(2)x=0.

Since Su (f2) is a scalar, we see that x and xuu are linearly dependent; i.e.:

xi xl,uu

= 0.

X2 x2,uu

21

But then:

xi _X2 xl ,ui _x2,u _(x1x2 ,u — x2x1,u)

XiX2,uu — x2xl,uu xl :X;l,uu X2 x2,uu SO:

xl x1 ,u A0

—

X2 X2 ,u where A0 E R* is constant. Observe that:

A0 =xlx2,u— x2xl,u xlx2 ,u — x2x1,u 1=

Ao

x1x2 X2Xi

1 + /IA01 ±vIAo1 ± /IAol +OaoI /

Thus, given a projective curve p(u) = [xi(u) : x2(u)] and the two solutions xi, x2 to the ODE (4.5) associated to p, if we let:

x1

P1 = ---

~I x1x2,u — x2x1,uI

X2

and p2 = ---

~/Ixix2 u — x2Xi uI

then the "lift" p(u) = (pi (u),p2(u)) of the projective curve p, from RP' to R2,

satisfies:

pip2,u — p2P1,u = 1•

This gives us the conformal Frenet frame:

conf

F:= [P Pu] E SL(2,R), which satisfies the conformal Frenet equation:

Fconf = [Pu Puu] d u

[Pu — 2 cP]

1 1 = Fconf 0 —21 0 ' (4.6)

where for notational convenience, c := Su (T ~) . In analogy with the Frenet

equation, we then see that the Schwarzian describes the "conformal" curvature of the lift p of the projective curve p, and the above ODE is equivalent to:

(Fconf)-1 d ,conf =0 '-2c E 51(2, I ).

du1 0

22

Before continuing on, we would like to make one remark regarding our use of projective coordinate expressions.

Remark 4.5 Throughout this section (and Appendix B), we have been only using the inhomogeneous coordinate expression [.f : 1] of the point [xi : x2] E R P1 with respect to the chart:

V+ = {[xi x2] E RP1 I x2 0, (xi, x2) E R2}

and the homeomorphism : V+ R defined by:

x2})x1 x2}) = — =:

X2 To be rigorous, we should also consider the chart:

V_ =-- fix]. : x2] E RP1 I xi 0, (x1, x2) E R2}

along with the homeomorphism : V_ R defined by:

X2

([xi : x2}) = — =: y. _xi

Then on V+ fl V_:

p_ 0)(x) = 1

and op+ 0 Ipil) (y) —1,

so {(V+, ik+), , )1 is a smooth atlas of RP', and the above discussion holds when we consider points with respect to the chart (V_ , as well. As such, when neither x1 nor x2 are 0 (i.e. on V+ n v._), then:

X1 X1 X1 X2

=

X2 X2 X2 XI

SO _{x2 xi} and it makes no difference which particular ratio is considered.

23

5 Deformations of Curves in RP' and the KdV Equation

We now investigate how curves in RP1 evolve with respect to a time variable, t, so we start with a definition:

Definition 5.1 Let M be an open interval. The function p : /I/ x R RP' is said to be the time evolution of a projective curve if p(u, t) is a projective

curve for each t G

Using the conformal Frenet frame, the time evolution pt of p can be ex- pressed as:

pt(u, t) = a(u, t)p(u, t) b(u, t)pu(u, t) ,(5.1)

for some smooth functions a, b:MxR R. Moreover, they must satisfy:

a Pi Pin 0

= — a

t P2 P2,u

= Pl ,tP2,u P2,tPl,u P1P2,ut P2P1,ut

=P1,t Pl,uP1 Pl,ut P2,t P2,u P2 P2,ut

(ap1 bP1,u) P1 ,u P1 (aupl aPl,u buP1,u bP1,uu) (ap2 + bP2,u) P2,u p2 (aup2 + ap2,u + buP2,u + bp2,uu)

= (2a + bu) P1 Piu ,b Pi Pi,uu P2 P2,u P2 P2,uu

= 2a + bu,

where we have used det(p,puu) = 0 at the very end. As such, we obtain the conformal (isoperimetric) condition:

bu(u,t) —2a(u,t).

Equivalently, the conformal Frenet frame satisfies the following PDE with respect to t:

_Fconf = [Pt Put]

at

= [(ap + bpu) ((au — P)c)p ± (a + bu)pu)]

_ Feonf au—bc2 a-1-

b a ± bui

We have the following proposition (which follows by direct calculation):

Proposition 5.2 Let A E Cl (I; SL(2, li1)) . Then A-1 ft- A E -51(2, 1), where:

.51(2,R) := {A E M2IR I trA 0}.

24

Since Fconf E SL(2, R), by Proposition 5.2, we obtain 2a+ b„ 0 again. Thus, in direct analogy with the isoperimetric condition (3.2), we see that "conformality"

of the frame is a Lie algebraic property.

Finally, by construction of p, we know that it satisfies Equation (4.5), so differentiating with respect to t gives:

Ptuu-3-c-tP= 0.

Inserting the time evolution expression then yields:

0 = (ap + bpu)uu + Ica) + .1c(ap + bpu)

= (aup + ap u + bupu + bpuu)u + letp + lc(ap + bpu) (Abp + + bpuu)u + CtP +c(—bp + bpu)

= (Abu. + Ct — buc)P + lbcpu + PuPuu bPuu.

= (-1-buuu + Ct — buc)p + -12-bcpu — lbcuP — (— ht.-r —buc—bcu)p

Therefore, the Schwarzian of p satisfies:

ct = buuu + 2buc + bc„.(5.2)

This can be equivalently derived by computing the integrability condition:

Vu —U + , V] = 0 for the simultaneous pair of PDEs satisfied by Fcmlf:

_a Fconf = Fconf

au a

and at Fconf = Fconf where:

—1ca a—bc

2V _-

10b a + bu •

Remark 5.3 As in Remark 3.5, there is a particularly important member of this class of PDEs which we have derived. If we let a(u, t) = --c„(u, t), then b(u, t) = c(u, t), and so the time evolution (5.1) is:

Pt (u, t) t)p(u, t) c(u, t)pu(u, t), and Equation (5.2) becomes:

ct = cuuu + 3ecu,(5.3)

which is the well-known KdV equation. Similarly, we can consider an integro- differential operator 5/ defined by:

hu. + 2ch + cu f Ii du,

25

and re-write the KdV equation as:

Ct = Clcu.

The conformal condition may also be (somewhat redundantly) re-written as:

2a+bu=0 b=-2fadu,

and if we define:

art -2cn-1Cu, bn -2 f an du,

where n E N, then the pairs {(an, bn)}nEN all satisfy the conformal condition, and for each n, c satisfies the n-th KdV equation:

Ct = bn,uuu + 2bn,uc + bneu

= 3 (-2an) + 2c(-2an) + cu ( — 2 f a„ du)

= (9u (Sr-1cu)+ 2c(~n-1Cu)+cu ff1Cudu

= (8,2,(-) + 2c(•) + cu f(•) du)S2 1 cu

= c cu.(5.4)

For example, the 2nd KdV equation can be shown to be:

Ct = Csssss + 5cuu•uC+ lOCuuCu + 2 CsC2• (5.5)

Remark 5.4 As before, we would like to consider some examples of solutions to the KdV equation. However, there is a well-known transformation of solutions of the mKdV equation into solutions of the KdV equation, called the Miura

transformation, which we introduce here.

Let us re-write the mKdV equation more generally as:

+ anss.s + OK2rcs = 0,

where a, 13 0. The Miura transformation of i, is then defined as follows:

p, := yic.s + 6n..2,

where 'y, 6 0. It can be shown that u solves the KdV equation:

µt + aµsss + bµµs = 0, a, b 0 if and only if:

6a a = a, 6 =,,y2= ---52 For example, we saw in Example 3.6 that:

n (s, t,) = 2 sech(s - t)

26

is a solution to the mKdV equation with a = 1 and [3 wave r is also a solution to the KdV equation:

+ Nsss + bµµs = 0 via the Miura transformation:

— 3

Thus, the solitary

—91 15;2

N_+b2~+2b.

Until now, we have not been explicit about whether everything is to be real- or complex-valued, but we see here, through a rather innocuous example, that no matter what b we choose, µ will be complex-valued. A choice of b = 6 gives the (complex) Miura transformation:

= +~s + 24-K,22 and (complex) KdV equation:

lit + EJsss + 6/a,u.s = 0.

Indeed, a complex KdV equation is quite natural, as our entire discussion in IIRP1 could have just as easily taken place in (CP1, in which case we would have derived a PDE for a complex-valued Schwarzian c. (This is the approach taken in [1] and [6].)

27

6 D-Modules and the KdV and mKdV Equa- tions

The point of view taken in [5] is that "the KdV equation arises from a certain universal construction with D-modules." In this section, we summarize the main results from [5] by studying the connection between the KdV equation and families of 2nd-order ODEs using the basic framework of D-modules. In this work, we will only consider 2nd-order ODEs of the form Ly :_ (a, -f- u)y = 0, where u is smooth, for the following reason: suppose we have the 2nd-order real-valued ODE:

(a + al Wax + ao(x))y = 0.

Then the substitution:

yze2-aidx =2 yields:

0 = yii + al y' + ao y

= eifaidx (z" + (ao _ )2‘a1 — 4ai)z)

2 = z" -I- uz,

where u(x) = a,o(x) — 2ai(x) — 4ai(x)2.

Previously, we encountered the KdV equation in the form:

ut = uxxx + 3uux ,

where u is a real-valued function. However, if we consider the equation:

ut = auxxx + Quux,(6.1)

where a, /3 E R*, then we can see that by scaling the variables t H aT and x be, for a, b E N*, we obtain:

T = t _ asa,13 u

) uu~ + b3 u ~

In other words, we can rescale the variables in our original equation and obtain nearly any pair of coefficients a and /3. As such, to avoid confusion about which pair of coefficients is "correct" or not, we will call any equation of the form (6.1) the KdV equation.

Now, for each t E R, let us consider the t-family of operators:

{L(t)}t := {a + u(x, t)}t C Dx, its corresponding t-family of rank 2 D-modules:

{Dx/(L(t))}t,

28

where (L(t)) is the left ideal generated by L(t), and a t-family of differential operators:

{P(t,)}t C Dx,

where the coefficients of P(t) are smooth functions of both x and t, but we assume that P has no terms in any power of at. -

We then have the following theorem (Theorem 7.1 of [5]):

Theorem 6.1 Given t-families of differential operators L(t) = a + u(x, t) and P(t) E Dx, the D-module Dx,t/(L, at - P) has rank 2 iff [at - P, L] = 0 mod L.

The proof relies on first identifying Dx/(L(0)), L(0) = ax + u(x, 0), as a module over C°° (x) (M; R), where M C IR is a fixed open interval, with the space of sections of the trivial bundle M x Rs+1, which has the flat connection Vox [ax] = [an in the module basis [1], [8],AL..., where s + 1 = deg(L) (for more general, monic L). This holds for each t, then, so this gives a t- family of rank s+ 1 D-modules, Dx/(L(t)). This entire t-family is equivalent to Dx ® C, °° (t)/([ ), which, itself, is equivalent to the space of sections of the (extended) trivial bundle Mx x Mt x ][gs+1 (Here, the notation Mx and Mt means that the interval M has local coordinate x or 1.)

Finally, by fixing a P E Dx,t as prescribed above, the connection on the bundle Mx x Rs+1 can be extended to a connection on the extended bundle Mx x Mt x Rs+1 by adding the relation:

Vat [ax] := MP].

It then follows that this extended connection is flat iff P satisfies [at — P, L] = 0 mod L, and it can be shown that this is equivalent to the module Dx,t/(L, at —P) having rank s + 1.

A corollary of direct relevance to our discussion on the KdV equation is the following:

Corollary 6.2 Let L = ax + u(x, t) and P E Dx,t such that P(x, t) = f (x•, t) + g(x, t)ax mod L. Then:

[at - P, L] = 0 mod L if and only if ut = 9xxx + 2g;lu + gux and fx=— .9xx•

Proof: We proceed by direct calculation:

[at — P, L] = 0 mod L L(at — P) = 0 mod L,

29

SO:

mat - P) = (a! + u)(at - f - gax)

= ata,2, - fx. - 2fxax - gxxax -2gzax2 - ga x3 +uat -uf — ugax

= at(—u) — fxx — 2 fxax — f (-u) - g.saw - 2gx(—u)

— gax(—u) + uat — uf — ugax

= —ut fxx — 2fxas gxxax + 2gxu + gux

= (—ut 2gmu + gum) + (-2f — gxx)ax

=0 mod L.

Therefore:

Ix — and:

ut= .1gxxx + 2gxu + gux. •

Thus, f (x , t) = g (x t) + h(t), for some smooth function h, and so we see that the differential operator P gx + h) + gax is sufficient to obtain the KdV equation, for if we let g —u, then P (lux + h) — wax gives the KdV

equation:

ut = 1-uxxx 3uux-

In the literature (for example, [7]), the choice of P is often made to be P

+ + for slightly different reasons; we will re-visit this choice of operator later.

Furthermore, if we now make the additional assumption that u = qx — q2, where q q(x , t) is a smooth function, then this is equivalent to expressing L

as:

u a2 q2

ax2 qx qax gas q2

= (ax q)(ax + q),

and the KdV equation becomes:

<=4>

Ut = tLxxx + 3uux

qxt 2qqt = qxxxx 6qqs2 3q2qxx 6q3qx-

0 = (qxt qXXXX 6qqX2 + 3q2q) -CX 2q(qt + 3q2qx)

<#. 0 = (ax 2q)(qt — 3q2qx).

Here, we recognize the second term as the mKdV equation; as a result, a solution q of the mKdV equation is also a solution of the KdV equation, by way of the Miura transformation u = qx — q2. If we consider u to be known, the Miura transformation cannot give us an explicit solution for q, in general,

30

but the idea of factoring a differential operator L into a product L1 L2 has a D-module interpretation, in that we have the following exact sequence of D-- modules:

0 —4 Dx/(L1) - * Dx/(L1L2) 4 Dx/(L2) 0, where the module homomorphisms a, ,3 are defined by:

cx(X + DxL1) := X L2 + DxLiL2 and 0(X + DxL1L2) := X + DL2.

Previously, we obtained the KdV equation by investigating whether the t,- family Dx/(L) can be extended to Dx,t/(L, at — P), so we would like to see whether the above t-family of exact sequences can be extended to an exact sequence:

0 Dx,t/(L1, at - P)Dx,t/(L1L2, at - P)0aDx,t/(L2,at - P) -4 0, where a' and /3' are module homomorphisms. The problem is, however, that because of the ideal (at — P), a' cannot be well-defined. Therefore, we restrict ourselves to asking if the sequence of t-families:

Dx/(L1L2) 4 Dx/(L2) 0

can be extended to the sequence:

Dx,t/(LiL2, at - P) 4 Dx,t/(L2, at - P) 0.

In fact, for the same reason that Theorem 6.1 holds, the following theorem also holds:

Theorem 6.3 Let L = a?, + qx — q2 = (ax — q)(ax + q) = L1L2 and P(t) in Dx,t be t-families of differential operators, where P has no non-zero terms in any power of at. Then the D-modules Dx,t/(L, at - P) and Dx,t/(L2, at - P) have ranks 2 and 1, respectively, if and only if:

at — P, L] = 0 mod L

and [at — P, L2] = 0 mod L2.

Given this result, let us show the following corollary:

Corollary 6.4 Let P = f + 9ax mod L. Then:

[at - P, L] = 0 mod L

and [at — P, L2] = 0 mod L2

if and only if:

1

ut = 29xxx + 2gxu + 9ux and qt = 29xx + gxq + 9qx

31

Proof: By Corollary 6.2, we immediately have that fs gxx; also, notice that P = f — gq mod L2. As such, we compute:

[a, - P, 11,2] = 0 mod L2 < L2 (at — P) 0 mod L2,

SO:

1,2(a, - p) = (ax + q)(0, - f + gq)

= at(-q) —f — f(—q)+ gxq + gqx + gqax + gat — qf +

—qt — fx+ qxq+ gqx

—qt + -lqxx + qxq + qqx

=0 mod L2 qt = 10:Ex + gxq gqx •

Thus, if we let g = —u q2 — qx, then the resulting PDE simplifies to:

qt = q2 xx (q2 q- )q q (q2 qx qx -

= qx2 qqxx qxxx 2q2qx qqxx q2qx q

= —0xxx 3q2fix) 1

which is the mKdV equation.

Here, we would like to recall the idea of a weighted homogeneous polynomial:

Definition 6.5 A polynomial p : Rn ---> R is said to be weighted homoge- neous whenever its variables x1,... xn are assigned weights w(xi) e Z such that the following equality holds for any e E R:

p(ca' xi, fan xn) = Eap (x xn), where a := Ei ai is the total weight of p.

Remark 6.6 There are two short observations to make about this definition:

the first is that, if w(xi) = a, then w(a) —ai; the other is that for E = 0, p(0, , 0) O.

Applying this idea to differential operators, let us consider two monic weighted homogeneous operators:

L Dx2 + u and

P 3 +Pn-laxn-1 + • • ±Plax PO;

where:

w(3) = = n — i; w(at) =n, =

and the coefficients pi are, themselves, weighted homogeneous differential poly- nomials in u (i.e. a polynomial expression in u and its derivatives).

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It is not difficult to show that for n = 1, we obtain the D-module:

Dx,t1(L, at — as) 4=> tit = ttx, and for n = 2, we obtain:

Dx,t/(L,at) 4=> ut 0-

As such, we would like to introduce the first non-trivial example, when n = 3.

Example 6.7 Let n = 3, so:

P = ax3 + h8 + gax + f

= (g — u)ax + (f — hu — ux) mod L.

By weighted homogeneity, w(f) = 3 or f 0, w(g) = 2 or g 0, and w(h) = 1 or h 0. By assumption that the coefficient functions are differential polyno- mials in u, where w (u) = 2, we must have Ii 0 and g = au for some a e R.

In addition, by Corollary 6.2, f = —laux, so:

P = 8,3„ + auax — aux

= (a — 1)uax — + 1)ux mod L, and the resulting PDE is the KdV equation:

ut --auxxx + 3auux.

As an interesting aside while we are considering 3rd-order differential opera- tors, let us return to our comment following Corollary 6.2. Instead of assuming that P is a weighted homogeneous operator, let us first make the stronger as- sumption that P and L satisfy:

[at — P, = O.

Then:

[at — F, U = If] — [P, = ut — [P, L} = 0, where:

[p, Li = (as3 h4 + + f)(a.2 + u) u)(ax3 h4 gax +1)

= (3u5 — hxs —2g5)4+ (3u55 + 2hu5 gxx — 2f5)&5 + (uxxx + huxx + gus — fxx).

Hence, ut [P, Id implies that:

hx 0 h constant,

3,u,x —2gx=0 u5,

3u55 + 2/nix gxx — 2f5 = 0 fx = hux + ;731- uxx , and uxxx + huxx + gux — fxx = Ut Ut = '1'11555 +

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which is again the KdV equation, corresponding to the choice of a = in the weighted homogeneous approach, and:

D_3a x+hx9x2++hu+ Ux

as is more commonly seen elsewhere. If we now additionally require P to be weighted homogeneous, of weight 3, under the restriction that w(u) 2 and w(ax) = 1, then h 0, which agrees precisely with the derivation of KdV via Lax forms (see [7]).

In fact, all of the discussion preceding Definition 6.5 has satisfied weighted homogeneity for n = 3, and the coefficients pi of P in Corollaries 6.2 and 6.4 turned out to be differential polynomials in u. Therefore, it would seem that the approach by D-modules and "mod" commutator relations gives another systematic approach to the derivation of integrable systems. In the next section, we will review the geometry associated to the mKdV and KdV equations, and draw some connections with the D-modules discussed above that generate these equations.

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7 Plane Curves and D-Modules

Let us first summarize the approach taken in Section 3, in deriving the mKdV equation (3.6) from deformations of the Frenet frame of a curve p E R2. Given p, we defined its arc length s, and by re-parameterizing p by s, its Frenet frame was found to be:

F [T = [p` Jp'] E SO(2, R).

This gave us the Frenet equation:

p" = kJp' <=> F-1F' = KJ E .51)(2,R),

and by Theorems 2.7 and 2.8, we saw that it is unique to p, up to action by SE(2). By then taking a t-deformation of p and expressing it with respect to F as:

Pt = fN + gT, we obtained the isoperimetric condition:

a

—F = Us + E .51)(2, !is =fit a t

and the integrability condition:

Kt = fss + gsh; + gKs.

The choice of f -ns implied gs --1Cn8 and g = tc2, which gave the mKdV equation (3.6):

= —nsss2Ks-

On the D-module side, we began with the module Dx /(L2), where L2 = ax+ q, and q is a smooth function of x and t. By taking a t-family of operators P =-- f - gq mod L2, where f, g are smooth functions of x and t as well, we saw that, by Theorem 6.3, the t-family of modules D ,/ (L2, at — P) has rank 1 if {at P, L2] = 0 mod L2, which is equivalent to the PDE:

qt = -.fx + gq + gqx•

The choice of fx --= - gxx and g q2 - qx then gave the mKdV equation:

qt = Aqxxx + 3g2gx.

Essentially, what is happening is that a curve p is being represented ab- stractly as a D-module Dx/(ax q), where the Frenet equation p" KJp' is the relation 3 + q 0. Although it means the same thing, the connection is perhaps slightly clearer if we consider the curve p E R2 as a curve p E C, so that the Frenet equation becomes (as — iK)ps = 0. As such, the time evolution Pt is a geometric expression of the t-family Dx,t/(L2, at - P), and the rank 1 condition on this extended D-module is simply the condition that Pt have a non-degenerate expression with respect to its frame for all t.

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With this in mind, we can see that the isoperimetric condition gs = f n is related to the operator P = f — gq mod L2, and the integrability condition on IC is identified with the "mod" commutator relation [at - P, L,2] = 0 mod L2.

However, there is one disparity to point out: given the isoperimetric condi- tion and the integrability condition, we need only choose f —Ks to obtain the mKdV equation; on the other hand, with the commutator relation mod L2, we

not only used the condition fx = Agxx from the KdV derivation, but we also

used the Miura transformation g = q2 — qx.

One possible interpretation of this is that we are implicitly working within the KdV/PSL(2, IR) framework without even realizing it in Section 3, whereas with the D-modules, we could only find mKdV after deriving KdV, by consid- ering an exact sequence of D-modules through the factorization L = L1 L2. Put differently, Dx,t/(L2, —P) is not (necessarily) a sub-module of D,/(L, —P), whereas 5o(2, R) C 5r(2, R). Another possibility is that choosing the arc length parameterization of p may correspond to a very specific form of P, which if properly accounted for in the D-module approach (i.e. for a certain choice of parameter x), would have led to the isoperimetric condition.

Let us now summarize the approaches taken in deriving the KdV equation.

Given a projective curve p = [x1 : x2] = : 1] E RP', parameterized on an interval M by a local coordinate u, the 2nd-order linear ODE:

x"(u) c(u)x(u) 0,

where c = Su(f) is the Schwarzian of f, was found to be uniquely associated to p up to linear fractional transformation. It was then shown that the above ODE has the fundamental solution set:

xi (u) = btof(u)

and x2(u)

If' (u)I Il(u)1

where p,i) E R* is a constant. We calculated that:

Ao = ^{Xi Xl,u}

X2 X2

,u

E R*

is constant, and we then "lifted" the projective curve p to the curve p E R2 \ {0}

defined by:

xi X2

P = (P1,732) :=( ±VIA or±VIA01))

and found the conformal Frenet frame of p to be:

Fcorif [p p'] E SL(2, R).

We then took a t-deformation of p E 1R2 \ 101 and expressed this in the

conformal Frenet frame as:

Pt = aP bPu,

36

where a, b conformal

are smooth condition:

(Fconf) —1

and again, c:

a at

functions.

Fconf

The Frenet frame

aau—be E 1(2

, R) b a+b u

we obtained an integ

The choice of a ^_ _{2c 2u}

rability condition,

was

this

shown

b•u =

time for ct = buuu + 2buc + bcu.

finally led us to the KdV equation:

to satisfy

—2a , the

the Schwarzian

Ct = cuuu + 3cc•u•

In contrast, using the 2nd-order differential operator L = a2, + u, where u is smooth in x and t, and the D-module Dx/(L), we considered a 1-family of operators P = f + gax mod L, for f, g smooth in x and t, and investigated the rank of the 1-family of D-modules Dx,t/(L, at — P). By Theorem 6.1, this family of D-modules has rank 2 for all t iff [at — P, L] = 0 mod L, which was found to be equivalent to the condition:

fx=-2gxx

and the PDE:

The choice of g

1 ut =a9

xxx+ 2gxu + gux.

_ —u was then shown to give the KdV equation:

ut = 2 uxxx + 3uux .

Much of the previous analysis holds in this situation, as well, for again, we see that a non-degenerate time evolution of a curve reflects the rank 2 property of the t-family of D-modules Dx,t/(L, at — P). In this case, the conformal and integrability conditions agree with the "mod" commutator relation; on the one hand, this may be a result of having not chosen any particularly distinguished parameterization of p E RP1, but on the other hand, we defined the conformal Frenet frame very carefully, by considering the solutions of the ODE associated to p.

This brings us to an interesting point regarding the Miura transformation.

In the D-module framework, we introduced the Miura transformation as a fac- torization of the operator L into:

L=ax2+u=LiL2

= (ax - q)(ax + q)

=a ..+qx-q2;

i.e. u = qx — q2 for some smooth function q. However, in Remark the context of projective curves and their lifts, we simply introduced algebraic formula, with little geometric meaning. We will explain this

5.4, it as here,

in an in

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