Aug 26, 2000 - Abstract. The motion of plane curves in Klein geometry is studied. It is shown that the KdV, Harry-Dym, Sawada-Kotera, Burgers, the defocusing ...
Integrable Equations Arising from Motions of Plane Curves Kai-Seng Chou
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Changzheng Qu
Department of Mathematics, Northwest University, Xi'an, 710069, P. R. China
August 26, 2000 Abstract
The motion of plane curves in Klein geometry is studied. It is shown that the KdV, Harry-Dym, Sawada-Kotera, Burgers, the defocusing mKdV hierarchies and the Kaup-Kupershmidt equation naturally arise from the motions of closed plane curves in SL(2)-, Sim(2)-, SA(2)- and A(2)-geometries. These local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area. Motions of curves in Euclidean, special linear and similarity geometries corresponding to the traveling wave solutions of the mKdV, KdV and Burgers equations are discussed. Keywords: Integrable equations motion of plane curves Klein geometry Lie algebra traveling wave
1. Introduction A variety of dynamics of shapes in physics, chemistry and biology are modeled in terms of motion of surfaces and interfaces, and some dynamics of shapes are reduced to motion of plane curves. These models are specied by velocity elds which are local or nonlocal functionals of the intrinsic quantities of curves. In physics, it is very interesting to describe motions of patterns such as interfaces, wave fronts and defects 1]. Applications include kinematics of interfaces in crystal growth 2, 3], deformation of vortex laments in inviscid uid 4], viscous ngering in a Hele-Shaw cell 5], etc. 1
The subject of how space curves evolve in time is of great interest and has been investigated by many authors. It was shown by Hasimoto 4] that the nonlinear Schrodinger equation arises from the dynamics of a string without stretching. Lamb 6] used the Hasimoto transformation to connect other motions of curves to the mKdV and Sine-Gordon equations. Langer and Perline 7] showed that the dynamics of non-stretching vortex lament in R 3 gives to the NLS hierarchy. Motions of curves in S 2 and S 3 were considered by Doliwa and Santini 8]. Lakshmanan 9] interpreted the dynamics of a nonlinear string of xed length in R 3 through the consideration of the motion of an arbitrary rigid body along it, deriving the AKNS spectral problem without spectral parameter. Motion of curves specied by accelerations in R 3 was considered by Tsurumi et al 10]. Recently, Nakayama 11] showed that the defocusing nonlinear Schroinger equation, the Regge-Lund equation, a coupled system of KdV equations and their hyperbolic type arise from motions of curves in hyperboloids in the Minkowski space. As compared to the motions of curves in space, only two types of integrable equations have been shown to be associated with motion of plane curves. In fact, Goldstein and Petrich 12] discovered that the dynamics of a non-stretching string on the plane produces the recursion operator of the mKdV hierarchy. Nakayama, Segur and Wadati 13] obtained the Sine-Gordon equation by considering a nonlocal motion. They also pointed out that the Serret-Frenet equations for curves in R 2 and R 3 are equivalent to the AKNS spectral problem without spectral parameter 14-18]. It is commonly believed that the KdV equation does not occur in the motion of plane curves. The purpose of this paper is to study the motion of curves on the plane in several Klein geometries. These geometries are characterized by their associated primitive Lie algebras of vector elds in R 2 . In fact, each primitive Lie algebra generates the isometry group of its corresponding Klein geometry. All primitive Lie algebras in the complex 2-plane was completely classied by Lie 19] and a corresponding classication in the real plane is also known 20]. Up to a change in independent and dependent variables, any primitive Lie algebra can be transformed into one of the following eight Lie algebras: e (2), so(3), sl(2), sim(2), sa(2), sa(2), so(3 1) and sl(3). We shall see that the KdV, Harry-Dym and Sawada-Kotera hierarchies and the Kaup-Kupershmidt equation naturally arise from the motions of plane curves in SL(2)-geometry. Motions of curves in Sim(2);, SA(2); and A(2)-geometries yield the Burgers, Sawada-Kotera and the defocusing mKdV hierarchies. The outline of this paper is as follows. In Section 2, we give a brief discussion on the Klein geometry. In Sections 3, 4, 5 and 6 we discuss motion laws of plane curves in SL(2)-, Sim(2)-, SA(2)- and A(2)-geometries. A complete classication for motions of plane curves which correspond to the traveling wave solutions of the mKdV equation is given in Section 7. In Section 8, we discuss the motion laws in 2
SL(2)- and Sim(2)-geometries which correspond to the traveling wave solutions of the KdV and the third-order Burgers equations. Finally, Section 9 contains some concluding remarks about this work.
2. Klein geometry In this section, we give a brief account of Klein geometry. Our basic references are 20] and 21]. The reader is referred to these works for denitions and proofs of results stated below. Let G be a Lie transformation group acts locally and eectively on the plane. Its Lie algebra g can be identied with a subalgebra of the Lie algebra of all smooth vector elds in R 2 under the usual Poisson bracket. According to the Erlanger Programme, every G or g determines a Klein geometry for plane curves via its invariants. To describe the invariants, let's assume a curve and its image under a typical element g in G are represented locally as graphs (x u(x)) and (y v(y)) over some intervals I and J respectively. A dierential invariant of g is a n-th smooth function dened on the n-jet space X U (n) for some n 1 satisfying (x u(x) u(n)(x)) = (y v(y) v(n)(y)) for all g 2 G . An invariant oneform, or more precisely a horizontal contact-invariant form, is a one-form dened in the n-jet space X U (n) , locally in the form d = P (x u(x) u(n)(x))dx, satisfying 0
Z
I
P (x u(x) u (x))dx =
for all g in G . Let
(n)
Z
J
P (y v(y) v(n)(y)dy
(1)
@ + (x u) @ v = (x u) @x @u
be an arbitrary vector eld in g. We denote its n-th prolongation vector eld on X U (n) by pr(n) v. The innitesimal criterion for the invariance of and d are given respectively by
pr(n) v() = 0
(2)
pr(n) v(P ) + P div = 0
(3)
and where div = x + uux. A basic result is
Theorem 2.1 21]. For any Lie transformation group acting locally and eectively
on the plane, there exist an invariant one-form d = Pdx and a dierential invariant 3
, both of lowest order such that every dierential invariant can be written as a function of and its derivatives D D2 where d: D = P1 dx Invariant one-forms of lowest order are unique up to the multiplication of a non-zero constant, and dierential invariants of lowest order are unique up to compositions of nonconstant functions. For each concrete geometry, we shall specify one such dierential form and call it the group arc-length element and one such invariant and call it the group curvature. Usually the latter is chosen so that it is linear in the highest order of u. As an illustration we look at the Euclidean geometry which is the Klein geometry associated to the Lie algebra generated by f@x @u u@x ; x@u g. p It is readily veried that one can choose its group arc-length element to be ds = 1 + u2x dx and group curvature k = uxx(1 + u2x) 3=2 : Next, consider the ane geometry which is associated to SA(2). The Lie algebra sa(2) contains all innitesimal Euclidean motions as its subalgebra. Here its group invariant is also a dierential invariant of Euclidean geometry. According to Theorem 2.1, it can be expressed as a function of k and its derivatives with respect to the Euclidean arc-length ds. In fact, one takes it to be k = k4=3 + 31 (k 5=3 ks)s. The ane arc-length element is given by d = k1=3 ds. In general, the order of derivatives involved in the group arc-length element and the group curvature increases with the dimension of the Lie algebra. ;
;
Since there are innitely many subalgebras of vector elds in the plane, it is important to introduce an equivalence relation to classify them. We call two subalgebras equivalent if one of them can be transformed to the other by a local dieomorphism in R 2 , i.e., by a local change of dependent and independent variables x and u. When restricted to those subalgebras whose corresponding Lie groups have no xed points, a classical theorem of Lie 19] classies all of them in the complex 2-plane. For the real plane, this has been done in 22], and the results can be found in Tables 1, 2, 3 and 6 in 22]. Let's focus on the class of primitive Lie algebras, which is the most interesting case. It turns out that there are precisely eight types of Lie algebras. We list basic facts concerning them in the following two tables. In Table 2 P4 = 3uxxuxxxx ; 5u2xxx Q5 = ks 2ksss ; 54 ks 3 kss2 ; k2ks 1 P5 = 9u2xxuxxxxx ; 45uxxuxxxuxxxx + 40u2xxx P7 = 13 u2xx(6P5Dx2 P5 ; 7(DxP5)2 ] + 2uxxuxxxP5DxP5 ; (9uxxuxxxx ; 7u2xxx)P52 ;
where k = uxx(1 + u2x) respectively.
;
3=2
;
;
and ds denote the Euclidean curvature and arc-length 4
In the following we shall consider the Klein geometries listed in these tables. For any parametrized curve , we dene its group tangent and group normal to be T = and N = respectively, where is the group arc-length. Notice that for Euclidean geometry, the group tangent is just the unit tangent, but the group normal is equal to the Euclidean curvature times the unit normal. On the other hand, the group tangent and group normal coincide with the ane tangent and ane normal in ane geometry 23]. A group invariant motion is of the form @ = f T + gN (4) @t where f and g are functions of the group curvature. We shall follow the procedure in 12] and 13] to derive group invariant motions whose corresponding equations for the group curvature are integrable. It can be described in the following four Name
Generators
Dimension Structure
E (2)
@x @u u@x ; x@u + (x@x + u@u)
3
R n R2
SL(2)
3
sl(2)
SO(3)
x@u u@x x@x ; u@u u@x ; x@u (1+ x2 ; u2 )@x +2xu@u 2xu@x + (1 ; x2 + u2)@u
3
so(3)
Sim(2)
@x @u x@x + u@u u@x ; x@u
4
R2 n R2
SA(2)
@x @u x@x ; u@u x@u u@x
5
sa(2)
6
a(2)
6
so(3 1)
8
sl(3)
A(2)
@x @u x@x u@u x@u u@x x@x + u@u u@x ; x@u SO(3 1) (x2 ;@ux2 )@@u +2 x xu@u 2xu@u +(u2 ; x2 )@u @x @u x@x u@x x@u u@u SL(3) x2 @x + xu@u xu@x + u2@u
Table 1: Primitive Lie algebras of vector elds in R2 steps. First, we determine the Serret-Frenet formulas of each geometry. It is of the 5
form
0 1 0 10 1 BB A B CC BB T CC @ B B@ T C C = A @ A@ A @ N
N
C D
(5)
In some occasions this system is the AKNS system without spectral parameter. Second, we compute the rst variation for the group perimeter
L=
I
d
for a closed curve driven under (4) to obtain dL = I Fd (6) dt where F depends on f and g in (4). By choosing f and g such that F vanishes pointwisely we ensure that @=@t @=@ ] = 0. Third, we compute the time evolution of T and N to get
0 1 0 10 1 BB A B CC BB T CC @B B@ T C C = A @ A@ A @t 0
N
C D
0
0
N
0
(7)
Finally, the compatibility condition between (5) and (7)
0 1 0 1 @2 B B@ T CCA = @2 BB@ T CCA @t@ @@t N
N
gives the general equation for the curvature. By choosing f and g suitably we obtain integrable equations. Let's recall how this procedure was applied to the Euclidean motion. We write, instead of (4), @ = f~n + g~t (8) @t where t and n are the Euclidean tangent and normal, We have the Serret-Frenet formulas
ts = kn ns = ;kt 6
(9)
Name
E (2) SL(2)
Arc-length
Curvature
p
exp( arctan ux) 1 + u2xdx exp(; arctan ux)(1 + u2x)3=2 uxx
ds~ = (xux ; u) 2uxxdx
= (xux ; u)3uxx1
;
SO(3) 2(1 + x2 + u2)
1
;
;
p1 + u2 dx
(1+x2 +u2 )uxx (1+u2x )3=2
x
+ 2(pu 1+xuux2 ) ;
x
Sim(2)
d = (1 + u2x) 1uxxdx
= (1 + u2x)uxxx ; 3uxu2xx]uxx2
SA(2)
d = u1xx=3 dx
= uxx8=3P4
A(2)
dl = uxx1 P4dx
SO(3 1)
;
;
p(1+u )uxxx 2
x
u
;
;
p
3ux u2xx
;
1+ 2x
= P4 3=2P5 ;
Q5
dx
uxx1P51=3dx
SL(3)
P7 P5 8=3 ;
;
Table 2: The arc-length and curvatures in the spaces of Table 1 and, by the formula
st = s(~gs ; kf~) the rst variation formula for the Euclidean perimeter reads as @L = I (~g ; kf~)ds: s @t So, we require g~ = @s 1 (kf~) and I ~ = 0: kfds ;
We also have
0 1 BB t CC @ A n
0 B =B @ t
0
;f~s ; kg~ 7
10 1 f~s + kg~ C B t C CA B@ CA 0
n
(10)
(11) (12)
(13)
From @=@s @=@t] = st @s = 0, the compatibility condition between (9) and (13) gives kt = f~ss + ksg~ + k2 f~ (14) and by (11)
kt = ; f~ (15) where = Ds2 + k2 + ks@s 1 k: By choosing f~ = n 1 ks and g~ = @s 1 k n 1 ks so that (12) holds, we get the mKdV heirarchy. ;
;
;
;
To conclude this section, we apply the same procedure to the nonstandard Euclidean geometry E (2), 6= 0. By an analogous computation, the E (2)curvature k satises the equation
k t = Ds2^ ; 2k s^ ; 2k Ds^ + (2 + 1)k2 + (2 + 1)k s^@s^ 1 k ]f ;
where s^ is the E (2) arc-length. Taking f = ;k s^ we get k t + k s^s^s^ ; 2(k k s^)s^ + 32 (2 + 1)k2 k s^ = 0: Unless = 0, the equation is not integrable. It seems that there are no integrable equations associated to E (2) when 6= 0. Things will change when we come to other Klein geometries.
3. Motions of curves in special linear geometry The isometries of the special linear geometry are the linear transformations from the special linear group SL(2). The SL(2) arc-length is given by
ds~ = kh 2ds ;
where h = ; n is the support function of , now regarded as a function of the parameter of the curve. Similarly, its curvature can be written as
= hk
3
The natural objects in special linear geometry are convex curves. The SL(2)tangent and normal are related to the Euclidean tangent and normal by the formulas
T = k 1 h2t N = k 1 h2(k 1 h2)st + k 1h4 n: ;
;
;
8
;
Using the Serret-Frenet formulas (9) in R 2 and the following identities 2 s~s~ 2 1 2 1 2 2 1 4 1 4 h k (h k )ss ; h (h k )s(h k )s ; h = ; 2 2s~ ; (h2 k 1)s + h 2 (h4k 1 )s = 2s~ ;
;
;
;
;
;
;
;
we obtain the SL(2) Serret-Frenet formulas 2 Ts~ = N Ns~ = 2s~ N + ( s~s~ ; 2 2s~ ; )T (16) Next we compute the rst variation of the SL(2)-perimeter. First of all, we express (4) in the form (8), where now f~ = k 1h4 f g~ = k 1h2 g + k 1 h2(k 1 h2)sf: By (10), (14) and ht = ;f~ + (f~s~ + kg~) t: we have dL = I (h 2k ; 2h 3kh + h 2ks 1 s )ds t t t dt I 2 = gs~ + fs~s~ + 2 s~ fs~ + 2( s~s~ ; 2s~ + )f ]ds~: We require I 2 (17) fs~s~ + 2 s~ fs~ + 2( s~s~ ; s2~ + )f ]ds~ = 0: and g = ;fs~ ; 2s~ f ; 2@s~ 1 (f ): The evolution of T and N is given by ;
;
;
;
;
;
;
;
;
0 1 BB T CC @ A N
where
t
0 10 1 B A B CC BB T CC =B @ A@ A C D
N
2 A = gs~ + ( s~s~ ; 2 2s~ ; )f B = fs~ + g + 2 s~ f 2 C = As~ + ( s~s~ ; 2 2s~ ; )B D = Bs~ + A + 2 s~ B:
9
(18)
The compatibility condition between (16) and (18) implies that
t = ;fs~s~ ; s~fs~ ; (s~s~ + 42 ; 2 12s~)f + s~g: ;
Substituting the expression for g into this equation, we nally obtain
t = ;fs~s~ ; 2s~fs~ ; (s~s~ + 42)f ; 2s~@s~ 1 (f ): ;
(19)
where f satises (17). We now consider several cases:
Case 1. f = u=. In this case, (19) becomes t = ;(Ds2~ + 4 + 2s~@s~ 1 )u: Example 1. u = ; n1 1 s~. We get the KdV hierarchy t = n1 s~ n 1 ;
;
where 1 = Ds2~ + 4 + 2s~@s~ 1 is the recursion operator of the KdV equation 24]. ;
The rst member is the KdV equation
t = s~s~s~ + 6s~ and the second one is the fth-order KdV equation
t = 5 + 103 + 2012 + 3021 which can also be obtained directly by taking f = ;(s~s~s~= + 6s~) in (19). Here and hereafter ui, i = 1 2 denotes the i-th derivative of u with respect to the indicated arc-length. Example 2. u = ;
;
3=2
s~@s~ 1 (q) + 21=2q. Letting = ;
1 =2
;
, satises
t = (Ds2~ ; s~Ds~ + s~s~ + 2s~s~s~@s~ 1 2) + 4]q: ;
;
Taking q = 0 in (20), we get the Harry-Dym equation
t = 3s~s~s~: Setting q = n2 1 s~, we get the Harry-Dym hierarchy ;
t = n2 s~ where 2 = 2 Ds2~ ; s~Ds~ + s~s~ + 3s~s~s~@s~ 1 2 + 4 ;
10
;
(20)
is a recursion operator of the Harry-Dym equation 25].
Case 2. f = us~s~s~= + us~. In this case, (19) becomes t = ;Ds5~ + 5Ds3~ + 4s~Ds2~ + (s~s~ + 42)Ds~ + 2s~@s~ 1 Ds~]u Taking u = ;, we have the Sawada-Kotera equation 26, 27] ;
t = 5 + 53 + 512 + 521 : (21) Next, we take u = ;@s~ 1 (Ds2~ + + s~@s~ 1 )q, q = n3 1 ()s~, where 3 () = (Ds3~ + 2Ds~ + 2Ds~)(Ds3~ + Ds2~@s~ 1 +@s~ 1 Ds2~ + 21 (2@s~ 1 + @s~ 1 2)) is the recursion operator of the Sawada-Kotera equation 28]. By a direct computation, the following identity holds (Ds4~ + 5Ds2~ + 4s~Ds~ + s~s~ + 42 + 2s~@s~ 1 )(Ds2~ + + s~@s~ 1 ) = (Ds3~ + 2Ds~ + 2Ds~)(Ds3~ + Ds2~@s~ 1 + @s~ 1 Ds2~ + 1 (2@s~ 1 + @s~ 1 2)): 2 Using this identity we see that satises the Sawada-Kotera hierarchy t = n3 ()s~ ;
;
;
;
;
;
;
;
;
;
;
;
;
Case 3. f = ;(s~s~s~= + 16s~). The resulting equation is the Kaup-Kupershmidt equation 29, 30] t = 5 + 203 + 5012 + 8021: It is very interesting to observe that the Sawada-Kotera, the fth-order KdV and the Kaup-Kupershmidt equations arise from (19) by the choices f = ;(s~s~s~= + s~) for = 1 6 and 16: In other words, they are embedded in the family of evolution equations t = 5 + ( + 4)3 + (3 + 2)12 + 521: It is not known whether this equation is integrable for = 10m + 6 m 2: Case 4. f = 4s~. We have ;
t = 12 ( 2)s~s~s~ + 3( 1)s~ which is an integrable equation 31]. ;
;
It is easy to see that in these four cases the motions also conserve the enclosed area of the curves. Notice that the area does not change under SL(2)-actions and so it make sense in special linear geometry. 11
4. Motions of curves in similarity geometry The similarity algebra is obtained by adding the dilatation vector eld to e0(2). Here the Sim(2) arc-length is given by d , where is the angle between the tangent and the x-axis. The curvature in similarity geometry is related to the Euclidean curvature by the Cole-Hopf transformation
= (ln k) = k 2ks: ;
Curves with constant similarity curvature are circles and logarithmic spiral waves given by
x(p) = a2 + a1e p( cos p ; sin p) y(p) = a3 + a1e p( sin p + cos p) ;
;
where ai , i = 1 2 3 and are constants. Using T = k 1 t and N = k 1n ; k 3kst, the Serret-Frenet formulas in similarity geometry are given by ;
;
;
T = N N = ;2 N ; ( + 2 + 1)T:
(22)
We express the motion (4) in the form (8), where now f~ = k 1f and g~ = k 1 (g ;
f ). The rst variation of the similarity perimeter is given by dL = I (k s + ks )dp t t dt I = f~ss + ( g)s]ds: ;
;
Hence dL=dt always vanishes for any closed curve. However, it still makes sense to set
g = ;f + 2 f + a a = const
(23)
so that @=@ @=@t] = 0 for any f and g related by (23). The evolution of the similarity tangent and normal are given respectively by
Tt = ;k 2 ktt + k 1tt = ;k 1 (Lf + a )t + ak 1 n = ;Lf T + aN ;
;
;
;
Nt = k 1(n ; t)t ; k 2 kt(n ; t) = ;(Lf + 2a )(N + T) ; (Lf + a ) ; (LF + a ) + a]T = ;(Lf + 2a )N ; (Lf + a ) + a( 2 + 1)]T: ;
;
12
Hence
0 1 BB T CC @ A N
0 10 1 B ;Lf a CC BB T CC =B @ A@ A t
Q
P
N
(24)
where Q = ;(Lf + a ) ; a( 2 + 1), P = ;(Lf + 2a ) and L = (@ ; )2 + 1 is a linear operator. The compatibility condition between (22) and (24) yields the following equation for the Sim(2)-curvature after using (23),
t = D3 ; 2 D2 ; (3 ; 2 ; 1)D ; ( ; 2
)]f + a :
(25)
where f is an arbitrary function. The simplest choice is f = ;1. Then (25) becomes the Burgers equation
t = ; 2
+ a : The next choice is f = , which yields the third order Burgers equation 31]
t = ; 3
; 3 2 + 3 2 + (a + 1) In general, setting f = @ 1 u, the equation is changed to ;
t = (D ; ; @ 1 )2 + 1]u + a : ;
When u = n4 2 , we obtain the Burgers hierarchy ;
t = ( n4 + n4 2 + a) ;
where 4 = D ; ; @ 1 is the recursion operator of the Burgers equation 24]. These equations can be linearized by the Cole-Hopf transformation = (ln ) , where is the reciprocal of the Euclidean curvature = 1=k. Indeed, the hierarchy is transformed to ;
t = Dn + Dn 2 + a: ;
It is noted that the motion conserves enclosed area of the curve only when n is odd.
13
5. Motion of curves in ane geometry The isometries of ane geometry contains all SL(2)-isometries plus the translations in x and y. The ane arc-length d and curvature are given in terms of the Euclidean arc-length and curvature by
d = k1=3 ds and
= k 34 + 13 (k 53 ks)s: ;
Curves with constant ane curvature are parabolas ( = 0), ellipses ( > 0) and hyperbolas ( < 0). One may consult 23] for a discussion on ane geometry. The ane Serret-Frenet formulas are given by
0 1 BB T CC @ A N
0 B =B @
0
10 1 1 CB T C CA B@ CA N
; 0
(26)
The ane tangent and normal are related to n and t via
t = kT n = 13 k ksT + k N: 1 3
;
5 3
;
1 3
It follows that (4) can be expressed in the form (8) where f~ = k 31 f g~ = k 31 g ; 13 k 53 ksf: By a direct computation f~ss = 31 (k 23 ks)sf + k 31 ksf + kf g~s ; kf~ = g ; 31 k 43 ks(g + f) ; f: ;
;
;
;
;
Substituting these equations into the evolution equations for s and k, (10) and (14) we have st = sg ; 31 k 43 ks(g + f) ; f ] kt = kf + k 43 ks(f + g) + f ]: ;
;
14
Hence, the rst variation of the ane perimeter satises dL = I ( kt + st )d dt I 31k s 2 = ( 3 f ; 3 f + g)d : We impose
I
and
( 31 f ; 32 f )d = 0
(27)
g = ; 13 f + 32 @ 1 ( f ):
(28)
;
On the other hand, we have
0 1 BB T CC @ A N
0 10 1 B g ; f f + g CC BB T CC =B @ A@ A
t
H1
H2
(29)
N
where H1 = g ; 2 f ; f ; g and H2 = f + 2g ; f . Under (27) and (28), @=@ @=@t] = 0, and so the compatibility condition between (26) and (29) implies
t = (D4 + 5 D2 + 4 D + + 4 2 + 2 @ 1 )f ;
(30)
after using (28). If we take f = in (30), we get the Sawada-Kotera equation again. Notice the analogy between the choices in Euclidean geometry and ane geometry suggests that the Sawada-Kotera equation is an ane version of the mKdV equation. If we take f = 3 + 2 1, we obtain a seventh-order Sawada-Kotera equation 3 t = 7 + 7 5 + 14 1 4 + 21 2 3 + 14 2 3 + 42 1 2 + 7 31 + 28 3 1 + a 1: In general, we may take f = (D2 + + @ 1 )u, where u = n3 1 ( ) , to obtain the Sawada-Kotera hierarchy again. ;
15
;
6. Motion of curves in fully ane space The isometries of fully ane geometry are compositions of the ane isometries and the dilatation (x u) ! (x u). Its arc-length is given by
l=
Zs 0
12 k 13 ds
and its curvature is
= 32 = 1 l:
(31)
;
;
The fully ane tangent and normal satisfy
T = k t N = 1k n + k ( k )st: ;
Hence
1 2
;
1 3
1 3
;
;
1 2
;
1 3
;
1 2
;
1 3
t = k T n = k N ; k ( k )sT: 1 2
We compute
1 3
;
1 3
;
1 3
;
1 2
;
1 3
Tl = ( k ( k )s)s ; 1k ]T + ;
1 2
;
1 3
;
1 2
;
1 3
4 3
;
12 k 13 ( 1k 13 )s + 1k 13 ( 21 k 31 )s] k 13 N ; k 31 ( 21 k 13 )sT] ;
;
;
;
;
;
;
;
;
;
By (31), we have 1 1 1 1 2 5 2 k 3 ( 2 k 3 )s]s = ;( 12 2k 3 s + 31 1k 3 ks)s = 14 2 ; 12 l ; 19 + 1k 34 + 12 12 k 43 ks ;
;
;
;
;
;
;
;
;
and
;
;
k 31 21 k 13 ( 1k 31 )s + 1k 13 ( 21 k 31 )s] = ; 23 : Thus the A(2) Serret-Frenet formulas now read as ;
;
0 1 BB T CC @ A N
;
0 B =B @
l
;
;
;
0
1
; 21 ( 2 + l + 29 ) ; 32
;
10 1 CC BB T CC : A@ A N
Motion of curves in SA(2)-geometry is specied by (4) or (8), where
f~ = 1k 31 f ;
g~ = 21 k 31 g + 21 k 31 ( 12 k 13 )sf: ;
;
;
16
;
;
;
(32)
The rst variation of the fully ane perimeter is dL = I ( t + kt + st )dl: dt 2 3k s As usual we require f and g are chosen to satisfy t + kt + st = 0 (33) 2 3k s By the denition of , we have 11 2 8 5 3 k )k ; 10k 3 k k + 3k 3 k t = (12k 31 ; 5k 38 kss + 40 k (34) t s st sst : s 3 By a tedious computation we have kt = k(fl ; 32 f )l + 12 k 13 ks(fl + g) + k 21 2 + l + 91 ; 32 21 k 43 ks ]f kst = 21 k 34 (fl ; 32 f )ll + ks(2fl ; 23 f )l + 12 21 k 43 ( 2 + l )l f + 21 k 31 ( 1 k 53 ; 3k3 + 5 k 1ks2 )(fl + g) 3 3 1 4 1 4 1 4 1 8 1 2 2 + 2 2 k 3 ( + l + 9 ; 3 2 k 3 ks )fl ; ( 21 2 k 3 ; 29 2 k 3 ) f 4 1 +ks( 2 ; 12 l + 92 ; 25 2 k 3 ks )f ;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
ksst = k 35 (fl ; 23 kf )lll + 12 (fl ; 23 f )ll + (fll ; 21 f )l + ( 2 + l )l fl + 21 ( 2 + l + 29 ; 3 21 k 34 ks )fll + 14 ( 2 + l + 92 ; 3 21 k 43 ks )fl ; fl + ( 21 ( 2 + l )ll + 14 ( 2 + l )l ; 21 l ; 21 k 43 ks + 91 )f ] + 21 k 13 ks( 10 fl ; 7 f )ll + 3( 5 k 1ks2 ; 3k3)(fl ; 1 f )l + 9 k3 ( f )l 3 2 3 2 2 1 1 1 5 5 1 7 2 5 1 35 +( 2 k 3 ks ; 19 2 k 3 ks + k 3 + 2 k 3 ks )(fl + g) 3 3 9 1 1 1 1 1 1 4 10 29 +( 12 2 k 3 ks 2 ; 3 2 k 3 ks l + 27 2 k 3 ks ; 5k 1ks2 + 92 k3 )fl 1 5 1 1 1 1 + 2 k 3 ks( 53 2 + 61 l )l ; k3( 92 2 + 1) ; 23 ( 2 k 3 ; 19 2 k 3 )ks 1 7 2 k 3 k 3 ]f + 52 ( 2 ; l + 29 )k 1ks2 ; 35 s 6 and st = sgl ; 23 k 34 ( 12 k s + 13 ks)(fl + g) + 41 ( 2 ; 2 l ; 49 + 2 21 k 34 ks )f ]: 17 ;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
Substituting these into (34) we get t = 3 A + 3 A + 2 A + 1 D 2 2 ll 4 l 3 2 and kt + st = D ; 2 A ; 1 D l 3k s 3 2 where A = (fl ; 23 f )l + 12 ( 2 + l + 29 )f D = g + fl ; 23 f: So (33) is equivalent to D l + 32 A ll + 34 A l = 0: In other words, we should choose (35) D = ; 32 A l ; 34 @l 1 ( A l ): To avoid lengthy computation, we shall derive the equation for the fully ane curvature directly. First of all, it follows from (33) that tl = ;2 l ( 3kkt + sst ) ; 2 ( 3kkt + sst )l : By the denition of SA(2) arc-length, t = ( k 31 s 1 )t = t ; ( 3kkt + sst ): Therefore
t = ; 23 25 t + 23 t ; ( 3kkt + sst )] = ; 3 1 t + 1 tl ; ( kt + st ) 2 3k s k t st = ;2( + )l 3k s = (3Dl2 ; 34 2 ; 34 l @l 1 + 34 )A l ;
;
;
;
;
;
;
;
where (35) has been used. Setting A~ = A l , we get the evolution for the fully ane curvature ~
t = 3Dl2 ; 14 2 ; 14 l @l 1 + 94 ]A: (36) 18 ;
Plugging A~ = 13 n5 1 l in (36) yields the defocusing mKdV hierarchy
t = n5 l where 5 = Dl2 ; 2=4 ; l @l 1 =4 + 4=9 is a recursion operator of the defocusing mKdV equaton
t = lll ; 38 2 l : So the defocusing mKdV hierarchy arises from a nonlocal motion of plane curves in fully ane geometry. ;
;
7. Motion of curves corresponding to traveling waves Just as in Euclidean geometry where the Euclidean curvature of a curve determines the curve itself up to a Euclidean motion, the curvature of a curve in any Klein geometry listed in Table 1 determines the curve itself up to an isometry of the geometry. In this sense the geometric invariant motion (4) and the integrable equations satised by its curvature are equivalent. However, it is not a straightforward matter to write down formulas relating the curvature and the curve except in the Euclidean case. This makes the study of various properties such as how symmetries and global solvability of the motion and the integrable equations are related a nontrivial issue. In this section, we discuss the motion of the curve whose curvature is a traveling wave (soliton) of the mKdV equation. This problem was studied in 13] and 32] using elliptic integrals when c0 vanishes. Here we use another approach and obtain results for all c0 . Recall that the mKdV equation
kt + ksss + 32 k2 ks = 0 is associated to the Euclidean invariant motion 2 t = ;ksn ; k2 t: In general, for any given function k(s t), we dene (s t) = and
x(s t) = y(s t) =
Zs 0
k( t)d + 0 (t)
Zs cos ( t)d + x0(t) 0 Zs 0
sin ( t)d + y0(t) 19
(37) (38) (39)
(40)
where 0 (t) x0 (t) and y0(t) are given functions of t. One can easily verify that for each xed t the curve (s t) = (x(s t) y(s t)) is parametrized by arc-length and its curvature given by k(s t). When k(s t) solves (37), we can determine 0 (t), x0 (t) and y0(t) by substituting (39) and (40) into (37) to nd Zt 0 (t) = ; (kss(0 ) + 21 k2 (0 ))d + 0 (41) 0 and Zt x0(t) = (sin (0 )ks(0 ) ; 12 cos (0 )k2(0 ))d + x0 Z0 t y0(t) = (; cos (0 )ks(0 ) ; 12 sin (0 )k2(0 ))d + y0: (42) 0 So (39)-(42) dene a curve (s t) = (x(s t) y(s t)) satises the mKdV ow (38). Here 0 , x0 and y0 are arbitrary constants, re ecting the fact that the curvature determines the curve up to a Euclidean motion. In the following, we usually choose x0 = y0 = 0 = 0 so that the initial curve passes through the origin with the unit tangent e1 . Now, suppose that k(s t) = k^(s ; t) is a traveling wave of the mKdV equation. Let ^ be the curve determined by k^:
Zs cos ^()d Z0 s
x^(s t) = y^(s t) = where
^(s) = Notice that k^ satises
sin ^()d
0
Zs 0
k^()d:
k^ ; k^ + 12 k^3 = c0
(43)
k^2 ; k^2 + 41 k^4 = 2c0k^ + c1
(44)
00
and
where = s ; t. It follows from (44) that all solutions k^ are periodic unless c0 = c1 = 0. Let k^ be a solution of (44) with minimal period T and let =
ZT 0
k^()d:
20
The number measures the total change in tangent angle of the curve with curvature k^ along a period. It is easy to see that ^ is closed if and only if there are some positive integers m and n such that mj j = 2n. And it has m leaves and total curvature 2n. On where the curvature does not vanish, we can use the tangent angle to parametrize ^. Let !( ) = k^2( ). It is easily to verify that ! satises the equation 2c0 : ! = 2 ; ! + p (45) ! Depending on the sign of c0, we consider three cases separately.
Case 1. c0 = 0. The general solution of (45) is given by ! = 2 + cos( ; 0 ), 0 2 R . In this case, clearly p it is sucient to consider 0 and 0 = 0. ^ The curvature is given by k = 2 + cos . When = 0, ^ is a circle of radius p 2=2. For any 2 (0 2), it is locally convex and consists of innitely many loops.
The loops are almost circular for small and, as increases, they are less circular and the distance between the loops become larger. At = 2, the solution has a single loop. In fact, it corresponds to the 1-soliton of the mKdV equation given by k^ = 4e2 =(1 + e4 ). As goes beyond to 2, k^ is well-dened in ; 1 1] where cos 1 = ;2=, 1 2 (=2 ). The corresponding curve is a convex arc ; which is symmetric with respect to the y-axis. It has total curvature 2 1 and the curvature vanishes at its endpoints. It can be extended to a complete curve with curvature k^ by putting innitely many copies of itself together. To describe the extension, we temporally take the right endpoint as the origin and assume the y-axis is along the tangent at this endpoint such that ; lies locally on the left of the y-axis. Let ; = f(x y) ;(x y) 2 ;g. It is easily to see that ; ; is a smooth arc whose curvature is k^. Similarly, we can extend the curve from the other endpoints. By doing this indenitely we obtain a complete curve ^ whose curvature is k^. Notice that the total curvature from the tip of ; to the tip of ; is . So, ^ is periodic along the whole x-axis provided the two endpoints of ; are distinct. When the two endpoints coincide, the complete curve closes up and become an eight-gure. It was shown in 32] that this really happen at = 5:480226 . As goes to 1, the total curvature of ; converges to . So eventually it comes very near to an entire graph over x-axis. The transition of ^ as increases from 0 to 1 is shown in Figure 1. 0
0
0
Case 2. c0 > 0. Let ! = ! be the solution of (45) subject to the initial conditions !(0) = > 0 and ! (0) = 0. It is an even function. Denote by 0 and 1 the respective positive roots of !3=2 ; 2!1=2 ; 2c0 = 0 and !3=2 ; 4!1=2 ; 8c0 = 0. Notice that 0 is always less than 1 . It is obvious from (45) that ! is concave in the region f! > 0g and convex in f! < 0 g. Any solution ! starting at 2 (0 1) is positive and periodic. The period = () produces a closed curve ^ if and only 0
21
Figure 1: Curves for c0 = 0 when =0, 0.8, 1.8, 2, 3,06, 5.48, 7,14, 16, 64.
22
if () = 2n=m for some positive integers n and m (Figure 2). Numerical studies show that () is not monotone in . For certain values of c0 , it rst increases and then decreases after goes beyond 0 . The solution ! 0 determines a circle p of radius 1= 0. The solution starting at = 1 touches the -axis tangentially at some 1 , i.e., ! ( 1) = 0. It is still periodic but now it has in ection points. However, when > 1 , ! hits the -axis at some 1 with ! ( 1) < 0. It determines a nite convex arc. To extend it to a complete curve we need to match it with suitable arcs arising from Case (3a) or Case (3b) below.
Case 3. c0 < 0. The curves y = x2 and y = 2 ; 2jc0jx intersect or are disjoint depending on jc0j. At c = 0:544331 they touch tangentially at one point.
Case 3a. jc0 j c . In this case ! is concave for all 2 (0 1) and has a rst zero 1 > 0. The set f 1 () : 2 (0 1)g has a universal upper bound depending on c0. As jc0j tends to c , this bound goes to innity. The curve corresponding to ! is a nite arc. It can be matched with similar arcs from Case 2 (for jc0j and
> 1) to form a complete curve ^. If we let 1 be the rst zero of ! from Case 2 which is attached to this arc, ^ is closed if and only if j 1 ; 1 j = 2n=m for some positive integers m and n. 0
0
Case 3b. jc0 j < c . Let !1 < !2 be the two positive roots of !3=2 ; 2!1=2 ; 2jc0j = 0. Then ! is concave in f! > !2 or ! < !1g and convex in f!1 < ! < !2g.
The curves corresponding to !1 and !2 are circles. It is easy to see that for any 2 (!1 1), the solution ! is positive and periodic. And the period could be arbitrarily large as comes close to !1 and tends to as goes to 1 (for u = != tends to u + u = 0 and u(0) = 1, u (0) = 0 whose solution is j cos j as ! 1). For any number n=m 2 (1=2 1) there exists a closed curve ^ with m leaves and total curvature 2n. Some of these curves are plotted in Figure 3. When 2 (0 !1), the solution hits the -axis at a rst zero 1 > 0 with ! ( 1) < 0 the same as in Case 3a. It can be matched with arcs from Case 2 (jc0j and > 1 ) to form a complete curve ^. Finally, we examine the motion of the curve determined by k^. By putting k(s t) = k^(s ; t) into (39)-(42) and using (43), we nd that (s t) is given by
(s t) = R(c0t)(^ (s ; t) ; X(t)) where
2 6 R(c0t) = 64
3 cos c0 t sin c0 t 7 75
; sin c0t cos c0t 23
(46)
Figure 2: Curves for c0 = 2 and (m n)=(6,5), (7,6) and (8,7).
Figure 3: Curves for (c0 m n) = (;0:5 2 3) (;0:5 3 5) and (;0:25 6 7).
0 1 B x0(t) CC X(t) = ^(;t) + R 1 (c0t) B @ A ;
y0(t)
and (x0 (t) y0(t)) is given by
x0(t) = y0(t) =
Z
t
; sin( ^( ) + c0 )k^ ( ) + 1 cos( ^( ) + c0 )k^2 ( )]d 2 t cos( ^( ) + c0 )k^ ( ) + 12 sin( ^( ) + c0 )k^2 ( )]d:
;
Z0 0
0
;
0
When c0 = 0, we can use the explicit form of ! to nd that X(t) = (;=2 0). Hence under this ow, translates along the x-axis with speed =2. On the other hand, when c0 6= 0, the motion is composed of a uniform rotation of speed c0 and a motion along the trajectory X (t). It is like a spinning top whose tip traces out a path. 24
In conclusion our study shows that symmetries of the integrable equation and the corresponding motion are related in some delicate way. For example, we do not know whether the special solution given in (46) is the group invariant solution for some generalized symmetry of the motion. It would be interesting to study other group invariant solutions such as the self-similar solutions for both equations as well. Finally we would like to point out that out approach can be applied to nonintegrable equations. For example, consider the generalized mKdV equation m + 3 km+2k + ck = 0 kt + (kmkss)s + m s s +2 which is obtained by setting f~ = kmks in (15) with n = 1. The equation corresponding to (45) is
! = (m + 2)(1 ; c) ; w ; (m +12)c0 w m+2 and we can use the same method to analyse it.
c0 = const:
8. The KdV and Burgers equations Now we examine the traveling wave solutions of the KdV equation. Let u = (x ; ct) be such a solution. It satises
uzz + 3u2 ; cu = D and
1 u2 + u3 ; 1 cu2 ; Du ; E = 0 2 z 2 for constants D and E . Only when the equation u3 ; 21 cu2 ; Du ; E = 0 has three real roots, the solution is bounded. Let's consider the generic case that there are three distinct roots c1 , c2 and c3, c1 < c2 < c3 rst. The solution is given by
u = u0 + Acn2 az where
z = x ; ct
u0 = 61 c2 A = 61 (c3 ; c2) c = 13 (c1 + c2 + c3) 1 (c ; c ) k2 = c3 ; c2 a2 = 12 3 1 c ;c 3
25
1
and k is the modulus of the Jacobi elliptic function cn z. The (real) period of cn z is given by 2K where Z =2 p d2 2 : K= 0 1 ; k sin Given a solution u of period " = 2K=a, we would like to determine the solution curve whose SL(2)-curvature is u. First we solve for its SL(2)-tangent, which by (16) satises the linear ODE 2 Tzz ; 2uuz Tz ; ( uuzz ; 2uu2z ; u)T = 0: Letting H = uT, H satises the Hill's equation
Hzz + uH = 0: Notice that the curve is given by
^(z) =
Zz 0
(47)
T(s)ds + ^(0):
(48)
Lemma 8.1. Let u be of period ". Then ^ closes up and has m leaves provided the function H given in (47) is of period m".
Notice that here ^ has m-leaves means that it can be decomposed as the disjoint union of m arcs which are SL(2)- copies of each other.
Proof. Clearly T is also of period m". It follows from (48) and (47) that (z + m") = (z) for arbitrary z 2.
Now the question of nding closed curve ^ becomes to nding m"-periodic solutions of H. Let's write (47) as
Hzz + (u0 + 2a2 k2) ; 2a2k2 sn2 z]H = 0: Further, letting y(x) be any component of H(x=a), y satises the equation y + ( ; 2k2 sn2 x)y = 0 = c 2;c3c :
(49)
00
3
1
This is the Lame's equation with m = 1 (x7.3 in 33]). Much is known for this equation. As relevant to our discussion, we need the fact that for each modulus k, there exists 0 such that (49) is stable (i.e. all solutions are bounded in z) for all 0 . More precisely, we let y1 and y2 be solution of (49) satisfying y1(0 ) = 1 y1(0 ) = 0 y2(0 ) = 0 y2(0 ) = 1 and #() = y1(2K ) + y2(2K ). Then j#()j 2 for all 0 and the elements in the sets fn : #(n ) = 2g and fn : #(n) = ;2g satisfy n n = O(n) as n ! 1. According to the Floquet's 0
0
0
0
0
0
26
Figure 4: A four-leaves closed KdV traveling wave at k=0.5 and =1.895, (m n)=(4,3). theorem 33], for any 0 satisfying j#()j < 2 and any independent solutions f1 and f2, there exist 2K -periodic functions p1 and p2 such that
f1 (z) = cos z p1 (z) f2 (z) = sin z p2(z) where
cos(2K) = #(2) : Proposition 8.2. For any m 3, there exist in nitely many j 0 such that the curve ^ determined by (49) is closed and has m-leaves.
Proof. For any m 3 and n 1, (n m) = 1, we can solve n ) = #() cos(2K mK 2
to obtain innitely many roots j 0. Clearly then f1 and f2 are 2mK -periodic. So any solution of H is of periodic m". By Lemma 8.1 it produces a closed curve of m leaves 2. An example of a closed ^ with four leaves can be found in Figure 4. Now, let's consider the limit case c1 = c2 < c3 and c = 1. The traveling wave is the one-soliton u = 21 sech2 12 z z = x ; t: We use the SL(2) Serret-Frenet formulas to obtain its SL(2)-tangent: T = t0(t)( 2z ; coth z2 ) sinh 2z cosh 3 z2 + n0(t) sinh z2 cosh 3 z2 : ;
27
;
y 1 t=0
0.8 0.6
t=2 t=4
0.4 0.2 -2.5 -2 -1.5 -1 -0.5
0.5 1
x
Figure 5: The motion of the curve corresponding to the KdV 1-soliton. To determine the vectors t0(t) and n0 (t), we note that the ow satises
s = T(s ; t t) Zst = T^ ( t)d + X(t): ;
;
t
Using the KdV ow
t = (s=)N ; (ss= + 2s =2 + 2)T we can show that t0(t) = 0 and n0(t) = 12 tt0 +n0. Taking t0 = (1 0) and n0 = (0 1), we get ^ = (; tanh 2z ; 12 z cosh 2 2z ; cosh 2 z2 ) and 2 3 1 ; 21 t 7 6 75 ^(x ; t) + 0: 6 (x t) = 4 0 1 0
;
;
Here 0 is a constant vector and translates under an SL(2)-action (see Figure 5). Finally we consider curves and their motions corresponding to the third-order Burgers equation with a = ;1. Letting = ( ; ct), c > 0, satises
zz ; 3
z + 3 + c = D z = ; ct where D is an arbitrary constant. Setting = ;vz =v, (50) is linearized by
vzzz + cvz + Dv = 0 28
(50)
whose general solution is given by
8 > < b0 + b1 cos pcz v (z ) = > : b0e2 z + b1e z cos z
if D = 0
(51)
if D 6= 0
;
Here 2 and ; i are the roots of the equation 3 + c + D = 0. It follows from the Sim(2) Serret-Frenet formulas that the tangent vector satises
Tzz + 2 T + ( 2 + + 1)T = 0: It can be written as
(v 1T)zz + v 1T = 0 ;
which has general solution
;
T = t0(t)v cos z + n0 (t)v sin z:
(52)
We consider the solution of (51) separately in the following three cases.
Case 1. v = b0 + b1 cos pcz and c = 1. In this case the motion is given by = (b0 sin z + 12 b1 z + 41 b1 sin 2z)t0(t) ; (b0 cos z + 14 b1 cos 2z)n0 (t) + 0(t):
From the Burgers ow
t = N + (2 2 ; z ; 1)T we nd
t0(t) = n0(t) = 0 0(t) = 0: 0
So we have
0
0
0 1 1 1 B b0 sin z + 2 b1z + 4 b1 sin 2z CC + ( t) = B @ A 0 ;b0 cos z ; 14 b1 cos 2z
The behavior of the curve is plotted in Figure 6.
Case 2. v = b0 + b1 cos pcz, c 6= 1. In this case, the motion is given by 0 10 1 p p cos( c ; 1)t sin( c ; 1)t x (z)
B
( t) = B @
; sin(pc ; 1)t cos(pc ; 1)t 29
CC BB A@
1
y1(z)
CC + A 0
Figure 6: A stationary curve corresponding to a traveling wave of the Burgers equation. where
p p x1 (z) = pc1+ 1 ;b0 cos z ; 2(pbc1+ 1) cos( c + 1)z + 2(pbc1; 1) cos( c ; 1)z] p p y1(z) = b0 sin z + 2(pbc1+ 1) sin( c + 1)z + 2(pbc1; 1) sin( c ; 1)z So the motion is a uniform rotation (see Figure 7).
Case 3. v = b0 e2 z + b1 e
z cos(z ),
;
consider = 1. The tangent vector is
T = cos z(b0 e2 z + b1e
c = 2 ; 32. Without loss of generality, we
z cos(z ))t
;
0
(t) + sin z(b0 e2 z + b1 e
;
z cos(z ))n
0
(t):
and the motion is given by ( t) = 42b0+ 1 (sin z + 2 cos z)e2 z + 2b+1 4 (sin 2z ; cos2 z ; 2 )e z ]t0(t) + 42b0+ 1 (; cos z + 2 sin z)e2 z ; 2(2b1+ 4) ( sin 2z + 2 cos 2z)e z ]n0(t) + 0(t): ;
;
It follows from the Burgers ow that b0 = 0, 0(t) = (0 0). Also t0(t) and n0 (t) satisfy 0
n0 ; 2t0(t) = (2 + 4)(22n0 (t) ; t0(t)) t0 + 2n0(t) = (2 + 4)(22t0(t) + n0 (t)) 0
0
0
0
Hence
( t) = e2
( 2
0 10 1 B cos 32 t ; sin 32 t CC BB x2(z) CC + +1)t B A@ A 0 @ sin 32 t cos 32 t 30
y2(z)
Figure 7: Curves of the Burgers ow for b0 = 1, b1 = pc + 1 and c=4, 9, and 25. where
x2 (z) = (sin 2z + 2 cos 2z)e
;
z
y2(z) = ; 1 (sin 2z ; cos2 z ; 2 )e z : ;
So the motion is a uniform rotation followed by a dilatation.
9. Concluding remarks We have shown that the approach in 12] and 13] can be applied to Klein geometries to link many integrable equations to geometric invariant motions. Work in progress have shown that similar results holds for the remaining Klein geometries| the SO(3)-, conformal (SO(3 1)-) and projective (SL(3)-) geometries. It is anticipated the same reasoning also works for transitive but not primitive Lie algebras. The equivalence between the integrable equations for the curvature and the invariant motion leads to new integrable equations. Since very often we can express a motion law as a single evolution equation for some quantity, in view of this equivalence this evolution equation should also be integrable. In general, there are many ways to reduce the motion (8) to a single equation (see Chapter 1 in 34]). As an illustration let's consider the mKdV ow (38). Suppose the ow can be expressed as the graph (x u(x t)) of some function u on the x-axis. Using the fact that the normal speed of , ut =(1+ u2x)1=2 , is given by ;ks, one nds that u satises ut = ( uxx2 3 )x: (53) (1 + ux) 2 Hence v = ux satises vt = ( vx 2 3 )xx: (54) (1 + v ) 2 Both (53) and (54) should be integrable. In fact, it turns out the integrability of (54) was established by Wadati, Konno and Ichikawa 35] who showed that it is the 31
compatibility condition of a certain WKI-scheme of inverse scattering transformation. This WKI-scheme for v is connected to the AKNS-scheme for k by a gauge transformation explicitly displayed in 36] (see also 37]). Thus the approach here gives a geometric interpretation of this correspondence. It is interesting to nd the inverse scattering scheme for other equations for the Euclidean and the other geometries. In particular, we mention that when the curve is locally convex, a convenient way to express (4) as a single equation is in terms of its support function h( t) (see Chapter 2 in 34]). For the mKdV ow, the equation takes the rather unconventional form ht = ks = 12 (k2 ) = 1 ( 1 )2] : 2 h + h We don't know if this equation arises from the AKNS- or the WKI-scheme. Finally, it is a challenging problem to nd physical interpretation for these motion laws. As for the mKdV- ow we recall that it is related to the approximate dynamics of vortex patches in ideal 2-D ows 38].
Acknowlegement: This work was supported by a Direct Grant of The Chinese University of Hong Kong. We are indebted to Prof. Luen-Chau Li who rst showed us the reference 12].
References 1] P. Pelc%e (ed.), Dynamics of Curved Fronts, Academic Press, New York, 1988. 2] J. S. Langer, Rev. Mod. Phys. 52 (1980) 1-28. 3] B. C. Brower, D. A. Kessler, J. Koplik and H. Levine, Phys. Rev. A 29 (1984) 1335-1342. 4] H. Hasimoto, J. Fluid Mech. 51 (1972) 477-485. 5] P. G. Saman and G. I. Taylor, Proc. R. Soc. London A 245 (1958) 312-329. 6] G. L. Lamb, Jr., J. Math. Phys. 18 (1977) 1654-1661. 7] J. Langer and R. Perline, J. Nonlinear Sci. 1 (1991) 71-93. 8] A. Doliwa and P. M. Santini, Phys. Lett. A 185 (1994) 373-384. 9] M. Lakshmanan, J. Math. Phys. 20 (1979) 1667-1672. 32
10] T. Tsurumi, H. Segur, K. Nakayama and M. Wadati, Phys. Lett. A 224 (1997) 253-263. 11] K. Nakayama, J. Phys. Soc. Jpn. 67 (1998) 3031-3037. 12] R. E. Goldstein and D. M. Petrich, Phys. Rev. Lett. 67 (1991) 3203-3206. 13] K. Nakayama, H. Segur and M. Wadati, Phys. Rev. Lett. 69 (1992) 2603-2606. 14] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Stud. Appl. Math. 53 (1974) 249-315. 15] V. E. Zakharov and A. B. Shabat, Func. Anal. Appl. 8 (1974) 226-235. 16] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philedaphia, 1981. 17] A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadephia, 1985. 18] M. J. Ablowitz, P. A. Clarkson, Solitions, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. 19] S. Lie, Theorie der Transformationsgruppen, B. G. Teubner, Leipzig, 1927. 20] P. J. Olver, Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge, 1995. 21] P. J. Olver, G. Sapiro and A. Tannenbaum, Dierential invariant signatures and ows in computer vision: A symmetry group approach, Geometry Driven Diusion in Computer Vision, B. M. Ter Haar Romeny, Ed., 205-306, Kluwer, Dordrecht, 1994. 22] A. Gonz%alez-L%opez, N. Kamran and P. J. Olver, American J. Math. 114 (1992) 1163-1185. 23] H. W. Guggenheimer, Dierential Geometry, Dover Publications, New York, 1973. 24] P. J. Olver, Applications of Lie groups to Dierential Equations, SpringerVerlag, New York, 1986. 25] M. Leo, R. A. Leo, G. Soliani, L. Solombrino and L. Martina, Phys. Rev. D 27 (1983) 1406-1408. 26] K. Sawada and T. Kotera, Prog. Theor. Phys. 51 (1974) 1335-1367. 27] P. J. Caudrey, R. K. Dodd and J. D. Gibbon, Proc. R. Soc. London A 351 (1976) 407-422. 33
B. Fuchssteiner and W. Oevel, J. Math. Phys. 23 (1982) 358-363. D. J. Kaup, Stud. Appl. Math. 62 (1980) 189-216. B. A. Kupershmidt, Phys. Lett. A 102 (1984) 213-215. A. V. Mikhailov, A. B. Shabat and R. I. Yamilov, What is Integrability? (V. E. Zakharov ed.) Springer-Verlag, Berlin, 1991. 32] K. Nakayama and M. Wadati, J. Phys. Soc. Jpn. 62 (1993) 473-479. 33] W. Magnus and S. Winkler, Hill's Equations, Dover Publications, New York, 1979. 34] K. S. Chou and X. P. Zhu, The Curve Shortening Problem, Chapman & Hall. CRC Press(UK), London, in press. 35] M. Wadati, K. Konno and Y. Ichikawa, J. Phys. Soc, Jpn. 47 (1979) 1698-1700. 36] Y. Ishimori, J. Phys. Soc. Jpn. 51 (1982) 3036-3041. 37] M. Wadati and K. Sogo, J. Phys. Soc. Jpn. 52 (1983) 394-398. 38] R. E. Goldstein and D. M. Petrich, Phys. Rev. Lett. 69 (1992) 555-558. 28] 29] 30] 31]
34
