INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES IN MINKOWSKI 3-SPACE

2004,24B(4):519-528

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1t'~JI1'fIl INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES IN MINKOWSKI 3-SPACE 1 Zuo Da!eng l ,2 ( £.Jt"t ) Chen Qingl ( F:t.9Jlr ) l Cheng Yi ( >fIt.) Zhou Kouhua l ,3 ( }!]~a~ ) 1.Department of Mathematics, University of Science and Technology of China, Hefei 230026, China 2.Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 3.Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Abstract In this paper, infinitesimal deformations of time-like surfaces are investigated in Minkowski 3-space R 2 ,1 . It is shown that some given deformations of the time-like surface can be described by 2+1 dimensional integrable systems. Moreover spectral parameters are introduced, and it is proved that deformation families are soliton surfaces' families. Key words

Time-like, deformation, 2 + 1 dimensional system

2000 MR Subject Classification

1

35Q51, 53A05

Introduction The relation between the deformation of surface and integrable system has been investi-

gated extensively in R 3[2-7]. In [2], U.Pinkall and LSterling studied infinitesimal deformations of constant mean curvature surfaces. B.G.Konopelchenko[8] established a connection between the integrable KdV hierarchy and curvature-preserving deformations of surfaces of revolution in 53. Note that the evolution equations for surfaces are highly redundant. It is not easy to link these equations to integrable systems. To reduce these equations, one way is to add an appropriate velocity of the motion[4-7j. The aim of this paper is to investigate deformations of certain class of time-like surfaces in Minkowski 3-space R 2 ,1 , including surface with constant Gaussian curvature K = 1 (CGC1 in brief) preserving Chebyshev-null coordinate systems and two kinds of time-like surfaces preserving time-like geodesic coordinates, that is, time-like Hasimoto surface and time-like extended Sinh-Gordon (E-SHG in brief) surface (defined later). Firstly we study these evolution systems for these deformations. Secondly we consider the relation between deformations with a given normal velocity and 2 + 1 dimensional integrable systems. Finally spectral parameters are introduced and Backlund transformations and Syrn formulas are obtained. 1 Received Janvary 14, 2002. science" .

This work was supported by NSFC(10301030) and 973 project "nonlinear

520

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The Time-like CGC-l Surface in R 2 ,1

Let R 2 ,1 denote the Minkowski 3-space endowed with linear coordinates (X 1,X2,X3 ) and the scalar product ( , ) given by -xl + Xi + Xl. The canonical bases of 8l(2,R) is given by

e,

=

~ (~1 ~), ,,~~ (~ ~), eo ~ ~ (~ ~1)'

(21)

We identify R 2 ,1 with 8l(2, R) [9] by

)

(2.2)

The scalar product of vectors in terms of matrices is then (X, Y) = 2trXY,

X, Y E 8l(2, R).

Under this identification, R 2 ,1 ::::: 8l(2, R) and 8l(2, R) acts isometrically and transitively on R 2 ,1 by (2.3) where q> E 8l(2, R) and X E R 2 ,1. In the following we shall use the matrix representation (2.2) instead of R 2 ,1. Let M be a simply connected domain and f : M --t R 2 ,1 be a time-like CGC-1 surface with two real distinct principal curvatures. Without loss of generality, assume that the principal vector corresponding to smaller principal curvature is space-like everywhere on M. For each regular point of i, one may use the time-like asymptotic line parametrisation (x, y) such that the first and the second fundamental forms are given by I

= dx 2 + 2 cosh Bdxdy + d y 2

{ II = 2 sinh Bdxdy.

(2.4)

Such local coordinate system (x, y) is called a Chebyshev-null coordinate (see [10] for details). With respect to this, the Gauss-Coddazzi equations of the surface are written as a sinh-Gordon(SHG) equation: (2.5) Bx y + sinhB = O. Consider a 8l(2, R)-valued 1 form

(2.6)

Note that dn = n 1\ n is equivalent to the GC equations. According to the Frobenius theorem, there exists a 8l(2, R)-valued smooth map such that d-1 = n. It is easily verified that d(-lhl- lh 2<1» = dfy and d(- l e3<1» = dN, where hI = cosh ~e2 - sinh ~el and h 2 = cosh ~e2 + sinh ~el' Then for a fixed point p E M and

521

Zuo et al: INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES

No.4

= {fx(p), fy (p), N(p)},

(p)-I{h1(p),h 2(p),e3(p)}(P)

the solution is unique up to ±I2.

Then we have Proposition 1 Let f : M -+ R 2 ,1 be a time-like CGC-l surface with two real distinct principal curvatures and {Ix, fy, N} be the frame field, then there exists a smooth 8l(2, R)valued map such that

f.

~ ~~-. ( e2~ !~-. (

fy=

_!1.

e 2

a

)~,

(2.7)

8

e2

08 e- 2

2

a

) e,

N = -l e3. Similar to [12], can be regarded as the frame of time-like surfaces.

3

Infinitesimal Deformations of Time-like CGC-l Surfaces in R 2 ,1

In the following we shall investigate infinitesimal deformation of the time-like CGC-1 surface. Assume that a smooth one-parameter family ft : M x (-t:,t:) -+ R 2 ,1 which preserves the Chebyshev-null system (x,y) and Gaussian curvature, where t: > a and t" = f. Let the corresponding frame be t and 0 = . For simplicity, we denote f = f(x, y, t) = P, = (x,y,t) = t and B = B(x,y,t) = Bt. Let us introduce the velocity field

ft

= ufx + vfy + wN = -1

(

U

8

-e 2

2

~

U

-e

_!1.

2

2

V!1. + -e

2

V-8

w

2

2

+ -e T

2

(3.1)

where {Ix, fy, N} is the frame field of the surface and u, v and ware some real functions. By using.Proposition 1, the motion of the corresponding frame is (3.2)

where

Ut =

and U

O

(

1 8 -e- 2

O.

2

~:!

= U,

V

B

x -4 O

). ( Vt =

- By 4

1 8 --e-2" 2

1 !1. --e 2

2 By 4

).

w=

(

W,

W2 - WI

WI +W2 -W3

), (3.3)

= V, WI = W2 =

1 f) (wx + w y + (u + v) sinh B), 4 cosh -

1

4 sinh

2 B (wy - Wx + (u - v) sinh 8),

2"

W3 = !(Bxu - Byv + (uy - v x) sinh B).

(3.4)

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Equation nt(x, y, t)

= dt( t)-l

Vol.24 Ser.B

can be solved if and only if (3.5)

that is, u~ - V;

+ rut,

vt]

o; - Wx + rut,

= 0,

W]

= 0,

v; - Wy + [vt,

W]

= O.

(3.6)

So we have the following Theorem 1 In the Chebyshev-null coordinate system, the deformation of the time-like CGC-1 surface depends on four functions u, v, wand () which satisfy the following seven fundamental equations(3.7-3.13): ()xy + sinh () = 0, (3.7) w xy + wcosh()

= 0,

(3.8)

+ V x cosh()

= 0,

(3.9)

Ux

u y cosh ()

()t - ()xu - ()yV

+ vy =

+ 2w + (u y + v x) sinh()

Proof Define '"

~ ~ (

+ Wxx + w -

(3.11)

= 0,

= 0,

(3.12)

()xwx coth() + ()xWy . 1 = 0 sm h()

(3.13)

2u y sinh () + Wyy + W - ()yW y coth () 2v x sinh()

(3.10)

0,

+ ()yW x

. 1 () sm h

). Note that (3.6) is equivalent to

(3.14) It is obvious that Eq.(3.7) is the GC equation which is equivalent to (2.7) and (3.1), we have fxx

1 sm

= coth()()xfx -

= (u x + coth()()xu -

fty

= (u y -

1

~h()()Yv

N, =

Since (ftx,fx) and we have

coth()w)fx

From (2.4),

fxy = sinh ()N,

~h()()xfy,

ftx

sin

wxy = wyx '

+ (v x -

1

1 1 . + ~h()w)fy + (w x + vsmh())N, sm sin

~h()()xU

+ ~h()w)fx + (v y + coth()()yv sm

coth()w)fy

+ (w y + usinh())N,

~ [(W x

- w y cosh () + v sinh () - u sinh () cosh ())fx sinh () +(w y - Wx cosh () + u sinh () - v sinh () cosh ())fy].

= (fty,f y) = 0, and we get () t _ () xU _ () yV + 2w _-

Eqs.(3.9, 3.10). Since (ftx,f y) + (fty,fx) uy

+ V x + U x cosh () + v y cosh () . h () sm

= ()tcoshB,

No.4

Zuo et al: INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES

Using (3.9) and (3.10), we get (3.11). From (fxy,N) = sinh 0, we know (fxyt,N)

= OtcoshO.

523

On the other hand,

(ftXY, N) = w xy + (Oxu + OyV - w) coshB + (u x + vy) sinhO.

°

Since fxyt = ftxy and (3.11), we get (3.8). Similarly using (fxx, N) = and (fyy, N) = 0, we may obtain Eq.(3.12) and Eq.(3.13) repectively. By a direct calculation, one can verify that Wxt = Wtx and Wyt = Wty always hold when Eqs.(3.8-3.13) hold. From the above system, one easily knows that tangential velocities u and v are determined by 0 and the normal velocity w in the sense. Hence from extrinsic views, infinitesimal deformations of time-like CGC-l surfaces are given by the normal vector velocity w. However for a given time-like CGC-l surface fO, the existence of the above infinitesimal deformation is equivalent to the systems (3.7-3.13) having a solution with an initial value. This is still unknown in general case. But if we may add an appropriate normal vector field velocity w, we may link the deformation systems to 2 + 1 dimensional integrable systems and use the method of integrable system to obtain some resluts. It is natural to ask how to choose w. Note that w satisfies the linearized Sinh-Gordon equation (3.8). Hence if 0 is a solution of Eq.(3.7), then Ox and Oy are solutions of Eq.(3.8). Theorem 2 In the Chebyshev-null coordinate system, if the normal velocity (3.15) where 0: and [3 are constants, then the deformation of the time-like CGC-l surface depends on o which satisfies the GC Eq.(2.5) and a 2 + 1 dimensional modified KdV system 1 1 1 1 0t = O:("20xxx - :t0x3- 3 "20x) + TJ(t}Ox + [3("2 0yyy - :tOy3- 3 "2 0y) + ((t)Oy,

(3.16

)

where TJ(t) and ((t) are two arbitrary functions of t. Proof Since (3.12) and (3.13), we have

[3 Oyy 0: ( 2) ( ) u=-2"sinhB+'4 20xxcothO-Ox +TJX,t, 0: Bxx [3 ( 2) ( ) v = -2" sinhB + '4 2B yycothB - By + ( x,t ,

where TJ(x, t) and ((x, t) are arbitrary functions. Substituting u and v into (3.9) and (3.10), we get

TJx

= 0,

(y

=

°

that is, TJ(x, t) = TJ(t) and ((x, t) = ((t). Substituting u, v and w into (3.11), we obtain (3.16). In the following, we consider two special cases. Case 1 Let 0:

= 1,

[3

= 0,

TJ(t) =

3

"2'

((t) = J.L (J.L is a constant),

(3.17)

then 0 satisfies the system: (3.18)

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ACTA MATHEMATICA SCIENTIA

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Its Lax pair with the spectral parameter A is

UA,

~~(~>')-1 =

~;(~>')-1

= v-,

(3.19)

w-,

~r(~>')-1 =

~>'I>.=o = ~(x, y, t),

where

~e-!)

U>'=(()X

~2:! ~

W>. = ~ ( 4

A

(A

3

2()x

+ ~()xxx -

+ A()xx

-

1

()4X

v>.=( -();

-~e-!

'

2A

~()~ -/-l()y

2

2A

2 A()x -

-1

ue

3

(A

-9!l.

)e 2

-

A()xx -

-A

2

()x -

-2\e!), ()y

.

4

~A(); 1

2()xXX

2A-

1/-le9)e-!).

(3.20)

1 3 + 4()x + /-l()y

Summarize the above results, we have Proposition 2(Sym formula) Let ~>. satisfy the linear system (3.19,3.20) and A = eA , A E R, then (3.22) describes the above time-like CGC-l surface family I

It.

Its fundamental forms are as follows

= dx 2 + 2 cosh ()(x, y, t)dxdy +d y 2 ,

{ II = 2 sinh ()(x, y, t)dxdy,

and the velocity field

By the theorem, for every fixed t E R, the surface can be viewed as a soliton surface in the deformation family[ll]. Using the Backlund transformation and Sym formula, one may obtain the exact solutions of the deformation of the time-like CGC-l surface. Proposition 3 The system (3.18) is invariant under the Backlund transformation

-

.

O+()

(() - ()) = -2A smh - x

(() + ())y = 2A -

-

(() - ())t

-1'

0- ()

smh -2-'

= A ()x 2

2 '

A()xx

O+()

cosh -2-

-2fJ(()y - A-I sinh

where

0=

B>. * ().

A

2

+ ("2()x

0; ()),

-

3.

o+()

A ) smh -2-

(3.22)

Zuo et al: INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES

No.4

Case 2 Let 'TJ(t)

= ((t) = 0, then 0 satisfies the system

Oxy {

525

e,

+ sinh 0 = 0, 1

1

(3.23)

1

1 = a(20xxx - 20x - 20x) + (3(20yyy - 2 0y3- 32 0y). 33

The system (3.23) is invariant under the transformation

(0 - O)x

. 0+0 = 2smh -2-'

0- 0 (0 + O)y = -2 sinh -2-' -

1

2

1

3

.

0+0

(0 - O)t = a[Ox - (2 0x + 2) smh -2+(3[20y + 20y - Oyyy

(3.24)

0+0

+ Oxx cosh -2-] 0-0

1

.

0- 0

+ Oyy cosh -2- + (2 + 20y2 smh -2-)]'

Using the transformation (3.24), we get one soliton solution:

0= 4tanh- 1 e",

I = ex - '!!..

e

+ [~( -3e + e3 ) + f!.(~ - ~)] t, 3 2

2 e

(3.25)

e

where e =I 1 is a constant. In this case, there isn't the Syrn formula to use. But we may integrate lx, I y and It to obtain the explicit expressions as follows:

~ ( 13 h + h) ' -2 h-h -13

Ih = (2 e

4e - 1)' sin h I cosh 28,

h

1 8 = -2 (-x

4e = (1 - e2)' sin h I sinh 8,

+ y + (a - (3)t), 4e

13 = -2(x + y) + ~1 coth v, e -

(3.26)

4 Infinitesimal Deformations of Time-Like Surfaces Preserving Timelike Geodesic Coordinates 4.1

The general csae Let M be a simply connected domain and I : M -+ R 2 ,1 be an orientable time-like immersion, there exists a local coordinate system (x, y) such that

Such local coordinate system (x, y) is called a time-like geodesic coordinate system. Assume that the second fundamental form is (4.2) Similar to Proposition 1 and choose a local orthnormal frame {Ix, g-l/y , N}, then there exists a smooth map : M -+ 5£(2, R) such that

(4.3)

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ACTA MATHEMATICA SCIENTIA

Obviously <1>xy

= <1>yx

Vol.24 Ser.B

is equivalent to the GC equations

+ U2 U3,

gxx : gui {

U2y - -2g xul - gUh,

(4.4)

Uly = Vlx - gx U2·

In the following, we consider two special cases: (1)u2 = g; (2)Ul = 1, U2 = Wx and 9 = -~Wy. 4.2 Infinitesimal deformations of time-like Hasimoto surfaces Consider Case 1. Assume that U2 = 9 and the fundamental forms of the surface I(x, y, t) be I = -dx 2 + g(x, y, t)2d y2, (4.5) { II = g(x, y, t)dx 2 - 2gUldxdy - gU3dy2. Similar to [1], the surface can be obtained by the motion of a time-like curve. Hence we call the surface to be the time-like Hasimoto surface. Analogous to Theorem 2, we have Theorem 3 In the time-like geodesic coordinate system, assume that the velocity field of the deformation (4.6) It = -"21 g 2 Ix + udy + gx N , then the deformation of the time-like Hasimoto surface depends on three functions Ul, U3, 9 which satisfy the following five equations: gxx 2 u3 = - Ul'

(4.7)

= -2gxul -

(4.8)

9

gy

gUlx,

Uly = U3x - gxg, gt Ult

= (gudy + gx IX u 1ydx,

= (-g;y +Ul

IX u 1ydx)x +ggy.

(4.9)

(4.10) (4.11)

r

Theorem 4 If q = ~ge-i u1dx, then q satisfies the defocusing NLS and the 2+1 dimensional repulsive breaking soliton equations (4.12) iq; = -qxy

+ 2q IX Iql~dx.

(4.13)

The Lax pair of (4.12,4.13) with the spectral parameter>. is

<1>"(<1>>')-1 x <1>;(<1>")-1

= U" , = v-,

<1>;(<1>")-1 = W", <1>"1,,=0

= <1> (x, y, t),

I)

Zuo et al: INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES

No.4

where

u>' = (A + Ul)el + ge2, V>' = (A2 + U3)el + g(A - Ude2 - gxe3,

527

(4.15)

W>' = ((A + Ul) jX u 1ydx _ g;y)e 1 + g jX Ulydxe2 + gye3 - AV'\

Proposition 4

Let >' satisfy the linear system (4.14, 4.15), then

I

I

=

x

1>.=0 = (
x

-1

a>'

aA

1>'=0,

describes a family of the above time-like Hasimoto surfaces. 4.3 Infinitesimal deformations of time-like E-SHG surfaces Consider Case 2. Assume that Ul = 1, U2 = Wx and g = -!wY ' Hence the fundamental forms of the surface I (x, y, t) are

I = -dx 2 + !w 2 d y 2 4 y , { I I = wxdx 2 + wydxdy where U3 =

Wl!~WUl!. w.

+ !wyu3dy2,

(4.16)

The GC equation is (

WXXy - WY) Wx

x

= wxwxy,

(4.17)

which is an extended sinh-Gordon equation. Hence we call the surface to be the time-like ESHG surface. In fact the time-like E-SHG surface is swept out by the purely bi-normal motion of an inextensible time-like curve of constant torsion. Assume that the velocity field of the deformation is It

= - coshwlx + (3ly + sinhwN,

(4.18)

where (3x = O. Theorem 5 If the deformation of the time-like E-SHG surface M with a given velocity as in (4.19), then w satisfies the following 2+1 dimensional extended sinh-Gordon system: (

WXXy - wY) Wx

Wxt

x

= wxw xy,

+ sinhw =

(3w xy.

(4.19) (4.20)

The Lax pair of (4.19, 4.20) with the spectral parameter A is A A(A)-1 - U, x ~(>')-1 =

;(>')-1 >'I>.=1

v-,

= WA,

(4.21)

= (x, y, t),

where (4.22)

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ACTA MATHEMATICA SCIENTIA

Proposition 5

Let

1>A

Vol.24 Ser.B

satisfy the linear system (4.21, 4.22), then

f

=

f AIA=1

(

= 1>

A)_1

81>A 8>'

1A=1

(4.23)

describes a family of the above time-like E-SHG surfaces. Proposition 6 The system (4.19, 4.20) is invariant under the Backlund transformation )

5

U3

=

W

w

-

(W-

w 2'\ w) .\ x y cosh W +. hW +2 \2 (W y + t/\w + W)Y = - 1 -+ t/\U3 sin -, + /\ 2 2

W x

(w + w)t ' were h

W

+ = 2'\· t/\S1n h --2-'

(W-

-w Y2 u y Wz

2i.

W-

(4.24)

W

= ""X" smh - 2 - + f3(w + w)y,

and f3x =

o.

Conclusion

In this paper, we have considered the relation between 2 + 1 dimensional integrable systems and infinitesimal deformations of time-like surfaces in R 2 ,1 . Note that as a deformation system with the Lax pair with the spectral parameter can be usually gauged to AKNS-type (by a suitable gauge transformation) and composed of two 1 + 1 dimensional systems. Hence we may use Darboux transformation and sym formula to obtain the explicit expressions of deformation. Moreover, to investigate the geometrical meaning of Backlund transformations of 2 + 1 dimensional integrable systems may be an interesting problem. References 1 Hasimoto H. A soliton on a vortex filament. J Fluid Mech, 1972, 51: 477-485 2 Pinkall D, Sterling I. On the classification of constant mean curvature tori. Ann Math, 1989, 130: 407-451 3 Bobenko A I. Surface in terms of 2 by 2 matrix, old and new integrable case. In: Fordy A P, ed. Harmonic Maps and Integrable Systems. Vieveg Germany, 1994. 83-127 4 Mclachlan R I, Segur H. A note on the motion of surfaces. Phys Lett, 1994, 194A: 165-172 5 Li Yishen. The motion of surface with constant negative Gaussian curvature. J Partial Diff Eqs, 1999, 12: 85-96(in Chinese) 6 Li Yishen, Chen Chunli. The motion of surfaces in geodesic coordinates in 2+ 1 dimensional breaking soliton equation. J Math Phys, 2000, 41: 2066-2076 7 Cao Xifang, Tian Chou. Integrable system and spacelike surfaces with prescribed mean curvature in Minkowski 3-spae. Acta Mathematica Scientia, 1999, 19(1): 91-96 8 Konopelchenko B G. Induced surfaces and their integrable dynamics. Stud Appl Math, 1996, 96: 9-51 9 Zuo Dafeng, Chen Qing, Cheng Yi. The surfaces and integrability in 3-dimensional Minkowski space. Chinese Ann Math, 2002, 23A(3): 331-338 10 Inoguchi J. Darboux transformation on time-like constant mean curvature surfaces. J Geom Phys, 1999, 32: 57-78 11 Sym A. Soliton surfces and their applications. Lect Notes Phys, 1985, 239: 154-231 12 Chen Qing, Cheng Yi. Spectral transformation of constant mean curvature in H 3 . Sci China, 2002, 45A(8): 1066-1075