冉 冊 冉 冊

Modified Korteweg–de Vries surfaces

Süleyman Tek^a兲

Department of Mathematics, Faculty of Science, Bilkent University, 06800 Ankara, Turkey

共Received 14 September 2006; accepted 13 November 2006;

published online 23 January 2007兲

In this work, we consider 2-surfaces inR³arising from the modified Korteweg–de Vries共mKdV兲 equation. We give a method for constructing the position vector of the mKdV surface explicitly for a given solution of the mKdV equation. mKdV surfaces contain Willmore-like and Weingarten surfaces. We show that some mKdV surfaces can be obtained from a variational principle where the Lagrange function is a polynomial of the Gaussian and mean curvatures. © 2007 American Institute of Physics. 关DOI:10.1063/1.2409523兴

I. INTRODUCTION

In this work we study the 2-surfaces in R³ arising from the deformations of the modified Korteweg–de Vries共mKdV兲 equation and its Lax pair. Deformation technique was developed by several authors. Here we mainly follow Refs.1–12.

Let u共x,t兲 satisfy the mKdV equation

ut= u_3x+3

2u²ux. 共1兲

Substituting the traveling wave ansatz ut−␣^ux= 0 in Eq.共1兲, where␣is an arbitrary real constant, we get

u_2x=␣^{u −u3}

2 . 共2兲

Here and in what follows, subscripts x, t, and␭ denote the derivatives of the objects with respect to x, t, and␭, respectively. The subscript nx stands for n times x derivative, where n is a positive integer, e.g., u_2xindicates the second derivative of u with respect to x. We use Einstein’s summa- tion convention on repeated indices over their range. Equation共2兲can be obtained from a Lax pair U and V, where

U = i

2

冉

冊

V = − i

2

冉

冊

and␭ is the spectral parameter. The Lax equations are given as

a兲Electronic mail: tek@fen.bilkent.edu.tr

48, 013505-1

0022-2488/2007/48共1兲/013505/17/$23.00 © 2007 American Institute of Physics

⌽x= U⌽, ⌽t= V⌽, 共5兲 where the integrability of these equations are guaranteed by the mKdV equation or the zero curvature condition

U_t− V_x+关U,V兴 = 0. 共6兲

A connection of the mKdV equation to surfaces inR³ can be achieved by defining su共2兲 valued 2⫻2 matrices A and B satisfying

At− Bx+关A,V兴 + 关U,B兴 = 0. 共7兲

Let F be an su共2兲 valued position vector of the surface S corresponding to the mKdV equation such that

y_j= F_j共x,t;␭兲, j = 1,2,3, F = i

兺

k=1 3

F_k␴k, 共8兲

where␴k’s are the Pauli sigma matrices

␴1=

冉

冊

冉

冊

冉

冊

The connection formula共connecting integrable systems to 2-surfaces in R³兲 is given by

Fx=⌽⁻¹A⌽, Ft=⌽⁻¹B⌽. 共10兲

Then at each point on S, there exists a frame 兵Fx, F_t,⌽⁻¹C⌽其 forming a basis of R³, where C

=关A,B兴/^储关A,B兴^储and关A,B兴 denotes the usual commutator 关A,B兴=AB−BA. The inner product 具,典 of su共2兲 valued vectors X and Y are given by 具X,Y典=−¹₂tr共XY兲. Hence^储X储=

冑

共dsI兲²⬅ gijdxⁱdx^j=具A,A典dx²+ 2具A,B典dxdt + 具B,B典dt²,

共dsII兲²⬅ hijdxⁱdx^j=具Ax+关A,U兴,C典dx²+ 2具At+关A,V兴,C典dxdt + 具Bt+关B,V兴,C典dt², 共11兲

where i , j = 1 , 2, x¹= x, and x²= t. Here gij and hij are coefficients of the first and second funda- mental forms, respectively. The Gauss and the mean curvatures of S are, respectively, given by K = det共g⁻¹h兲 and H=¹₂tr共g⁻¹h兲, where g and h denote the matrices 共gij兲 and 共hij兲, and g⁻¹stands for the inverse of the matrix g.

In order to calculate the fundamental forms in Eq.共11兲and the curvatures K and H, one needs the deformation matrices A and B. Given U and V, finding A and B from Eq.共7兲is a difficult task in general. However, there are some deformations which provide A and B directly. They are given as follows:

• Spectral parameter␭ invariance of the equation:

A =␮⳵^U

⳵␭, B =␮⳵^V

⳵␭, F =␮⌽⁻¹⳵⌽

⳵␭, 共12兲

where␮is an arbitrary function of␭.

• Symmetries of the共integrable兲 differential equations:

A =␦^{U, B =}␦^{V, F =}⌽⁻¹␦⌽, 共13兲

where␦ represents the classical Lie symmetries and共if integrable兲 the generalized symme- tries of the nonlinear partial differential equations共PDEs兲.

• Gauge symmetries of the Lax equation:

A = Mx+关M,U兴, B = Mt+关M,V兴, F = ⌽⁻¹M⌽, 共14兲 where M is any traceless 2⫻2 matrix.

There are some surfaces which may be obtained from a variational principle. For this purpose, we consider a functionalF which is defined by

F ⬅

冕

E共H,K兲dA + p

冕

dV, 共15兲

whereE is some function of the curvatures H and K, p is a constant, and V is the volume enclosed by the surface S. For open surfaces, we let p = 0. The first variation of the functionalF gives the following Euler-Lagrange equation for the Lagrange functionE:^13–16

共ⵜ²+ 4H²− 2K兲⳵E

⳵^H+ 2共ⵜ · ⵜ¯ + 2KH兲⳵E

⳵^K− 4HE + 2p = 0, 共16兲

whereⵜ²andⵜ·ⵜ¯ are defined as ⵜ²= 1

冑

⳵

⳵^xi

冉 ^冑

冊

冑

⳵

⳵^xi

冉 ^冑

冊

and g˜ = det共gij兲, where g^ijand h^ijare the inverse components of the first and second fundamental forms, x¹= x, x²= t. The following are examples of surfaces derived from a variational principle:

共i兲 Minimal surfaces:E=1, p=0;

共ii兲 constant mean curvature surfaces:E=1;

共iii兲 linear Weingarten surfaces:E=aH+b, where a and b are some constants;

共iv兲 Willmore surfaces:E=H²;^17,18 and

共v兲 surfaces solving the shape equation of lipid membrane: E=共H−c兲², where c is a constant.13–16,19–21

The surfaces obtained from the solutions of the equation

ⵜ²H + aH³+ bHK = 0 共18兲

are called Willmore-like surfaces, whereⵜ²is the Laplace-Beltrami operator defined on the surface and a , b are arbitrary constants. Unless a = 2 and b = −2, these surfaces do not arise from a varia- tional problem. The case a = −b = 2 corresponds to the Willmore surfaces. For compact 2-surfaces, the constant p may be different than zero, but for noncompact surfaces we assume it to be zero.

For the latter, we require asymptotic conditions, where K goes to a constant and H goes to zero.

This requires that the mKdV equation have solutions decaying rapidly to zero as兩x兩→ ±⬁. Soliton solutions of the mKdV equation satisfy this requirement. In this work, using solitonic solutions of the mKdV equation, we find the corresponding 2-surfaces and then solve the Euler-Lagrange equation关Eq. 共16兲兴 for polynomial Lagrange functions of H and K, i.e.,

E = aNH^N+ ¯ + b10KH + b₁₁KH²+ ¯ + e1K + . . . . 共19兲 For each N, we find the constants a_l, b_nk, and e_m in terms of others and the parameters of the surface.

From a solution of the mKdV equation, we first find the fundamental forms in Eq.共11兲and the curvatures K and H of the corresponding 2-surface S. To find the position vector y共x,t兲 of S, we use Eq.共10兲. To solve this equation, we need the matrix⌽ satisfying the Lax equation 关Eq.共5兲兴 for a given function u共x,t兲. Hence, in general, our method for constructing the position vector y of integrable surfaces consists of the following steps:

共i兲 Find a solution u共x,t兲 of the mKdV equation.

共ii兲 Find a solution of the Lax equation关Eq.共5兲兴 for a given u共x,t兲.

共iii兲 Find the corresponding deformation matrices A, B, and find F from Eq.共10兲.

In this work more specifically, starting with one soliton solution of the mKdV equation and following the steps above, we solve the Lax equations and find the corresponding SU共2兲 valued function⌽共x,t兲. Then using the spectral deformations and combination of the gauge and spectral deformations, we find the parametric representations共position vectors兲 of the mKdV surfaces and plot some of them for some special values of constants. We show that there are some Weingarten and Willmore-like mKdV surfaces obtained from spectral deformations. Surfaces arising from a combination of the gauge and spectral deformations do not contain Willmore-like surfaces. We study also the mKdV surfaces corresponding to the symmetry deformations. We determine all geometric quantities in terms of the function u共x,t兲 and the symmetry ␾共x,t兲. For the simplest symmetry ␾= ux, the surface turns out to be the surface of the sphere with radius 兩共␣␮兲/共2␭兲兩, where␭ is the spectral parameter and␣ and␮ are constants.

II. mKdV SURFACES FROM SPECTRAL DEFORMATIONS

In this section, we find surfaces arising from the spectral deformation of Lax pair for the mKdV equation. We start with the following proposition.

Proposition 1: Let u satisfy (which describes a traveling mKdV wave) Eq. (2). The corre- sponding su共2兲 valued Lax pair U and V of the mKdV equation are given by Eqs. (3) and (4), respectively. Then, su共2兲 valued matrices A and B are

A = i

2

冉

冊

B = − i

2

冉

冊

where A =␮^U_␭^{, B =}␮^V_␭^, ␮is a constant, and␭ is the spectral parameter. The surface S, gener- ated by U, V, A and B, has the following first and second fundamental forms共j,k=1,2兲:

共dsI兲²= gjkdx^jdx^k=␮²

4 共关dx + 共␣+ 2␭兲dt兴²+ u²dt²兲, 共22兲

共dsII兲²= h_jkdx^jdx^k=␮u

2 共dx + 共␣⁺␭兲dt兲²+␮u

4 共u²− 2␣兲dt², 共23兲 with the corresponding Gaussian and mean curvatures

K = 2

␮^{2共u2− 2}␣兲, H = 1

2␮^{u共3u2+ 2共␭2−}␣兲兲, 共24兲 where x¹= x , x²= t.

By using U, V, A, and B and the method given in Sec. I, Proposition 1 provides the first and second fundamental forms, and the Gaussian and mean curvatures of the surface corresponding to spectral deformation. The following proposition gives a class of surfaces which are Willmore-like.

Proposition 2: Let u_x²=␣u²− u⁴/ 4. Then the surface S, defined in Proposition 1, is a Willmore- like surface, i.e., the Gaussian and mean curvatures satisfy Eq.(18), where

a =4

9, b = 1, ␣⁼␭², 共25兲

and␭ is an arbitrary constant.

It is important to search for mKdV surfaces arising from a variational principle.^13–16For this purpose, we do not need a parametrization of the surface. The fundamental forms and the Gauss and mean curvatures are enough to look for such mKdV surfaces. The following proposition gives a class of mKdV surfaces that solves the Euler-Lagrange equation关Eq.共16兲兴.

Proposition 3: Let u_x²=␣^{u2− u4}/ 4. Then there are mKdV surfaces defined in Proposition 1 satisfying the generalized shape equation [Eq.(16)] whenE is a polynomial function of H and K.

Here are several examples:

Example 1: Let deg 共E兲=N, then 共i兲 for N = 3:

E = a1H³+ a₂H²+ a₃H + a₄+ a₅K + a₆KH,

␣⁼␭², a₁= − p␮⁴

72␭⁴, a₂= a₃= a₄= 0, a₆= p␮⁴ 32␭⁴, where␭⫽0, and ␮, p, and a₅are arbitrary constants;

共ii兲 for N = 4:

E = a1H⁴+ a₂H³+ a₃H²+ a₄H + a₅+ a₆K + a₇KH + a₈K²+ a₉KH²,

␣⁼␭², a₂= − p␮⁴

72␭⁴, a₃= − 8␭²

15␮^{2共27a1− 8a8兲, a4= 0,}

a₅= ␭⁴

5␮^{4共81a1+ 16a8兲, a7=} p␮⁴

32␭⁴, a₉= − 1

120共189a1+ 64a₈兲, where␭⫽0,␮⫽0, and p, a1, a₆, and a₈are arbitrary constants;

共iii兲 for N = 5:

E = a1H⁵+ a₂H⁴+ a₃H³+ a₄H²+ a₅H + a₆+ a₇K + a₈KH + a₉K²+ a₁₀KH²+ a₁₁K²H + a₁₂KH³,

␣=␭², a₃= − 1

504␮²␭²共␭⁶关4212a1+ 256a₁₁兴 + 7p␮⁶兲,

a₄= − 8␭²

15␮^{2共27a2− 8a9兲, a5=} 6␭⁴

7␮^{4共135a1− 88a11兲,}

a₆= ␭⁴

5␮^{4共81a2+ 16a9兲, a8=} 1

32␮²␭²共␭⁶关− 324a1+ 512a₁₁兴 + p␮⁶兲,

a₁₀= − 1

120共189a2+ 64a₉兲, a12= − 1

756共1053a1+ 512a₁₁兲, where␭⫽0,␮⫽0, and p, a1, a₂, a₇, a₉, and a₁₁ are arbitrary constants;

共iv兲 for N = 6:

E = a1H⁶+ a₂H⁵+ a₃H⁴+ a₄H³+ a₅H²+ a₆H + a₇+ a₈K + a₉KH + a₁₀K²+ a₁₁KH² + a₁₂K²H + a₁₃KH³+ a₁₄K³+ a₁₅K²H²+ a₁₆KH⁴,

␣⁼␭²,

a₄= − 1

504␮²␭²共␭⁶关4212a2+ 256a₁₂兴 + 7p␮⁶兲,

a₅= − ␭⁴

900␮⁴共− 359 397a1+ 191 488a₁₄− 203 472a₁₆兲 − 8␭²

15␮^{2共27a3− 8a10兲,}

a₆= 6␭⁴

7␮^{4共135a2− 88a12兲,}

a₇= ␭⁶

25␮^{6共29 889a1− 9856a14+ 11 664a16兲 +}

␭⁴

5␮^{4共81a3+ 16a10兲,}

a₉= 1

32␮²␭⁴共␭⁶关− 324a2+ 512a₁₂兴 + p␮⁶兲,

a₁₁= − ␭²

1800␮^{2共59 778a1− 13 312a14+ 23 328a16兲 −} 1

120共189a3+ 64a₁₀兲,

a₁₃= − 1

756共1053a2+ 512a₁₂兲,

a₁₅= − 1

2880共5103a1+ 2048a₁₄+ 3888a₁₆兲,

where␭⫽0,␮⫽0, and p, a1, a₂, a₃, a₈, a₁₀, a₁₂, a₁₄, and a₁₆ are arbitrary constants.

For general N, from the above examples, the polynomial function E takes the form

• for odd N:

E =

兺

l=0 M

alH^2l+1+

兺

n=1

M

冉

^兺

冊

where N = 2M + 1 , M = 1 , 2 , 3. . .;

• for even N:

E =

兺

l=0 M

alH^2l+

兺

n=1 共M−1兲

冉

^兺

冊

^兺

where N = 2M, M = 2 , 3 , 4 , . . .. In both cases al, bkn, and emare constants.

A. The parametrized form of the three parameter family of mKdV surfaces

In the previous section, we found possible mKdV surfaces satisfying certain equations. In this section, we find the position vector

y =共y1共x,t兲,y2共x,t兲,y3共x,t兲兲 共26兲 of the mKdV surfaces for a given solution of the mKdV equation and the corresponding Lax pair.

To determine y, we use the following steps:

共i兲 Find a solution u of the mKdV equation with a given symmetry: Here we consider Eq.共2兲, which is obtained from the mKdV equation by using the traveling wave solutions ut

=␣^ux, where␣= −1 / c, c⫽0 are arbitrary constants.

共ii兲 Find the matrix ⌽ of the Lax equation 关Eq. 共5兲兴 for given U and V: In our case, the corresponding su共2兲 valued U and V of the mKdV equation are given by Eqs.共3兲and共4兲. Consider the 2⫻2 matrix ⌽

⌽ =

冉

冊

By using this and Eq.共3兲 for U, we can write⌽x= U⌽ in matrix form as

冉

冊

冉

冊

Combining共⌽11兲x=¹₂i␭⌽11−¹₂iu⌽21 and共⌽21兲x= −¹₂i␭⌽21−¹₂iu⌽11, we get

共⌽21兲xx−ux

u共⌽21兲x+

冋

册

Similarly, a second order equation can be written for⌽₂₂ by using the first order equa- tions of⌽12 and ⌽22. By solving the second order equation 关Eq. 共29兲兴 of ⌽21 and the equation for⌽22, we determine the explicit x dependence of⌽21,⌽22and also⌽11,⌽12. The components of⌽t= V⌽ read

共⌽11兲t= − i

2

冋

册

共⌽21兲t= i

2

冋

册

and

共⌽12兲t= − i

2

冋

册

共⌽22兲t= i

2

冋

册

By solving these equations, we determine the explicit t dependence of⌽11,⌽21,⌽12, and

⌽22. This way we completely determine the solution ⌽ of the Lax equations.

共iii兲 We use Eq.共10兲to find F. For our case, A and B are given by Eqs.共20兲and共21兲, which are obtained by the spectral deformation of U and V, respectively. Integrating Eq.共10兲, we get F.

Now by using a given solution of the mKdV equation, we find the position vector of the mKdV surface. Let u = k₁sech␰^,␰^{= k}1共k12t + 4x兲/8, be one soliton solution of the mKdV equation, where␣^{= k}₁²/ 4. By substituting u into the second order equation关Eq.共29兲兴 and using the notation u_x= k₁u_␰/ 2 ,共⌽21兲x= k₁共⌽21兲_␰/ 2, we find the solution of⌽x= U⌽ as follows:

⌽21= iA₁共t兲共tanh␰+ 1兲^i␭/2k1共tanh␰− 1兲^−i␭/2k1sech␰^{+ B}1共t兲共k1tanh␰+ 2i␭兲

⫻共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1, 共34兲

⌽22= iA₂共t兲共tanh␰+ 1兲^i␭/2k1共tanh␰− 1兲^−i␭/2k1sech␰^{+ B}2共t兲共k1tanh␰+ 2i␭兲

⫻共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1, 共35兲

⌽11= − i

k₁A₁共t兲共2␭ + ik1tanh␰兲共tanh␰+ 1兲^i␭/2k1共tanh␰− 1兲^−i␭/2k1+ ik₁B₁共t兲

⫻共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1sech␰^, 共36兲

⌽12= − i

k₁A₂共t兲共2␭ + ik1tanh␰兲共tanh␰^{+ 1}兲^i␭/2k1共tanh␰^{− 1}兲^−i␭/2k1+ ik₁B₂共t兲

⫻共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1sech␰^. 共37兲 Hence one part共⌽x= U⌽兲 of the Lax equations has been solved. By using these solutions in Eqs.

共30兲–共33兲obtained from⌽t= V⌽, we find

A₁共t兲 = A1e^{i共k12+4␭2兲t/8} and B₁共t兲 = B1e^{−i共k12+4␭2兲t/8}, 共38兲

A₂共t兲 = A2e^{i共k12+4␭2兲t/8} and B₂共t兲 = B2e^{−i共k12+4␭2兲t/8}, 共39兲 where A₁, A₂, B₁, and B₂are arbitrary constants. We solved the Lax equations for a given U, V and a solution u of the mKdV equation关Eq. 共2兲兴. The components of ⌽ are

⌽11= − i

k₁A₁e^{i共k12+4␭2兲t/8}共2␭ + ik1tanh␰兲共tanh␰^{+ 1}兲^i␭/2k1共tanh␰^{− 1}兲^−i␭/2k1

+ ik₁B₁e^{−i共k12+4␭2兲t/8}共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1sech␰^, 共40兲

⌽12= − i

k₁A₂e^{i共k12+4␭2兲t/8}共2␭ + ik1tanh␰兲共tanh␰+ 1兲^i␭/2k1共tanh␰− 1兲^−i␭/2k1

+ ik₁B₂e^{−i共k12+4␭2兲t/8}共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1sech␰^. 共41兲

⌽21= iA₁e^{i共k12+4␭2兲t/8}共tanh␰^{+ 1}兲^i␭/2k1共tanh␰^{− 1}兲^−i␭/2k1sech␰^{+ B}1e^{−i共k12+4␭2兲t/8}共k1tanh␰^{+ 2i}␭兲

⫻共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1, 共42兲

⌽22= iA₂e^{i共k12+4␭2兲t/8}共tanh␰^{+ 1}兲^i␭/2k1共tanh␰^{− 1}兲^−i␭/2k1sech␰^{+ B}2e^{−i共k12+4␭2兲t/8}共k1tanh␰^{+ 2i}␭兲

⫻共tanh␰− 1兲^i␭/2k1共tanh␰+ 1兲^−i␭/2k1. 共43兲

Here we find that det共⌽兲=关共k₁²+ 4␭²兲/k1兴共A1B₂− A₂B₁兲⫽0.

Inserting A, B, and ⌽ in Eq.共10兲, and solving the resultant equation and letting A1= A₂, B₁

=共A1e^␲␭/k1兲/k1, and B₂= −B₁, we obtain a three parameter 共␭,k1,␮兲 family of surfaces param- etrized by

y₁= − 1

4k₁共e^2␰+ 1兲R₁共E共e^2␰+ 1兲 + 32k1兲, 共44兲

y₂= − 4R₁cos G sech␰^, 共45兲

y₃= 4R₁sin G sech␰^, 共46兲

where

R₁= − ␮^k1

2共k₁²+ 4␭²兲, 共47兲

G = t

冉

冊

E =共t关8␭ + k₁²兴 + 4x兲共k₁²+ 4␭²兲, 共49兲

␰^=k1

3

8

冉

2

冊

This surface has the following first and second fundamental forms:

共dsI兲²=1

4␮²

冋 冉

冋

册

冊

册

共dsII兲²=1

2␮^k1sech␰

冋

冉

冊

册

and the Gaussian and mean curvatures, respectively, are

K = k₁²

␮^{2共2 sech2}␰− 1兲, 共52兲

H = 1

4␮^k1sech␰^共6k12sech2␰⁺共4␭²− k₁²兲兲. 共53兲 Proposition 4: The surface which is parametrized by Eqs. (44)–(46) is a cubic Weingarten surface, i.e.,

4␮^2H2共2关␮^{2K + k}12兴兲 − 9␮^{4K2− 12}␮²共k₁²+ 2␭²兲K − 共k₁²+ 2␭²兲²= 0. 共54兲 When k₁= 2␭ in Eqs.(52)and(53), it reduces to a quadratic Weingarten surface, i.e.,

8␮²H²− 9␮²K − 36␭²= 0. 共55兲

B. The analyses of the three parameter family of mKdV surfaces

In general, y₂and y₃are asymptotically decaying functions, and y₁approaches ±⬁ as␰^tends to ±⬁. For some small intervals of x and t, we plot some of the three parameter family of surfaces for some special values of the parameters k₁,␭, and␮^{in Figs. 1–4.}

Example 2: By taking k₁= 2 ,␭=1, and␮= −8 in Eqs.共44兲–共46兲, we get the surface 共Fig.1兲.

The components of the position vector of the surface are

y₁= − E₁− 8/共e^2␰+ 1兲, y2= − 4 cos G sech␰^{, y}3= − 4 sin G sech␰^, 共56兲 where E₁= 4共x+3t兲,G=x+3t, and ␰= x + t. As ␰ tends to ±⬁, y₁ approaches ±⬁, and y₂ and y₃ approach zero. This can also be seen in Fig. 1. For small values of x and t, the surface has a twisted shape.

Example 3: By taking k₁= 2,␭=0, and␮= −4 in Eqs.共44兲–共46兲, we get the surface共Fig.2兲.

The components of the position vector of the surface are

y₁= − E₁− 8/共e^2␰+ 1兲, y2= − 4 cos G sech␰^{, y}3= − 4 sin G sech␰^, 共57兲 where E₁= 2共x+3t兲, G=t, and␰= x + t. As␰^{tends to ±}⬁, y1approaches ±⬁, and y2and y₃tend to zero. This can also be seen in Fig.2. Asymptotically, this surface and the surface given in Example 2 are the same. However, for small values of x and t, they are different.

FIG. 1. 共x,t兲苸关−3,3兴⫻关−3,3兴.

FIG. 2. 共x,t兲苸关−6,6兴⫻关−6,6兴.

Example 4: By taking k₁= 3,␭=1/10, and␮= −452/ 75 in Eqs.共44兲–共46兲, we get the surface 共Fig.3兲. The components of the position vector of the surface are

y₁= − E₁− 8/共e^2␰+ 1兲, y2= − 4 cos G sech␰^{, y}3= − 4 sin G sech␰^, 共58兲 where E₁= −共5537t+2260x兲/750, G=17共497t+20x兲/200, and ␰⁼共12x+27t兲/8. Asymptotically, this surface is similar to the previous two surfaces. For small values of x and t, the surface looks like a shell.

Example 5: By taking k₁= 1,␭=−1/10, and␮= −52/ 25 in Eqs.共44兲–共46兲, we get the surface 共Fig.4兲. The components of the position vector of the surface are

y₁= − E₁− 8/共e^2␰+ 1兲, y2= − 4 cos G sech␰^{, y}3= − 4 sin G sech␰^, 共59兲 where E₁= −13共20x+t兲/250, G=共47t−20x兲/200, and␰⁼共4x+t兲/8. Asymptotically, this surface is similar to the previous three surfaces.

III. mKdV SURFACES FROM SPECTRAL-GAUGE DEFORMATIONS

In this section, we find surfaces arising from a combination of the spectral and gauge defor- mations of the Lax pair for the mKdV equation.

Proposition 5: Let u satisfy (which describes a traveling mKdV wave) Eq. (2). The corre- sponding su(2) valued Lax pair U and V of the mKdV equation are given by Eqs. (3) and (4), respectively. The su(2) valued matrices A and B are

A = i

冉 ^共

^兲

共

兲 冊

FIG. 3. 共x,t兲苸关−6,6兴⫻关−6,6兴.

B = i

冉

共

兲

^共

^兲 冊

where A =␮U_␭+␯关␴2, U兴, B=␮V_␭+␯关␴2, V兴, ␭ is the spectral parameter,␮and ␯ are constants, and␴2is the Pauli sigma matrix. The surface S , generated by U, V, A, and B, has the following first and second fundamental forms共j,k=1,2兲:

共dsI兲²= gjkdx^jdx^k, 共62兲

共dsII兲²= h_jkdx^jdx^k, 共63兲

where

g₁₁=1

4␮²⁺␯共␯关u²+␭²兴 −␮^u兲, 共64兲

g₁₂=1

4共␣^{+ 2}␭兲␮²⁺¹

4␯共␯关2共␭ + 2␣兲u²+ 4共␭³+␣␭ + ␭²␣兲兴 − 4␮共␣⁺␭兲u兲, 共65兲

g₂₂=1

4共u²+共2␭ +␣兲²兲␮²⁺␯

冉

冋

册

−1

2␮u³−␮共␣²+共2␭ − 1兲␣+␭²兲u

冊

h₁₁=1

2␮u −␯共u²+␭²兲, 共67兲

FIG. 4. 共x,t兲苸关−20,20兴⫻关−20,20兴.

h₁₂=1

2␮共␣⁺␭兲u −␯

冉

冊

h₂₂=1

4␮共u³+ 2关␣²⁺共2␭ − 1兲␣⁺␭²兴u兲 −␯

冉

冊

共69兲 and the corresponding Gaussian and mean curvatures are

K = 2u共u²− 2␣兲

␯共2␯^u关u²− 2␣兴 − 3␮^{u2− 2}␮共␭²−␣兲兲 +␮^{2u, 共70兲}

H = ␮共3u²+ 2共␭²−␣兲兲 − 4u␯共u²− 2␣兲

2␯共2␯u关u²− 2␣兴 − 3␮^{u2− 2}␮共␭²−␣兲兲 + 2␮^{2u, 共71兲} where x¹= x and x²= t.

A. The parametrized form of the four parameter family of mKdV surfaces

We apply the same technique that we used in Sec. II to find the position vector of the corresponding surface. Let u = k₁sech␰^,␰^{= k}1共k₁²t + 4x兲/8, be the one soliton solution of the mKdV equation, where␣^{= k}12/ 4. The Lax pair U and V are given by Eqs.共3兲and共4兲, respectively, which is the same as in the spectral deformation case. So we can use the solution of the Lax equation 关Eq. 共5兲兴 that we found in the spectral deformation case. By solving Eq. 共10兲, we obtain the position vector, where the components of ⌽ are given by Eqs.共40兲–共43兲and A, B are given by Eqs.共60兲and共61兲, respectively. Here we choose A1= A₂, B₁=共A1e^␲␭/k1兲/k1, and B₂= −B₁to write F in the form F = i共␴1y₁+␴2y₂+␴3y₃兲. Hence we obtain a four parameter 共␭,k1,␮^,␯兲 family of surfaces parametrized by

y₁= R₂ e^2␰− 1

共e^2␰+ 1兲sech␰^{+ R}3E˜ + R₄ 1 e^2␰+ 1,

y₂=

冋

册

y₃=

冋

册

where

R₂= 2k₁²␯

k₁²+ 4␭², R₃=␮

8, 共73兲

R₄= 4␮^k12

k₁²+ 4␭², R₅=␯共k12− 4␭²兲

k₁²+ 4␭² , 共74兲

R₆=␯共4␭²+ 3k₁²兲

2共k12+ 4␭²兲 , R₇= 4␭k₁²␯

k₁²+ 4␭², 共75兲

G = t

冉

冊

E˜ = 共t关8␭ + k12兴 + 4x兲, 共77兲

␰^=k1

3

8

冉

冊

Thus the position vector y =共y1共x,t兲,y2共x,t兲,y3共x,t兲兲 of the surface is given by Eq. 共72兲. This surface has the following first and second fundamental forms共j,k=1,2兲:

共dsI兲²= g_jkdx^jdx^k, 共79兲

共dsII兲²= hjkdx^jdx^k, 共80兲

where

g₁₁=1

4␮²+␯共␯关k₁²sech²␰⁺␭²兴 −␮k₁sech␰兲,

g₁₂=1

4共␣+ 2␭兲␮²+1

4␯共␯关2k₁²共␭ + 2␣兲sech²␰⁺共4␭³+ 4␣␭ + 4␭²␣兲兴 − 4␮共␣+␭兲k1sech␰兲,

g₂₂=1

4共k₁²sech²␰⁺共2␭ +␣兲²兲␮²+␯

冉

冋

册

−1 2␮^k1

3sech³␰⁻␮^k1共␣²⁺共2␭ − 1兲␣⁺␭²兲sech␰

冊

h₁₁=1

2␮^k1sech␰⁻␯共k₁²sech²␰⁺␭²兲,

h₁₂=1

2␮共␣⁺␭兲k1sech␰⁻␯

冉

冊

h₂₂=1

4␮共k₁³sech³␰^{+ 2k}1

2关␣²+共2␭ − 1兲␣+␭²兴sech²␰兲

−␯

冉

冊

and the corresponding Gaussian and mean curvatures are

K = 2k₁sech␰共k₁²sech²␰^{− 2}␣兲

␯共2␯^k1sech␰关k12sech²␰^{− 2}␣兴 − 3␮^k12sech²␰^{− 2}␮共␭²−␣兲兲 +␮^2k1sech␰^,

H = ␮共3k12sech²␰^{+ 2}共␭²−␣兲兲 − 4␯^k1sech␰共k12sech²␰^{− 2}␣兲

2␯共2␯^k1sech␰关k₁²sech²␰^{− 2}␣兴 − 3␮^k12sech²␰^{− 2}␮共␭²−␣兲兲 + 2␮^2k1sech␰^, where x¹= x , x²= t, and␣⁼₄^1k₁^2.

B. The analyses of the four parameter family of surfaces

Asymptotically, y₁ approaches ±⬁, y2 approaches R₅cos G ± R₇sin G, and y₃ approaches

−R₅sin G ± R₇cos G as␰tends to ±⬁. For some small intervals of x and t, we plot some of the four parameter family of surfaces for some special values of the parameters k₁,␭,␮, and␯^{in Figs.}

5–7.

Example 6: By taking k₁= 2,␭=0,␮^{= −4, and} ␯^{= 1 in Eq.}共72兲, we get the surface共Fig.5兲.

The components of the position vector are

y₁= 2 sech␰共e^2␰− 1兲/共e^2␰+ 1兲 − E2− 8/共e^2␰+ 1兲, 共81兲

y₂=关− 4 sech␰⁺共e^4␰+ 1兲/共e^2␰+ 1兲²−共3/2兲sech²␰兴cos G, 共82兲

y₃=关4 sech␰⁻共e^4␰+ 1兲/共e^2␰+ 1兲²+共3/2兲sech²␰兴sin G, 共83兲 where E₂= −2共x+t兲,G=t and␰= x + t. As␰tends to ±⬁, y1 approaches ±⬁, y2 approaches cos t, and y₃approaches −sin t.

Example 7: By taking k₁= 2 , ␭=1,␮= 1 / 10, and␯^{= 1 in Eq.}共72兲, we get the surface共Fig.6兲.

The components of the position vector are

y₁= sech␰共e^2␰− 1兲/共e^2␰+ 1兲 + E2+ 1/共10共e^2␰+ 1兲兲, 共84兲

y₂=关共1/20兲sech␰^{− sech2}␰兴cos G + sin G共e^2␰− 1兲/共e^2␰+ 1兲, 共85兲

y₃=关− 共1/20兲sech␰^{+ sech2}␰兴sin G + cos G共e^2␰− 1兲/共e^2␰+ 1兲, 共86兲 where E₂=共x+3t兲/20, G=共x+3t兲, and ␰= x + t. As␰tends to ±⬁, y1 tends to ±⬁, y2 approaches sin共x+3t兲, and y3 approaches cos共x+3t兲.

Example 8: By taking k₁= 1,␭=−1/10,␮= −52/ 25, and␯= −1 in Eq.共72兲, we get the surface 共Fig.7兲. The components of the position vector are

y₁= −共25/13兲sech␰共e^2␰− 1兲/共e^2␰+ 1兲 − E2− 8/共e^2␰+ 1兲, 共87兲

FIG. 5. 共x,t兲苸关−4,4兴⫻关−4,4兴.

y₂=关− 4 sech␰⁻共12/13兲共e^4␰+ 1兲/共e^2␰+ 1兲²+共19/13兲sech²␰兴cos G

+共5/13兲sin G共e^2␰− 1兲/共e^2␰+ 1兲, 共88兲

y₃=关4 sech␰⁺共12/13兲共e^4␰+ 1兲/共e^2␰+ 1兲²−共19/13兲sech²␰兴sin G + 共5/13兲cos G共e^2␰− 1兲/共e^2␰+ 1兲, 共89兲 where E₂= 13共20x+t兲/250, G=共47t−20x兲/200, and␰⁼共4x+t兲/8. As ␰ ^{tends to ±}⬁, y1 tends to

±⬁, y2approaches cos t, and y₃ approaches −sin t.

IV. CONCLUSION

In this work, we considered mKdV 2-surfaces by using two deformations, spectral deforma- tion and a combination of gauge and spectral deformations of mKdV equation and its Lax pair. We found the first and second fundamental forms, and the Gaussian and mean curvatures of the

FIG. 6. 共x,t兲苸关−6,6兴⫻关−6,6兴.

FIG. 7. 共x,t兲苸关−20,20兴⫻关−20,20兴.