SURFACES FROM DEFORMATION OF PARAMETERS

Transcription

1 Seventeenth International Conference on Geometry, Integrability and Quantization June 5 10, 2015, Varna, Bulgaria Ivaïlo M. Mladenov, Guowu Meng and Akira Yoshioka, Editors Avangard Prima, Sofia 2016, pp doi: /giq SURFACES FROM DEFORMATION OF PARAMETERS SÜLEYMAN TEK and METIN GÜRSES Department of Mathematics, University of the Incarnate Word, 4301 Broadway San Antonio, TX 78209, USA Department of Mathematics, Faculty of Sciences, Bilkent University Ankara, Turkey Abstract. We construct surfaces from modified Korteweg-de Vries mkdv and sine-gordon SG soliton solutions by the use of parametric deformations. For each case there are two types of deformations. The first one gives surfaces on spheres and the second one gives highly complicated surfaces in three dimensional Euclidean space R 3. The SG surfaces that we obtained are not the critical points of functional where the Lagrange function is a polynomial function of the Gaussian K and mean H curvatures of the surfaces. We also give the graph of interesting mkdv and SG surfaces arise from parametric deformations. MSC: 53A05, 53C42, 35Q51, 35Q53 Keywords: Deformations of matrix Lax pair, integrable equations, soliton surfaces CONTENTS 1. Introduction General Theory Surfaces From a Variational Principle Surfaces in R mkdv Surfaces From Deformation of Parameters The Parameterized Form of the mkdv Surfaces Graph of Some of the mkdv Surfaces Sine-Gordon Surfaces SG surfaces are not the Critical Points of Functionals The Parameterized Form of the SG Surfaces Graph of Some of the SG Surfaces Conclusion

2 Surfaces From Deformation of Parameters Acknowledgments References Introduction Soliton surfaces or Integrable surfaces are the surfaces in R 3 obtained by a technique which uses the symmetries of both the integrable nonlinear partial differential equations and their Lax equations. In this technique, the main step is to associate a deformation corresponding to each symmetry. These are the scaling of the spectral parameter, gauge transformations and the classical and generalized symmetries of the integrable equations. By the use of corresponding deformations of different symmetries of different integrable equations such as mkdv, SG, nonlinear Schrödinger NLS, and KdV equations, several surfaces have been obtained by several authors [3 7], [12 20], [24 29]. In this work, we present a new deformation which corresponds to the integration parameters of the solutions of the integrable equations. These deformations are given by the formulas A i = δ i U = µ 2 U k i, B i = δ i V = µ 2 V k i, 1 Φ F i = µ 2 Φ 1 k i where i = 0, 1, n and k i are parameters of the solution ux, t, k 0,..., k n of the integrable equations, µ 2 is a constant. Here x and t are independent variables and n is the number of integration parameters [28]. Some of these deformations may generate exactly the same surface generated by other deformations such as symmetry and spectral parameter deformations but we have different surfaces corresponding to the deformation for different parameters. For a review of the integrable surfaces see [16]. 2. General Theory Let Φ be a group G-valued and U and V be the corresponding algebra g-valued functions satisfying the Lax equations Φ x = U Φ, Φ t = V Φ. 2 In the Lax equations all functions such as Φ, U and V are functions of x, t and the spectral parameter λ. Here and in what follows, subscripts x and t denote the derivatives of the objects with respect to x and t, respectively. The integrability condition of the Lax equations is the well known equation called the zero curvature condition [1], [2], [9] has the following form U t V x + [U, V ] = 0. 3

3 320 Süleyman Tek and Metin Gürses This equation is the source of all integrable nonlinear partial differential equations. There are symmetries which leave both the Lax equations and the integrable nonlinear partial differential equations invariant. They are i scaling the spectral parameter λ, ii gauge transformations, iii transformations whose infinitesimals are the generalized symmetries, iv scaling of the parameters of the solutions. For each transformations we can define infinitesimal symmetries. Let δ denotes such infinitesimal deformation and let δu = A and δv = B. Deformation operator δ commutes with D x and D t. Both A and B are g-valued functions. Taking δ deformation of the Lax equations in equation 2, we obtain the following equations δφ x = AΦ + UδΦ, δφ t = BΦ + V δφ 4 From the integrability condition of the equations in equations 4 or from the zero curvature condition given in equation 3, we get the following equation A t B x + [U, B] + [A, V ] = 0. 5 In addition, the equations in equations 4 imply the followings F x = Φ 1 A Φ, F t = Φ 1 B Φ 6 where F = Φ 1 δφ + F 0. 7 Notice that tracef = δdetφ. It is clear that, from the Lax equations given in equation 2, detf does not depend on x and t but it may depend on the spectral parameter λ and also other variables. Hence the constant term F 0 in equation 7 is added to make tracef = 0. In the theory of soliton surfaces, the g-valued function F is taken as the parametrization of the surfaces embedded in R 3 or in M 3 three dimensional Minkowski space. We give four types of these deformations below. The first three were given by Sym [24 26], Fokas and Gel fand [12], Fokas et al [5, 13] and Cieśliński [7] and others [27 29], [15, 16]. The last one is a new one and introduced in [28]. i Spectral parameter λ invariance of the equation U A = δu = µ 1 λ, B = δv = µ V 1 λ, F = µ 1 Φ 1 Φ λ where µ 1 is an arbitrary function of λ. This kind of deformation was first used by Sym [24 26]. ii Symmetries of the integrable differential equations A = δu, B = δv, F = Φ 1 δφ where δ represents the classical Lie symmetries and if integrable the generalized symmetries of the nonlinear PDE s [5, 7, 12, 13].

4 Surfaces From Deformation of Parameters 321 iii The Gauge symmetries of the Lax equation A = δu = M x + [M, U], B = δv = M t + [M, V ], F = Φ 1 MΦ where M is any traceless 2 2 matrix [7, 12, 13]. iv The deformation of parameters for solution of integrable equation A = δu = µ 2 U k i, B = δv = µ 2 V k i, 1 Φ F = µ 2 Φ k i where i = 0, 1,..., n and k i are parameters of the solution ux, t, k 0,..., k n of the PDEs, µ 2 is constant. Here x and t are independent variables [28]. Surfaces corresponding to integrable equations are called integrable surfaces and a connection formula, relating integrable equations to surfaces, was first established by Sym [24 26]. His formula gives a relation between family of immersions and Lax pairs defined in a Lie algebra. In this work, we consider deformation of parameters to construct new surfaces. This means that we take A i = δ i U = µ 2 U/ k i, B i = δ i V = µ 2 V/ k i where k i i = 0, 1, 2, n are the arbitrary parameters in the solutions the integrable equations. For the sake of completeness, we now give a brief summary that gives the relation between solutions of the integrable equations and surfaces. For the differential geometry of surfaces in R 3 see [10], [11] and [8]. Let F : U R 3 be an isometric immersion of a domain U R 2 into R 3, where R 2 is the 2-plane. Let x, t U. The surface F x, t is uniquely defined up to rigid motions by the first and second fundamental forms. Let Nx, t be the normal vector field defined at each point of the surface F x, t. Then the triple {F x, F t, N} at a point p S defines a basis of the tangent space at p, T p S, where S is the surface parameterized by F x, t. The first and the second fundamental forms of S are ds 2 I = g ij dx i dx j = F x, F x dx F x, F t dx dt + F t, F t dt 2 = A, A dx A, B dx dt + B, B dt 2 ds 2 II = h ij dx i dx j = F xx, N dx F xt, N dx dt + F tt, N dt 2 = A x + [A, U], C dx 2 2 A t + [A, V ], C dx dt B t + [B, V ], C dt 2 where i, j = 1, 2, x 1 = x and x 2 = t, A, B = 1/2traceAB, [A, B] = AB BA, A = A, A, and C = [A, B]/ [A, B]. A frame on this surface S is Φ 1 AΦ, Φ 1 BΦ, Φ 1 CΦ. The Gauss and the mean curvatures of S are given by K = detg 1 h, H = 1 2 traceg 1 h.

5 322 Süleyman Tek and Metin Gürses The function Φ, which is defined by equations given in equation 2 exists if and only if U and V satisfy the equation given in equation 3 [12]. In other words, equation 3 is the compatibility condition of equation 2. The equations in equation 6 define a surface F if and only if A and B satisfy the equation given in equation 5 [12]. Namely, equation 5 is the condition to define a surface F in Lie algebra g which is obtained from the equation given by equation 7. Furthermore, to have regular surfaces, F x and F t or A and B must be linearly independent at each point of the surface S. This is the regularity condition of the mapping F : S R 3. Hence the commutator [A, B] is nowhere zero on the surface. This ensures that the three vectors A, B and C form a triad at each point of the surface. For the given U and V, finding A and B from the equation in equation 5 is in general a difficult task. However, there are some deformations which provide A and B directly. Some of these deformations are given by Sym [24 26], Fokas and Gel fand [12], Fokas et al [13] and Cieśliński [7]. In order to obtain the surfaces using the given technique, we have to find position vector F which is given by [13] F = Φ 1 Φ µ 1 λ + µ Φ 2 + δφ + M Φ. k i In order to calculate F explicitly, the Lax equations provided by equation 2 need to be solved for a given solution of an integrable equation Surfaces From a Variational Principle Let H and K be the mean and Gaussian curvatures of a surface S either in M 3 or in R 3 then we have the following definition. Definition 1. Let S be a surface with its Gaussian K and mean H curvatures. A functional F is defined by F = S EH, KdA + p dv 8 V where E is some function of H and K, p is a constant and V is the volume enclosed within the surface S. The following proposition gives the first variation of the functional F. Proposition 2. Let E be a twice differentiable function of H and K. Then the Euler-Lagrange equation for F is given by the following equation [22, 23, 30 33] 2 + 4H 2 2K E H KH E 4HE + 2p = 0 9 K

6 Surfaces From Deformation of Parameters 323 where 2 and are the differential operators defined in the following way 2 = 1 gg ij g x i x j, = 1 gkh ij g x i x j and g = det g ij, g ij and h ij are components of the inverse of the first and second fundamental forms, i, j = 1, 2, and we assume Einstein s summation convention on repeated indices over their ranges. Example 3. The following are some examples of the surfaces in Proposition 2 for different choices of E i Minimal surfaces [21]: E = 1, p = 0. ii Constant mean curvature surfaces: E = 1. iii Linear Weingarten surfaces: E = ah+b, where a and b are some constants. iv Willmore surfaces: E = H 2 [34, 35]. v Surfaces solving the shape equation of lipid membrane: E = H c 2, where c is a constant [22, 23, 30 33]. Definition 4. The surfaces obtained from the solutions of the equation 2 H + ah 3 + bh K = 0 10 are called Willmore-like surfaces, where a and b are arbitrary constants. Remark 5. The case a = b = 2 in equation 10 corresponds to the Willmore surfaces. In this work, we assume p = 0. In addition, for surfaces derivable from a variational principal, we require asymptotic conditions such that H goes to a constant value and K goes to zero asymptotically. This is consistent with vanishing of boundary terms in obtaining Euler-Lagrange equation given by equation 9. This requires that the soliton equations such as KdV, mkdv, SG and NLS equations must have solutions decaying rapidly to zero at x ±. For this purpose, we shall calculate H and K for all surfaces obtained by mkdv and SG equation and look for possible solutions surfaces of the Euler-Lagrange equation [equation 9]. 3. Surfaces in R 3 In this section, closely following [5, 12, 13] and [16] we give the immersion of a surface in R 3 in terms of a group representation explicitly. For this purpose, we use Lie group SU2 and its Lie algebra su2 with basis e j = i σ j, j = 1, 2, 3, where σ j denote the usual Pauli sigma matrices i 1 0 σ 1 =, σ =, σ i 0 3 =

7 324 Süleyman Tek and Metin Gürses Define an inner product on su2 in the following way X, Y = 1 2 tracexy where X, Y su2 valued vectors. su2 valued representation F of a vector Y in R 3 can be written as 3 F x, t = i Y k σ k 12 where Y k, k = 1, 2, 3 are components of the vector F. We can write the vector F given in equation 12 more explicitly as Y F = i 3 Y 1 iy 2 Y 1 + iy 2 Y 3. k=1 In this representation, the inner product of two vectors is defined as F, G = 1 2 tracef G where F, G su2. The length of the vector is defined as F = F, F. 13 If F is the su2 representation of the position vector Yx, t, then F x and F t are the su2 representation of the tangent vectors Y,x and Y,t. If we let su2 representation of unit normal vector N as Z, then we find Z as Z = [F x, F t ] [F x, F t ] Here [.,.] denotes the usual commutator. Hence we can give the su2 representation of a triad defined at every point of a surface as {F x, F t, Z}. In soliton theory, surfaces are developed in this way. Fundamental forms are given as Fx, F g = x F x, F t Fx, Z, h = x F x, Z t F x, F t F t, F t F t, Z x F t, Z t and the Gaussian and mean curvatures take the following forms K = det g 1 h, H = 1 2 traceg 1 h.

8 Surfaces From Deformation of Parameters mkdv Surfaces From Deformation of Parameters In [5], [14 16], [27 29], we considered spectral parameter deformation and combination of spectral and Gauge deformations to develop surfaces. In this section, we consider the mkdv surfaces arising from deformations of parameters of the integrable equations solution. Let ux, t satisfy the mkdv equation u t = u xxx u2 u x. 14 Substituting the travelling wave ansatz u t α u x = 0 in the equation given by equation 14, we get u xx = αu u where α is an arbitrary real constant and integration constant is taken to be zero. mkdv equation in equation 15 can be obtained from the following Lax pairs U and V U = i 2 λ u u λ V = i 1 2 u2 α + αλ + λ 2 α + λu iu x 2 α + λu + iu x 1 2 u2 + α + αλ + λ 2 and λ is a spectral parameter. Consider the one soliton solution of mkdv equation 15 as 16 u = k 1 sech ξ 1 17 where α = k 2 1 /4, ξ 1 = k 1 k 2 1 t + 4x/8 + k 0, and k 0 and k 1 are arbitrary constants. The following Proposition gives the mkdv surfaces arising from deformation of parameter k 0. Proposition 6. Let u which describes a travelling mkdv wave, given by equation 17, satisfy the equation 15. The corresponding su2 valued Lax pairs U and V of the mkdv equation are provided by equations 16. The su2 valued matrices A and B are obtained as 0 ϕ0 A = iµ 2 B = iµ 2 ϕ 0 0 u ϕ 0 k1 2/4 + λϕ 0 i ϕ 0 x k1 2/4 + λϕ 0 + i ϕ 0 x u ϕ 0

9 326 Süleyman Tek and Metin Gürses where A = µ U/ k 0, B = µ V/ k 0, ϕ 0 = u/ k 0, k 0 is a parameter of the one soliton solution u, and µ is a constant. Then the surface S, generated by U, V, A and B, has the following first and second fundamental forms j, k = 1, 2 where ds 2 I g jk dx j dx k, ds 2 II h jk dx j dx k g 11 = 1 4 µ2 ϕ 2 0, g 12 = g 21 = 1 16 µ2 ϕ 2 0k λ g 22 = 1 64 µ2 16 ϕ 0 2 x + ϕ u 2 + k λ 2 h 11 = 16 1 λ u ϕ 2 0 h 12 = 4 1 ϕ 0 4 ϕ 0 x u x + u ϕ 0 2 u 2 k1λ λ 2 h 22 = 1 u ϕ 2 0 k λ 2u 2 + 4λ 2 + k1λ ϕ 0 4uϕ 0 xt ϕ 0 x k λu x + 4u t +4uϕ 0 x ϕ 0 x k λ 4ϕ 0 t 1 = µ 32 ϕ 0 2 x + u 2 ϕ 2 0 1/2 and the corresponding Gaussian and mean curvatures have the following form where x 1 = x, x 2 = t. K = 16λ2 k 2 1 µ2, H = 4λ k 1 µ The mkdv surfaces obtained from k 0 deformation given in Proposition 6 are spheres. Another parameter of the one soliton solution of mkdv equation is k 1. We use k 1 parameter deformation to construct new mkdv surfaces in the following proposition. Proposition 7. Let u, given by equation 17, satisfy the equation 15. The corresponding su2 valued Lax pairs U and V of the mkdv equation are given by equations 16. The su2 valued matrices A and B are obtained as A = iµ 0 ϕ1 2 ϕ 1 0 B = iµ 8 4 u ϕ1 2k 1 λ + 1 τ 4iϕ 1 x τ + 4iϕ 1 x 4 u ϕ 1 + 2k 1 λ

10 Surfaces From Deformation of Parameters 327 where A = µ U/ k 1, B = µ V/ k 1, τ = 2k 1 u + k λϕ 1 and ϕ 1 = u/ k 1, k 1 is a parameter of the one soliton solution u, and µ is a constant. Then the surface S, generated by U, V, A and B, has the following first and second fundamental forms j, k = 1, 2 ds 2 I g jk dx j dx k, ds 2 II h jk dx j dx k where g 11 = 1 4 µ2 ϕ 2 1, g 12 = g 21 = 1 16 µ2 ϕ 1 2 k 1 u + ϕ 1 k λ g 22 = 1 64 µ2 4k ϕ 2 1u k 1 k1 2 4u ϕ ϕ 1 2 x +k λ 2 ϕ k1λ h 11 = µ 3 λ ϕ 2 1 h 12 = h 21 = µ 3 ϕ 2 1 k 1 λ u ϕ 1 8 ϕ 1 x u x + k 1 λ + 1 2uϕ 1 22λ 2 u 2 + k1λ h 22 = 1 { µ 3 ϕ 1 8 ϕ 1 x 2 k 1 u u x + k λ ϕ 1 u x uϕ 1 x +4uϕ 1 t } + k 1 λ u ϕ 1 { 16 ϕ 1 xt 4 k 1 uu λ +ϕ 1 k λ 2u λ 2 + k 2 1λ + 1 }. The Gaussian and mean curvatures are K = ϕ 5 1 µ 10 4ϕ 1 2 x λϕ 1 xt 16ϕ 1 ϕ 1 2 xu 2 x +2 3 u x ϕ 1 x 4ϕ 1 u 2 + 2λk 1 u ϕ 1 λ + 2 k λ 2 { + 3 8λϕ 1 ϕ 1 x u t 2λuϕ 1 x ϕ 1 x k λ 4ϕ 1 t 3 /4 2u 2 ϕ 1 2u 2 3λ + 2 k1 2 12λ 2 +4λk 1 uu 2 + 2λ + k 2 1ϕ λ k λ 2}

11 328 Süleyman Tek and Metin Gürses H = µ 5 ϕ ϕ 1 ϕ 1 x 4uϕ 1 t 2 k 1 u u x k λϕ 1 u x { ϕ 1 ϕ xt + 2 k 1 u ϕ 1 2 u 2 8 λ 2 2k λ ϕ 1 2 x + 2 u 2 2 k1 2 λ + ϕ 2 13 k λ + 4 λ k1λ } k1 2 ϕ 2 1k λ 24 k 1 λ 3 u ϕ 1 where x 1 = x, x 2 = t. Here 2 and 3 are in the following form 8 2 = 2 µ 2 ϕ 1 u ϕ1 k 1 λ /2, 3 = k 1 λ + 1 2uϕ ϕ 2 x When u, given by equation 17, is substituted into K and H, they take the following forms where K = H = η 0 = sech ξ 1 1 k 1 η 1 tanh ξ 1, 1 7 µ 2 η 0 4 η η3 2 2 Q l sech ξ 1 l l= µ η 0 4 η η3 2 3/2 Z m sech ξ 1 m m=0 η 1 = 3 k 2 1 t + 4 x/8 η 2 = 3 k 3 1 π tk 2 1 t + 8 x/16, η 3 = k 1 π k 1 /π 2 k 1 η 0 η 4 = k 1 /2k 1 η 1 sech ξ 1 + 2η 0 tanh ξ 1 Q 1 = k1 6 π k 1 2 ππ + 4 k /π 4 Q 2 = k π 2 π k 1 2 η 1 sinh ξ 1 /π 4 Q 3 = 4 k1π 6 k 1 k 1 π π 2 k π k1 /π 4 Q 4 = 4Q 2 k1 π 2 6 k π / π k π 2 Q 5 = 4 k k 2 1 η1 2 + πk 1 2η /π 3 Q 6 = 4 k1η 8 1 sinh ξ πk1 4η /π 3, Q 7 = 48 k1η 9 1/π 2 3 Z 0 = 4 k 4 1k 1 π 3 /π 4, Z 1 = 0 Z 2 = 4 k 4 1k 1 π6 k 1 + πk 2 1 5/π 3 Z 3 = 24 k1η 5 1 sinh ξ 1 k 1 π 2 /π 3 Z 4 = 4 k1 5 k1 π2 + 3η k1η /π 2 Z 5 = 8 k1η 6 1 sinh ξ 1 k1 π1 + 2η /π 2, Z 6 = 56 k1 7 η1/π 2 2.

12 Surfaces From Deformation of Parameters The Parameterized Form of the mkdv Surfaces In the previous section, we constructed mkdv surfaces. We found the first and second fundamental forms, Gaussian and mean curvatures of the surfaces. But we did not find the position vectors of these surfaces. In this section, we explore the position vector y = y 1 x, t, y 2 x, t, y 3 x, t of the mkdv surfaces for a given solution of the mkdv equation and the corresponding Lax pairs. In order to find immersion function F explicitly, we need to find the solution Φ of the Lax equations given by equation 2. Consider the one soliton solution of mkdv equation u = k 1 sech ξ 1, where α = k1 2/4, ξ 1 = k 1 k1 2t + 4x/8 + k 0, and k 0 and k 1 are arbitrary constants. We solve the Lax equations [equation 2] for the given U and V by equations 16, respectively and a solution of the mkdv equation. The components of the 2 2 matrix Φ are where Φ 11 = 4 k 1 A 1 2 λi k 1 tanh ξ 1 Exp ik λ 2 t/8 Ξ 1 i k 2 1B 1 sech ξ 1 Exp ik λ 2 t/8 Ξ 2 Φ 12 = 4 k 1 A 2 2 λi k 1 tanh ξ 1 Exp ik λ 2 t/8 Ξ 1 i k 2 1B 2 sech ξ 1 Exp ik λ 2 t/8 Ξ 2 Φ 21 = 4 i A 1 sech ξ 1 Exp ik λ 2 t/8 Ξ 1 +B 1 2 λi + k 1 tanh ξ 1 Exp ik λ 2 t/8 Ξ 2 Φ 22 = 4 i A 2 sech ξ 1 Exp ik λ 2 t/8 Ξ 1 +B 2 2 λi + k 1 tanh ξ 1 Exp ik λ 2 t/8 Ξ 2 Ξ 1 = tanh ξ iλ/2k 1 tanh ξ 1 1 iλ/2k 1 Ξ 2 = tanh ξ 1 1 iλ/2k 1 tanh ξ iλ/2k 1 4 = k 1 /k λ2. Here we find the determinant of the matrix Φ as detφ = A 1 B 2 A 2 B

13 330 Süleyman Tek and Metin Gürses Immersion function of the mkdv surfaces obtained using k 0 deformation We find the immersion function F of the mkdv surface obtained using k 0 deformation by using the following equation 1 Φ F = ν Φ + k 0 r11 r 12 r 21 r 22 from which we obtain the position vector, where the components of Φ are given by equations 19, respectively. Here we choose A 1 = k 1 B 2 Exp λ π/k 1, A 2 = k 1 B 1 Exp λ π/k 1, r 11 = r 22 = 0, r 12 = r 21 to write F in the form F = iσ 1 y 1 + σ 2 y 2 + σ 3 y 3. Hence we obtain a family of mkdv surfaces parameterized by y 1 = W 6 sech 2 ξ 1 W 3 coshξ 1 cosω 1 +W 4 sinhξ 1 sinω 1 + 4λ W 8 2W1 cosh2ξ 1 + W 7 y 2 = 1 sech 2 ξ 1 W 10 sinhξ 1 cosω 1 W 5 W 11 coshξ 1 sinω 1 + W 9 cosh 2 ξ 1 y 3 = W 6 sech 2 ξ 1 W 13 coshξ 1 cosω 1 W 12 sinhξ 1 sinω 1 + 2λ W 2 2W1 cosh2ξ 1 + W 7 20 where Ω 1 = k 2 1λ + 1/4 + λ 2 t + λ x + 2λ k 0 /k 1 ξ 1 = k 1 k 2 1t + 4x/8 + k 0 W 1 = k λ 2 /8, W 2 = B 2 1 B 2 2, W 3 = W 2 k 3 1 W 4 = 2λ W 2 k 2 1, W 5 = k λ 2, W 6 = ν/ W 5 k 1 B B 2 2 W 7 = 5k 2 1/4 + λ 2, W 8 = B 1 B 2, W 9 = W 5 r 21, W 10 = 2 λν k 1 W 11 = νk 2 1, W 12 = 4 λ W 8 k 2 1, W 13 = 2W 8 k 3 1. Thus the position vector y = y 1 x, t, y 2 x, t, y 3 x, t of the surface is given by equations 20.

14 Surfaces From Deformation of Parameters Immersion function of the mkdv surfaces obtained using k 1 deformation We find the immersion function F of the mkdv surface obtained using k 1 deformation by using the following equation 1 Φ F = ν Φ + k 1 r11 r 12 r 21 r 22 from which we obtain the position vector, where the components of Φ are given by equations 19, respectively. Here we choose A 1 = k 1 B 2 Exp λ π/k 1, A 2 = k 1 B 1 Exp λπ/k 1 r 11 = r 22 = ν πλ + k 1 B2 2 B2 1 k1 2B2 1 + B2 2 r 12 = r 21 k 2 1 B2 1 + B νB 1B 2 πλ + k 1 k 2 1 B2 1 + B2 2 in order to write F in the form F = iσ 1 y 1 + σ 2 y 2 + σ 3 y 3. Hence we obtain a family of mkdv surfaces parameterized by y 1 = W 14 sech 2 ξ 1 W 15 2Ω 2 sinhξ 1 16/3 coshξ 1 sinω 1 +W 16 Ω 2 coshξ 1 cosω 1 +W 8 2Ω 3 cosh2ξ 1 + 2k1λ 2 sinh2ξ 1 + Ω 4 y 2 = W 14 sech 2 ξ 1 W 17 2Ω 2 sinhξ 1 16/3 coshξ 1 cosω 1 21 W 18 Ω 2 coshξ 1 sinω 1 + W 19 cosh2ξ y 3 = W 14 sech 2 ξ 1 W 20 2Ω 2 sinhξ 1 16/3 coshξ 1 sinω 1 W 21 Ω 2 coshξ 1 cosω 1 +W 2 /2 2Ω 3 cosh2ξ 1 + 2k1λ 2 sinh2ξ 1 + Ω 4

15 332 Süleyman Tek and Metin Gürses where Ω 2 = t k x k 1 /3, Ω 3 = 4 λ 2 + k1 k 2 1λ 3 + 1t 4 λ k 0 Ω 4 = tk3 1 4 λ 2 λ k λ λ k 12 2 x k 1 k 0 4 λ 2 k 0 W 14 = W 6 /k 1, W 15 = 3λ W 2 k1/8, 2 W 16 = 3 W 2 k1/8 3 W 17 = λ k1b B2, 2 W 18 = k1b B2 2 W 19 = 4 r 21 k1b B2/ν 2 + W 8 k 1 + λ π W 5 /3 W 20 = 3 λ W 8 k 2 1/4, W 21 = W 8 k 3 1. Thus the position vector y of the surface is given by equations Graph of Some of the mkdv Surfaces Graph of some of the mkdv surfaces from k 0 deformation Example 8. Taking λ = 2.7, ν = 1, B 1 = 1, B 2 = 1, k 0 = 0.3, k 1 = 1.5 and r 21 = 1 in the equations provided by equations 20, we get the surface given in Fig. 1. Figure 1. x, t [ 5, 5] [ 5, 5]

16 Surfaces From Deformation of Parameters Graph of some of the mkdv surfaces from k 1 deformation Example 9. Taking λ = 0.03, ν = 1, B 1 = 1, B 2 = 1, k 0 = 0, k 1 = 0.1 and r 21 = 1 in the equations provided by equations 21, we get the surface given in Fig. 2. Figure 2. x, t [ 3000, 3000] [ 3000, 3000] Figure 3. x, t [ 100, 100] [ 100, 100] Example 10. Taking λ = 0, ν = 1, B 1 = 1, B 2 = 1, k 0 = 0, k 1 = 0.7 and r 21 = 1 in the equations provided by equations 21, we get the surface given in Fig. 3. Example 11. Taking λ = 0.8, ν = 1, B 1 = 1, B 2 = 1, k 0 = 0, k 1 = 0.2 and r 21 = 1 in the equations provided by equations 21, we get the surface given in Fig. 4. Example 12. Taking λ = 0.8, ν = 1, B 1 = 1, B 2 = 1, k 0 = 5, k 1 = 0.2 and r 21 = 1 in the equations provided by equations 21, we get the surface given in Fig. 5.

17 334 Süleyman Tek and Metin Gürses Figure 4. x, t [ 20, 20] [ 20, 20] Figure 5. x, t [ 20, 20] [ 20, 20] Example 13. Taking λ = 0.1, ν = 1, B 1 = 1, B 2 = 1, k 0 = 4, k 1 = 0.2 and r 21 = 1 in the equations provided by equations 21, we get the surface given in Fig. 6. Figure 6. x, t [ 500, 500] [ 500, 500] Figure 7. x, t [ 80, 80] [ 80, 80] Example 14. Taking λ = 0.4, ν = 1, B 1 = 1, B 2 = 1, k 0 = 0, k 1 = 0.2 and r 21 = 1 in the equations provided by equations 21, we get the surface given in Fig. 7.

18 Surfaces From Deformation of Parameters Sine-Gordon Surfaces Let ux, t be the solution of the sine-gordon equation u xt = sin u. 22 The matrix Lax pairs U and V of the SG equation given by equation 22 are U = i λ ux, V = 1 i cos u sin u 23 2 u x λ 2 λ sin u i cos u were λ is a spectral constant. One soliton solution of the SG equation given in equation 22 has the following form u = 4 arctan e ξ Here ξ 2 = x/k 3 + k 3 t + k 2, where k 2 and k 3 are arbitrary constants. We develop SG surfaces using deformation of parameter k 2 in the following Proposition. Proposition 15. Let u, provided by equation 24, satisfy the SG equation given in equation 22. The corresponding su2 valued Lax pairs U and V of the SG equation are provided by equations 23. The su2 valued matrices A and B are obtained as A = i µ 2 0 ϕ2 x ϕ 2 x 0, B = µ i sin u ϕ2 cos u ϕ 2 2 λ cos u ϕ 2 i sin u ϕ 2 where A = µ U/ k 2, B = µ V/ k 2, ϕ 2 = u/ k 2, k 2 is a parameter of the one soliton solution u, and µ is a constant. Then the surface S, generated by U, V, A and B, has the following first and second fundamental forms j, k = 1, 2 1 ds 2 I g jk dx j dx k = µ 2 sech 2 ξ 2 k3 2 tanh 2 ξ 2 dx λ 2 dt2 ds 2 II h jk dx j dx k = 2 µ sech 2 ξ 2 λ 2 tanh 2 ξ 2 dx 2 + k3 2 dt 2 λ k 3 and the corresponding Gaussian and mean curvatures are 2 λ 2 k3 K =, H = 2 λ k 3 µ µ where x 1 = x, x 2 = t. These surfaces given in Proposition 15 are also sphere in R 3. Similar to k 2 deformation, the following proposition gives new SG surfaces by using deformation of the other parameter k 3 of the one soliton solution given in equation 24.

19 336 Süleyman Tek and Metin Gürses Proposition 16. Let u, provided by equation 24, satisfy the SG equation given in equation 22. The corresponding su2 valued Lax pairs U and V of the SG equation are given by equation 23 respectively. The su2 valued matrices A and B are obtained as A = i µ 2 0 ϕ3 x ϕ 3 x 0, B = µ i sin u ϕ3 cos u ϕ 3 2 λ cos u ϕ 3 i sin u ϕ 3 where A = µ U/ k 3, B = µ V/ k 3, ϕ 3 = u/ k 3, k 3 is a parameter of the one soliton solution u,and µ is a constant. Then the surface S, generated by U, V, A and B, has the following first and second fundamental forms j, k = 1, 2 ds 2 I g jk dx j dx k, ds 2 II h jk dx j dx k 25 and the corresponding Gaussian and mean curvatures are where K = 4m 2 sinh ξ 2 0 ξ 3 k 3 cosh ξ 2 ξ 3 sinh ξ 2 H = m 0 k 3 cosh ξ 2 2 ξ 3 sinh ξ 2 ξ 3 k 3 cosh ξ 2 ξ 3 sinh ξ ξ 2 = x/k 3 + k 3 t + k 2, ξ 3 = x t k3, 2 m 0 = λ k3 3 µ g 11 = µ2 2, µ 2 k3 6 sech 4 ξ3 ξ 2 k 3 cosh ξ 2 ξ 3 sinh ξ 2 g22 = λ k3 2 sech 2 ξ 2 h 11 = 2µ λ k 3 3 h 22 = 2µ ξ 3 λ k 3 sech 2 ξ 2. sinh ξ 2 sech 4 ξ 2 k 3 cosh ξ 2 ξ 3 sinh ξ SG surfaces are not the Critical Points of Functionals Let us consider a polynomial Lagrange as functions of the curvatures H and K E = a N0 H N a 11 H K + a 21 H 2 K a 01 K where a ij with i, j = 0, 1, 2, N are all constants. Here H and K are mean and Gauss curvatures of the SG surfaces given in equations 26 and 27 with the metrics equation 25. When we use equation 28 in the Euler-Lagrange equation [equation 9] the direct calculation yields all the constants a nl = 0 and p = 0, where n, l = 0, 1, 2,... and N = 3, 4, 5,.... This leads to an important theorem on SG surfaces.

20 Surfaces From Deformation of Parameters 337 Theorem 17. The SG surfaces obtained in Proposition 16 are not the critical points of functional E in the generalized shape equation given by equation 9 when E is a polynomial function of H and K The Parameterized Form of the SG Surfaces In this section, we find the position vector y of the SG surfaces obtained using k 2 and k 3 deformation. For this purpose, first we need to solve the Lax equations given by equation 2 using the given Lax pairs. Using one soliton solution given by equation 24 and the Lax pairs U and V provided by equations 23 we solve the Lax equations. The components of the solution 2 2 matrix Φ are obtained as Φ 11 = 1 2 Ξ 4 A 1 Ξ 3 Exp ξ 4 + 2B 1 Expξ 4 Φ 12 = 1 2 Ξ 4 A 2 Ξ 3 Exp ξ 4 + 2B 2 Expξ 4 1 Φ 21 = Ξ 4 2 A 1 Exp ξ 4 B 1 Ξ 3 Expξ 4 1 Φ 22 = Ξ 4 2 A 2 Exp ξ 4 B 2 Ξ 3 Expξ 4 where Ξ 3 denotes the complex conjugate of Ξ 3 and Ξ 3 = λ k 3 coshξ 2 + i sinhξ 2, Ξ 4 = sechξ 2 ξ 4 = i k3 tk 3 ξ 2 λ 2 + t. 2λ Here we find the determinant of the matrix Φ as detφ = 1 2 λ 2 k A 1 B 2 A 2 B Immersion function of the SG surface obtained using k 2 deformation We find the immersion function F of the SG surface obtained using k 2 deformation by using the following equation 1 Φ r11 r F = ν Φ + 12 k 2 r 21 r 22 from which we obtain the position vector, where the components of Φ are given by equations 29, respectively. Here we choose A 1 = 2 B 2, A 2 = 2 B 1, r 11 = r 22 = 0, r 12 = r 21 to write F in the form F = iσ 1 y 1 + σ 2 y 2 + σ 3 y 3. Hence

21 338 Süleyman Tek and Metin Gürses we obtain components of the position vector of the SG surfaces as y 1 = D 5 sech 2 ξ 2 D 1 coshξ 2 cosζ 1 D 2 sinhξ 2 sinζ 1 + D D4 cosh 2 ξ 2 y 2 = 1 sech 2 ξ 2 D 7 sinhξ 2 cosζ 1 D 6 ν coshξ 2 sinζ 1 + D 8 cosh 2 ξ 2 y 3 = D 5 sech 2 ξ 2 D 9 coshξ 2 cosζ 1 D 3 sinhξ 2 sinζ 1 + D D4 cosh 2 ξ 2 30 where ξ 2 = x/k 3 + k 3 t + k 2, ζ 2 = t/λ x + k 2 k 3 λ D 1 = B 2 1 B 2 2, D 2 = λ k 3 D 1, D 3 = λ k 3 D 9 D 4 = D 6 /2, D 5 = ν/b B 2 2 D 6, D 6 = λ 2 k D 7 = ν λ k 3, D 8 = D 6 r 21, D 9 = 2 B 1 B 2. Thus the position vector y of the SG surface is given by equations Immersion function of the SG surface obtained using k 3 deformation We find the immersion function F of the SG surface obtained using k 3 deformation by using the following equation 1 Φ F = ν Φ + k 3 r11 r 12 r 21 r 22 from which we obtain the position vector, where the components of Φ are given by equations 29, respectively. Here we choose A 1 = 2 B 2, A 2 = 2 B 1, r 11 = r 22 = ν k 3 λ 2 / λ 2 k , r 12 = r 21 to write F in the form F = iσ 1 y 1 + σ 2 y 2 + σ 3 y 3. Hence we obtain components of the position vector of the SG

22 Surfaces From Deformation of Parameters 339 surfaces y 1 = D 10 sech 2 ξ 2 D 2 ξ3 sinhξ 2 k 3 coshξ 2 sinζ 1 +D 1 ξ 3 coshξ 2 cosζ 1 D 3 D11 cosh 2 ξ 2 ξ 3 + k 3 /2 sinh2ξ 2 y 2 = D 12 sech 2 ξ 2 D 7 ξ3 sinhξ 2 k 3 coshξ 2 cosζ 1 +ν ξ 3 coshξ 2 sinζ 1 + D 13 cosh 2 ξ 2 y 3 = D 10 sech 2 ξ 2 D 3 ξ3 sinhξ 2 k 3 coshξ 2 sinζ 1 D 9 ξ 3 coshξ 2 cosζ 1 +D 2 D11 cosh 2 ξ 2 ξ 3 + k 3 /2 sinh2ξ 2 31 where ξ 2 = x/k 3 + k 3 t + k 2, ξ 3 = x t k 2 3 ζ 1 = t/λ x + k 2 k 3 λ, D 10 = D 5 /k 2 3 D 11 = k 2 k 3 D 6 /2, D 12 = 1/[D 6 k 2 3], D 13 = D 8 k 2 3. Thus the position vector y of the SG surface is given by equations Graph of Some of the SG Surfaces Graph of some of the SG surfaces from k 2 deformation Example 18. Taking λ = 2, ν = 1, B 1 = 2, B 2 = 2, k 2 = 0, k 3 = 3 and r 21 = 1 in the equations provided by equations 30, we get the surface given in Fig Graph of some of the SG surfaces from k 3 deformation Example 19. Taking λ = 4, ν = 1, B 1 = 1, B 2 = 1, k 2 = 0, k 3 = 2 and r 21 = 0 in the equations provided by equations 31, we get the surface given in Fig. 9. Example 20. Taking λ = 1.4, ν = 1, B 1 = 1, B 2 = 1, k 2 = 0, k 3 = 1 and r 21 = 1 in the equations provided by equations 31, we get the surface given in Fig. 10.

23 340 Süleyman Tek and Metin Gürses Figure 8. x, t [ 8, 8] [ 8, 8] Figure 9. x, t [ 4, 4] [ 4, 4] Example 21. Taking λ = 0.4, ν = 1, B 1 = 1, B 2 = 1, k 2 = 0, k 3 = 1 and r 21 = 1 in the equations provided by equations 31, we get the surface given in Fig. 11. Figure 10. x, t [ 10, 10] [ 10, 10] Figure 11. x, t [ 5, 5] [ 5, 5] 5. Conclusion In this work, we introduce a new deformation, namely, deformation of parameters of solution of integrable equations to develop surfaces from integrable equations. Using this deformation, we construct surfaces from modified mkdv and

24 Surfaces From Deformation of Parameters 341 SG equations. For each integrable equation, we considered two types of deformations arising from parameters of soliton solutions of the corresponding integrable equation. One of them gives surfaces on spheres and the other one gives highly complicated surfaces in R 3. We obtain the quantities such as first and second fundamental forms, Gaussian and mean curvatures of the mkdv and SG surfaces. We find the position vector of mkdv and SG surfaces using the immersion function F. Furthermore, we provide the graph of interesting mkdv and SG surfaces. The SG surfaces that we obtained are not the critical points of functional where the Lagrange function is a polynomial function of the Gaussian and mean curvatures of the SG surfaces. 6. Acknowledgments This work is partially supported by the Scientific and Technological Research Council of Turkey TUBITAK. References [1] Ablowitz M. and Segur H., Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge [2] Blaszak M., Multi-Hamilton Theory of Dynamical Systems, Springer, Berlin [3] Bobenko A., Surfaces in Terms of 2 by 2 Matrices, Old and New Integrable Cases in Harmonic Maps and Intgerable Systems, A. Fordy and J. Wood Eds, Aspects of Mathematics, Friedr. Vieweg, Braunscweig 1994, pp [4] Bobenko A., Integrable Surfaces, Funct. Anal. Prilozh [5] Ceyhan Ö., Fokas A., and Gürses M., Deformations of Surfaces Associated with Integrable Gauss-Minardi-Codazzi Equations, J. Math. Phys [6] Cieśliński J., Goldstein P. and Sym A., Isothermic Surfaces in E 3 as Soliton Surfaces, Phys. Lett. A [7] Cieśliński J., A Generalized Formula for Integrable Classes of Surfaces in Lie Algebraas, J. Math. Phys [8] Do Carmo M., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs [9] Drazin P., Solitons: An Introduction, Cambridge Univ. Press, New York [10] Eisenhart L., A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York [11] Eisenhart L., Riemannian Geometry, Princeton Univ. Press, Princeton [12] Fokas A. and Gelfand I., Surfaces on Lie Groups, on Lie Algebras, and Their Integrability, Commun. Math. Phys [13] Fokas A., Gelfand I., Finkel F. and Liu Q.-M., A Formula for Constructing Infinitely Many Surfaces on Lie Algebras and Integrable Equations, Selecta Math. New Ser

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