Surfaces from Deformation ParametersXVIIth International Conference

Geometry, Integrability and QuantizationVarna, Bulgaria

Suleyman TekUniversity of the Incarnate Word, San Antonio, TX, USA

Metin GursesBilkent University, Ankara, Turkey

June 5-10, 2015

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 1 / 49

Introduction

Surface theory in R3 plays a crucial role in differential geometry,partial differential equations (PDEs), string theory, general theory ofrelativity, and biology [Parthasarthy and Viswanathan, 2001] -[Ou-Yang et. al., 1999].

Soliton equations play a crucial role for the construction of surfaces.

The theory of nonlinear soliton equations was developed in 1960s.

For details of integrable equations one may look [Drazin, 1989],[Ablowitz and Segur, 1991], and the references therein.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 2 / 49

Introduction

Lax representation of nonlinear PDEs consists of two linear equationswhich are called Lax equations

Φx = U Φ, Φt = V Φ, (1)

and their compatibility condition

Ut − Vx + [U, V ] = 0, (2)

where x and t are independent variables. Here U and V are called Laxpairs.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 3 / 49

Introduction

The relation of 2-surfaces and integrable equations is established by theuse of Lie groups and Lie algebras.

Using this relation, soliton surface theory was first developed by Sym[Sym, 1982]-[Sym, 1985]. He obtained the immersion function by usingthe deformation of Lax equations for integrable equations.

Fokas and Gel’fand [Fokas and Gelfand, 1996] generalized Sym’s resultand find more general immersion function.

Soliton surface technique is an effective method to develop surfaces inR3 and in M3.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 4 / 49

Introduction

The relation of 2-surfaces and integrable equations is established by theuse of Lie groups and Lie algebras.

Using this relation, soliton surface theory was first developed by Sym[Sym, 1982]-[Sym, 1985]. He obtained the immersion function by usingthe deformation of Lax equations for integrable equations.

Fokas and Gel’fand [Fokas and Gelfand, 1996] generalized Sym’s resultand find more general immersion function.

Soliton surface technique is an effective method to develop surfaces inR3 and in M3.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 4 / 49

Introduction

The relation of 2-surfaces and integrable equations is established by theuse of Lie groups and Lie algebras.

Using this relation, soliton surface theory was first developed by Sym[Sym, 1982]-[Sym, 1985]. He obtained the immersion function by usingthe deformation of Lax equations for integrable equations.

Fokas and Gel’fand [Fokas and Gelfand, 1996] generalized Sym’s resultand find more general immersion function.

Soliton surface technique is an effective method to develop surfaces inR3 and in M3.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 4 / 49

Introduction

The relation of 2-surfaces and integrable equations is established by theuse of Lie groups and Lie algebras.

Using this relation, soliton surface theory was first developed by Sym[Sym, 1982]-[Sym, 1985]. He obtained the immersion function by usingthe deformation of Lax equations for integrable equations.

Fokas and Gel’fand [Fokas and Gelfand, 1996] generalized Sym’s resultand find more general immersion function.

Soliton surface technique is an effective method to develop surfaces inR3 and in M3.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 4 / 49

Introduction

The relation of 2-surfaces and integrable equations is established by theuse of Lie groups and Lie algebras.

Using this relation, soliton surface theory was first developed by Sym[Sym, 1982]-[Sym, 1985]. He obtained the immersion function by usingthe deformation of Lax equations for integrable equations.

Fokas and Gel’fand [Fokas and Gelfand, 1996] generalized Sym’s resultand find more general immersion function.

Soliton surface technique is an effective method to develop surfaces inR3 and in M3.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 4 / 49

Introduction

In this method, one mainly uses the deformations of the Lax equationsof the integrable equations [Sym, 1982]-[Gurses and Tek, 2014],

Sine Gordon (SG) equation

Korteweg de Vries (KdV) equation

Modified Korteweg de Vries (mKdV) equation

Nonlinear Schrodinger (NLS) equation

There are many attempts to find new examples of two surfaces.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 5 / 49

Soliton Surface Technique

We follow Fokas and Gel’fand [Fokas and Gelfand, 1996] approach todevelop surfaces using integrable equations.

Lax equation

Φx = U Φ , Φt = V Φ. (3)

Compatibility condition

Ut − Vx + [U, V ] = 0, (4)

Deformation matrices A and B

Let δU = A, δV = B, where A and B satisfy

At −Bx + [A, V ] + [U,B] = 0. (5)

Soliton Surface: Let 〈, 〉 defines an inner product in g.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 6 / 49

Soliton Surface Technique

We follow Fokas and Gel’fand [Fokas and Gelfand, 1996] approach todevelop surfaces using integrable equations.

Lax equation

Φx = U Φ , Φt = V Φ. (3)

Compatibility condition

Ut − Vx + [U, V ] = 0, (4)

Deformation matrices A and B

Let δU = A, δV = B, where A and B satisfy

At −Bx + [A, V ] + [U,B] = 0. (5)

Soliton Surface: Let 〈, 〉 defines an inner product in g.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 6 / 49

Soliton Surface Technique

We follow Fokas and Gel’fand [Fokas and Gelfand, 1996] approach todevelop surfaces using integrable equations.

Lax equation

Φx = U Φ , Φt = V Φ. (3)

Compatibility condition

Ut − Vx + [U, V ] = 0, (4)

Deformation matrices A and B

Let δU = A, δV = B, where A and B satisfy

At −Bx + [A, V ] + [U,B] = 0. (5)

Soliton Surface: Let 〈, 〉 defines an inner product in g.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 6 / 49

Soliton Surface Technique

We follow Fokas and Gel’fand [Fokas and Gelfand, 1996] approach todevelop surfaces using integrable equations.

Lax equation

Φx = U Φ , Φt = V Φ. (3)

Compatibility condition

Ut − Vx + [U, V ] = 0, (4)

Deformation matrices A and B

Let δU = A, δV = B, where A and B satisfy

At −Bx + [A, V ] + [U,B] = 0. (5)

Soliton Surface: Let 〈, 〉 defines an inner product in g.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 6 / 49

Soliton Surface Technique

We follow Fokas and Gel’fand [Fokas and Gelfand, 1996] approach todevelop surfaces using integrable equations.

Lax equation

Φx = U Φ , Φt = V Φ. (3)

Compatibility condition

Ut − Vx + [U, V ] = 0, (4)

Deformation matrices A and B

Let δU = A, δV = B, where A and B satisfy

At −Bx + [A, V ] + [U,B] = 0. (5)

Soliton Surface: Let 〈, 〉 defines an inner product in g.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 6 / 49

Soliton Surface Technique

First fundamental form

(dsI)2 ≡ gij dxi dxj = 〈A,A〉 dx2 + 2〈A,B〉 dx dt+ 〈B,B〉 dt2,

Second fundamental form

(dsII)2 ≡ hij dxi dxj = 〈Ax + [A,U ], C〉 dx2

+ 2〈At + [A, V ], C〉 dx dt+ 〈Bt + [B, V ], C〉 dt2,

[A,B] = AB −BA, ||A|| =√|〈A,A〉|, and C = [A,B]

||[A,B]|| .

Gaussian and mean curvatures

K = det(g−1 h) , H =1

2trace(g−1 h), g = (gij), h = (hij).

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 7 / 49

Soliton Surface Technique

First fundamental form

(dsI)2 ≡ gij dxi dxj = 〈A,A〉 dx2 + 2〈A,B〉 dx dt+ 〈B,B〉 dt2,

Second fundamental form

(dsII)2 ≡ hij dxi dxj = 〈Ax + [A,U ], C〉 dx2

+ 2〈At + [A, V ], C〉 dx dt+ 〈Bt + [B, V ], C〉 dt2,

[A,B] = AB −BA, ||A|| =√|〈A,A〉|, and C = [A,B]

||[A,B]|| .

Gaussian and mean curvatures

K = det(g−1 h) , H =1

2trace(g−1 h), g = (gij), h = (hij).

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 7 / 49

Soliton Surface Technique

First fundamental form

(dsI)2 ≡ gij dxi dxj = 〈A,A〉 dx2 + 2〈A,B〉 dx dt+ 〈B,B〉 dt2,

Second fundamental form

(dsII)2 ≡ hij dxi dxj = 〈Ax + [A,U ], C〉 dx2

+ 2〈At + [A, V ], C〉 dx dt+ 〈Bt + [B, V ], C〉 dt2,

[A,B] = AB −BA, ||A|| =√|〈A,A〉|, and C = [A,B]

||[A,B]|| .

Gaussian and mean curvatures

K = det(g−1 h) , H =1

2trace(g−1 h), g = (gij), h = (hij).

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 7 / 49

Soliton Surface Technique

First fundamental form

(dsI)2 ≡ gij dxi dxj = 〈A,A〉 dx2 + 2〈A,B〉 dx dt+ 〈B,B〉 dt2,

Second fundamental form

(dsII)2 ≡ hij dxi dxj = 〈Ax + [A,U ], C〉 dx2

+ 2〈At + [A, V ], C〉 dx dt+ 〈Bt + [B, V ], C〉 dt2,

[A,B] = AB −BA, ||A|| =√|〈A,A〉|, and C = [A,B]

||[A,B]|| .

Gaussian and mean curvatures

K = det(g−1 h) , H =1

2trace(g−1 h), g = (gij), h = (hij).

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 7 / 49

Soliton Surface Technique

Since our aim is finding a class of surfaces which correspond tointegrable equations, we need to find A and B that satisfy thefollowing equation

At −Bx + [A, V ] + [U,B] = 0. (6)

But in general, solving that equation is not simple. However there aresome deformations which provide A and B directly.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 8 / 49

Soliton Surface Technique

Since our aim is finding a class of surfaces which correspond tointegrable equations, we need to find A and B that satisfy thefollowing equation

At −Bx + [A, V ] + [U,B] = 0. (6)

But in general, solving that equation is not simple. However there aresome deformations which provide A and B directly.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 8 / 49

Soliton Surface Technique

Spectral parameter λ invariance of the equation

A = µ1∂U

∂λ, B = µ1

∂V

∂λ, F = µ1 Φ−1

∂Φ

∂λ, (7)

That kind of deformation was first used by Sym[Sym, 1982]-[Sym, 1985].

Gauge symmetries of the Lax equation

A = Mx + [M,U ], B = Mt + [M,V ], F = Φ−1MΦ, (8)

where M is any traceless 2× 2 matrix. [Fokas and Gelfand, 1996],[Fokas et. al., 2000], [Cieslinski, 1997].

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 9 / 49

Soliton Surface Technique

Spectral parameter λ invariance of the equation

A = µ1∂U

∂λ, B = µ1

∂V

∂λ, F = µ1 Φ−1

∂Φ

∂λ, (7)

That kind of deformation was first used by Sym[Sym, 1982]-[Sym, 1985].

Gauge symmetries of the Lax equation

A = Mx + [M,U ], B = Mt + [M,V ], F = Φ−1MΦ, (8)

where M is any traceless 2× 2 matrix. [Fokas and Gelfand, 1996],[Fokas et. al., 2000], [Cieslinski, 1997].

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 9 / 49

Soliton Surface Technique

Spectral parameter λ invariance of the equation

A = µ1∂U

∂λ, B = µ1

∂V

∂λ, F = µ1 Φ−1

∂Φ

∂λ, (7)

That kind of deformation was first used by Sym[Sym, 1982]-[Sym, 1985].

Gauge symmetries of the Lax equation

A = Mx + [M,U ], B = Mt + [M,V ], F = Φ−1MΦ, (8)

where M is any traceless 2× 2 matrix. [Fokas and Gelfand, 1996],[Fokas et. al., 2000], [Cieslinski, 1997].

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 9 / 49

Soliton Surface Technique

Symmetries of the (integrable) differential equations

A = δU, B = δV, F = Φ−1δΦ, (9)

where δ represents the classical Lie symmetries and (if integrable) thegeneralized symmetries of the nonlinear PDE’s[Fokas and Gelfand, 1996], [Fokas et. al., 2000], [Cieslinski, 1997].

Deformation of parameters of solution of integrable equation

A = µ2 (∂U/∂ki) , B = µ2 (∂V/∂ki) , F = µ2Φ−1 (∂Φ/∂ki) , (10)

where i = 1, 2 and ki are parameters of the solution u(x, t, k1, k2) of thePDEs, µ2 is constant. [Gurses and Tek, 2015]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 10 / 49

Soliton Surface Technique

Symmetries of the (integrable) differential equations

A = δU, B = δV, F = Φ−1δΦ, (9)

where δ represents the classical Lie symmetries and (if integrable) thegeneralized symmetries of the nonlinear PDE’s[Fokas and Gelfand, 1996], [Fokas et. al., 2000], [Cieslinski, 1997].

Deformation of parameters of solution of integrable equation

A = µ2 (∂U/∂ki) , B = µ2 (∂V/∂ki) , F = µ2Φ−1 (∂Φ/∂ki) , (10)

where i = 1, 2 and ki are parameters of the solution u(x, t, k1, k2) of thePDEs, µ2 is constant. [Gurses and Tek, 2015]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 10 / 49

Soliton Surface Technique

Symmetries of the (integrable) differential equations

A = δU, B = δV, F = Φ−1δΦ, (9)

where δ represents the classical Lie symmetries and (if integrable) thegeneralized symmetries of the nonlinear PDE’s[Fokas and Gelfand, 1996], [Fokas et. al., 2000], [Cieslinski, 1997].

Deformation of parameters of solution of integrable equation

A = µ2 (∂U/∂ki) , B = µ2 (∂V/∂ki) , F = µ2Φ−1 (∂Φ/∂ki) , (10)

where i = 1, 2 and ki are parameters of the solution u(x, t, k1, k2) of thePDEs, µ2 is constant. [Gurses and Tek, 2015]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 10 / 49

mKdV Surfaces from Deformation of Parameters

In this section, we obtain the immersions of 2-surfaces in R3.

For this purpose, we use Lie group SU(2) and its Lie algebra su(2) withbasis ej = −i σj , j = 1, 2, 3, where σj denote the usual Pauli sigmamatrices

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (11)

Define an inner product on su(2) as

〈X,Y 〉 = −1

2trace(XY ), (12)

where X, Y ∈su(2) valued vectors.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 11 / 49

mKdV Surfaces from Deformation of Parameters

In this section, we obtain the immersions of 2-surfaces in R3.

For this purpose, we use Lie group SU(2) and its Lie algebra su(2) withbasis ej = −i σj , j = 1, 2, 3, where σj denote the usual Pauli sigmamatrices

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (11)

Define an inner product on su(2) as

〈X,Y 〉 = −1

2trace(XY ), (12)

where X, Y ∈su(2) valued vectors.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 11 / 49

mKdV Surfaces from Deformation of Parameters

In [Tek, 2007], we considered spectral parameter deformation andcombination of spectral and Gauge deformations of mKdV equation.

In this section, we consider the mKdV surfaces arising fromdeformations of parameters of the it’s soliton solution.Let u(x, t) satisfy the mKdV equation

ut = uxxx +3

2u2ux. (13)

Substituting the travelling wave ansatz ut − αux = 0 in Eq. (13), weget

uxx = αu− u3

2. (14)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 12 / 49

mKdV Surfaces from Deformation of Parameters

In [Tek, 2007], we considered spectral parameter deformation andcombination of spectral and Gauge deformations of mKdV equation.

In this section, we consider the mKdV surfaces arising fromdeformations of parameters of the it’s soliton solution.Let u(x, t) satisfy the mKdV equation

ut = uxxx +3

2u2ux. (13)

Substituting the travelling wave ansatz ut − αux = 0 in Eq. (13), weget

uxx = αu− u3

2. (14)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 12 / 49

mKdV Surfaces from Deformation of Parameters

In [Tek, 2007], we considered spectral parameter deformation andcombination of spectral and Gauge deformations of mKdV equation.

In this section, we consider the mKdV surfaces arising fromdeformations of parameters of the it’s soliton solution.Let u(x, t) satisfy the mKdV equation

ut = uxxx +3

2u2ux. (13)

Substituting the travelling wave ansatz ut − αux = 0 in Eq. (13), weget

uxx = αu− u3

2. (14)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 12 / 49

mKdV Surfaces from Deformation of Parameters

Lax pairs U and V are given as

U =i

2

(λ −u−u −λ

), (15)

V = − i2

1

2u2 − (α+ αλ+ λ2) (α+ λ)u− iux

(α+ λ)u+ iux −1

2u2 + (α+ αλ+ λ2)

,(16)

and λ is a spectral parameter.

Consider the one soliton solution of mKdV equation [Eq. (14)] as

u = k1 sech ξ1, (17)

where α = k21/4, ξ1 = k1(k21t+ 4x)/8 + k0, and k0 and k1 are arbitrary

constants.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 13 / 49

mKdV Surfaces from Deformation of Parameters

Lax pairs U and V are given as

U =i

2

(λ −u−u −λ

), (15)

V = − i2

1

2u2 − (α+ αλ+ λ2) (α+ λ)u− iux

(α+ λ)u+ iux −1

2u2 + (α+ αλ+ λ2)

,(16)

and λ is a spectral parameter.

Consider the one soliton solution of mKdV equation [Eq. (14)] as

u = k1 sech ξ1, (17)

where α = k21/4, ξ1 = k1(k21t+ 4x)/8 + k0, and k0 and k1 are arbitrary

constants.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 13 / 49

mKdV Surfaces from Deformation of Parameters

First we consider mKdV surfaces arising from deformation ofparameter k0.

Proposition

Let u be a travelling wave solution of mKdV equation given by Eq.(17). The corresponding su(2) valued Lax pairs U and V of the mKdVequation are given by Eqs. (15) and (16), respectively. su(2) valuedmatrices A and B are

A = − iµ2

(0 φ0φ0 0

), (18)

B = − iµ2

(uφ0 (k21/4 + λ)φ0 − i (φ0)x

(k21/4 + λ)φ0 + i (φ0)x −uφ0

),(19)

where A = µ (∂U/∂k0) , B = µ (∂V/∂k0), φ0 = ∂ u/∂k0; k0 is aparameter of the one soliton solution u, and µ is a constant.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 14 / 49

mKdV Surfaces from Deformation of Parameters

First we consider mKdV surfaces arising from deformation ofparameter k0.

Proposition

Let u be a travelling wave solution of mKdV equation given by Eq.(17). The corresponding su(2) valued Lax pairs U and V of the mKdVequation are given by Eqs. (15) and (16), respectively. su(2) valuedmatrices A and B are

A = − iµ2

(0 φ0φ0 0

), (18)

B = − iµ2

(uφ0 (k21/4 + λ)φ0 − i (φ0)x

(k21/4 + λ)φ0 + i (φ0)x −uφ0

),(19)

where A = µ (∂U/∂k0) , B = µ (∂V/∂k0), φ0 = ∂ u/∂k0; k0 is aparameter of the one soliton solution u, and µ is a constant.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 14 / 49

mKdV Surfaces from Deformation of Parameters

Proposition

Then the surface S, generated by U, V,A and B, has the following firstand second fundamental forms (j, k = 1, 2)

(dsI)2 ≡ gjk dxj dxk, (20)

(dsII)2 ≡ hjk dxj dxk, (21)

where

g11 =1

4µ2φ20, g12 = g21 =

1

16µ2φ20(k

21 + 4λ), (22)

g22 =1

64µ2(

16 (φ0)2x + φ20

[16u2 + (k21 + 4λ)2

] ), (23)

h11 = −16 ∆1 λuφ20, (24)

h12 = 4 ∆1φ0

(4 (φ0)x ux + uφ0

[2u2 − k21(λ+ 1)− 4λ2

] ), (25)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 15 / 49

mKdV Surfaces from Deformation of Parameters

Proposition

h22 = −∆1

(uφ20 (k21 + 4λ)

[2u2 + 4λ2 + k21(λ+ 1)

](26)

+ 4φ0

[4u(φ0)xt − (φ0)x

([k21 + 4λ]ux + 4ut

)](27)

+ 4u(φ0)x

[(φ0)x(k21 + 4λ)− 4(φ0)t

])∆1 =

µ

32(

(φ0)2x + u2φ20

)1/2 (28)

and the corresponding Gaussian and mean curvatures are

K =16λ2

k21µ2, H = − 4λ

k1µ, (29)

where x1 = x, x2 = t.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 16 / 49

mKdV Surfaces from Deformation of Parameters

Proposition

h22 = −∆1

(uφ20 (k21 + 4λ)

[2u2 + 4λ2 + k21(λ+ 1)

](26)

+ 4φ0

[4u(φ0)xt − (φ0)x

([k21 + 4λ]ux + 4ut

)](27)

+ 4u(φ0)x

[(φ0)x(k21 + 4λ)− 4(φ0)t

])∆1 =

µ

32(

(φ0)2x + u2φ20

)1/2 (28)

and the corresponding Gaussian and mean curvatures are

K =16λ2

k21µ2, H = − 4λ

k1µ, (29)

where x1 = x, x2 = t.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 16 / 49

mKdV Surfaces from Deformation of Parameters

Another parameter of the one soliton solution of mKdV equation is k1.Now we give mKdV surfaces arising from k1 parameter deformation.

Proposition

Let u be the soliton solution of mKdV equation and the Lax pairs Uand V are given by Eqs. (15) and (16), respectively. su(2) valuedmatrices A and B are

A = − iµ2

(0 φ1φ1 0

), (30)

B = − iµ8

(4uφ1 − 2k1(λ+ 1) τ − 4i(φ1)x

τ + 4i(φ1)x −4uφ1 + 2k1(λ+ 1)

), (31)

where A = µ (∂U/∂k1) , B = µ (∂V/∂k1), τ = 2k1u+ (k21 + 4λ)φ1 andφ1 = ∂ u/∂k1; k1 is a parameter of the one soliton solution u, and µ isa constant.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 17 / 49

mKdV Surfaces from Deformation of Parameters

Proposition

Then the surface S, generated by U, V,A and B, has the following firstand second fundamental forms (j, k = 1, 2)

(dsI)2 ≡ gjk dxj dxk, (32)

(dsII)2 ≡ hjk dxj dxk, (33)

where

g11 =1

4µ2φ21, g12 = g21 =

1

16µ2φ1

(2 k1 u+ φ1[k

21 + 4λ]

), (34)

g22 =1

64µ2(

4[k21 + 4φ21

]u2 + 4 k1 (k21 − 4)uφ1 + 16(φ1)

2x (35)

+ (k21 + 4λ)2φ21 + 4 k21(λ+ 1)2), (36)

h11 =1

16∆2 µ

3 λφ21

(k1[λ+ 1]− 2uφ1

), (37)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 18 / 49

mKdV Surfaces from Deformation of Parameters

Proposition

h12 = h21 =1

64∆2 µ

3φ21

(8 (φ1)xux

+[k1(λ+ 1)− 2uφ1

][2(2λ2 − u2) + k21(λ+ 1)

]), (38)

h22 =1

256∆2 µ

3 φ1

(8 (φ1)x

2 k1 uux + (k21 + 4λ)

[φ1 ux − u(φ1)x

]+ 4(uφ1)t

+[k1(λ+ 1)− 2uφ1

]16 (φ1)xt − 4 k1 u(u2 + 2λ)

+ φ1(k21 + 4λ)

(2[u2 + 2λ2] + k21[λ+ 1]

)). (39)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 19 / 49

mKdV Surfaces from Deformation of Parameters

Proposition

The Gaussian and mean curvatures are

K =1

µ2 η0(4 η24 + η23

)2 7∑l=1

Ql (sech ξ1)l, (40)

H =1

4µ η0(4 η24 + η23

)3/2 7∑m=0

Zm (sech ξ1)m, (41)

where η0, . . . , η4, Q1, . . . , Q7, Z1, . . . , Z6 are functions of x and t.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 20 / 49

The Parameterized Form of the mKdV Surfaces

In this section, we explore the position vector

−→y = (y1(x, t), y2(x, t), y3(x, t)) , (42)

of the mKdV surfaces that we obtain using deformation of parameters.

Consider the one soliton solution of mKdV equation

u = k1 sech ξ1, (43)

where α = k21/4, ξ1 = k1(k21t+ 4x)/8 + k0, and k0 and k1 are arbitrary

constants.We solve the Lax equations Φx = U Φ and Φt = V Φ using Lax pairs Uand V , and a solution of the mKdV equation.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 21 / 49

The Parameterized Form of the mKdV Surfaces

In this section, we explore the position vector

−→y = (y1(x, t), y2(x, t), y3(x, t)) , (42)

of the mKdV surfaces that we obtain using deformation of parameters.Consider the one soliton solution of mKdV equation

u = k1 sech ξ1, (43)

where α = k21/4, ξ1 = k1(k21t+ 4x)/8 + k0, and k0 and k1 are arbitrary

constants.

We solve the Lax equations Φx = U Φ and Φt = V Φ using Lax pairs Uand V , and a solution of the mKdV equation.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 21 / 49

The Parameterized Form of the mKdV Surfaces

In this section, we explore the position vector

−→y = (y1(x, t), y2(x, t), y3(x, t)) , (42)

of the mKdV surfaces that we obtain using deformation of parameters.Consider the one soliton solution of mKdV equation

u = k1 sech ξ1, (43)

where α = k21/4, ξ1 = k1(k21t+ 4x)/8 + k0, and k0 and k1 are arbitrary

constants.We solve the Lax equations Φx = U Φ and Φt = V Φ using Lax pairs Uand V , and a solution of the mKdV equation.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 21 / 49

The Parameterized Form of the mKdV Surfaces

The components of the 2× 2 matrix Φ are

Φ11 = −∆4

k1

[A1(2λi− k1 tanh ξ1) · exp

(i(k21 + 4λ2)t/8

)· Ξ1

−i k21B1 sech ξ1 · exp(− i(k21 + 4λ2)t/8

)· Ξ2

], (44)

Φ12 = −∆4

k1

[A2(2λi− k1 tanh ξ1) · exp

(i(k21 + 4λ2)t/8

)· Ξ1

−i k21B2 sech ξ1 · exp(− i(k21 + 4λ2)t/8

)· Ξ2

], (45)

Φ21 = ∆4

[i A1 sech ξ1 · exp

(i(k21 + 4λ2)t/8

)· Ξ1

+B1(2λi+ k1 tanh ξ1) · exp(− i(k21 + 4λ2)t/8

)· Ξ2

], (46)

Φ22 = ∆4

[i A2 sech ξ1 · exp

(i(k21 + 4λ2)t/8

)· Ξ1

+B2(2λi+ k1 tanh ξ1) · exp(− i(k21 + 4λ2)t/8

)· Ξ2

], (47)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 22 / 49

The Parameterized Form of the mKdV Surfaces

where

Ξ1 = (tanh ξ1 + 1)iλ/2k1(tanh ξ1 − 1)−iλ/2k1 , (48)

Ξ2 = (tanh ξ1 − 1)iλ/2k1(tanh ξ1 + 1)−iλ/2k1 , (49)

∆4 =√k1/(k21 + 4λ2). (50)

Here we find the determinant of the matrix Φ as

det(Φ) = (A1B2 −A2B1) 6= 0. (51)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 23 / 49

The Parameterized Form of the mKdV Surfaces

where

Ξ1 = (tanh ξ1 + 1)iλ/2k1(tanh ξ1 − 1)−iλ/2k1 , (48)

Ξ2 = (tanh ξ1 − 1)iλ/2k1(tanh ξ1 + 1)−iλ/2k1 , (49)

∆4 =√k1/(k21 + 4λ2). (50)

Here we find the determinant of the matrix Φ as

det(Φ) = (A1B2 −A2B1) 6= 0. (51)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 23 / 49

The Parameterized Form of the mKdV Surfaces

Immersion function of the mKdV surface obtained usingk0 deformation

We find the immersion function F of the mKdV surface obtained usingk0 deformation by using the following equation

F = ν Φ−1∂Φ

∂k0+

(r11 r12r21 r22

), (52)

from which we obtain the position vector.

Using Φ given in the previous slides and choosingA1 = −k1B2 exp(−λπ/k1), A2 = k1B1 exp(−λπ/k1), r11 = r22 = 0,r12 = −r21 to write F in the form F = −i(σ1y1 + σ2y2 + σ3y3).

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 24 / 49

The Parameterized Form of the mKdV Surfaces

Immersion function of the mKdV surface obtained usingk0 deformation

We find the immersion function F of the mKdV surface obtained usingk0 deformation by using the following equation

F = ν Φ−1∂Φ

∂k0+

(r11 r12r21 r22

), (52)

from which we obtain the position vector.

Using Φ given in the previous slides and choosingA1 = −k1B2 exp(−λπ/k1), A2 = k1B1 exp(−λπ/k1), r11 = r22 = 0,r12 = −r21 to write F in the form F = −i(σ1y1 + σ2y2 + σ3y3).

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 24 / 49

The Parameterized Form of the mKdV Surfaces

Hence we obtain a family of surfaces parameterized by

y1 = W6 · sech2(ξ1)[W3 · cosh(ξ1) cos(Ω1)

+W4 · sinh(ξ1) sin(Ω1) + 4λW8

(2W1 cosh(2ξ1) +W7

)],(53)

y2 =1

W5sech2(ξ1)

[W10 · sinh(ξ1) cos(Ω1)

−W11 · cosh(ξ1) sin(Ω1) +W9 · cosh2(ξ1)], (54)

y3 = W6 · sech2(ξ1)[W13 · cosh(ξ1) cos(Ω1)

−W12 · sinh(ξ1) sin(Ω1) + 2λW2

(2W1 cosh(2ξ1) +W7

)],(55)

where Ω1 =(k21(λ+ 1)/4 + λ2

)t+ λx+ 2λ k0/k1,

ξ1 = k1(k21t+ 4x)/8 + k0 and W1, . . . ,W13 are constants.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 25 / 49

The Parameterized Form of the mKdV Surfaces

Immersion function of the mKdV surface obtained usingk1 deformation

We find the immersion function F of the mKdV surface obtained usingk1 deformation by using the following equation

F = ν Φ−1∂Φ

∂k1+

(r11 r12r21 r22

), (56)

from which we obtain the position vector.

Here we use the solution, Φ, of Lax equations and we choose thefollowings

A1 = −k1B2 exp(−λπ/k1), A2 = k1B1 exp(−λπ/k1), (57)

r11 = −r22 =ν (πλ+ k1)

(B2

2 −B21

)k21(B2

1 +B22)

, (58)

r12 = −r21 k21(B2

1 +B22) + 2νB1B2(πλ+ k1)

k21(B21 +B2

2), (59)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 26 / 49

The Parameterized Form of the mKdV Surfaces

Immersion function of the mKdV surface obtained usingk1 deformation

We find the immersion function F of the mKdV surface obtained usingk1 deformation by using the following equation

F = ν Φ−1∂Φ

∂k1+

(r11 r12r21 r22

), (56)

from which we obtain the position vector.Here we use the solution, Φ, of Lax equations and we choose thefollowings

A1 = −k1B2 exp(−λπ/k1), A2 = k1B1 exp(−λπ/k1), (57)

r11 = −r22 =ν (πλ+ k1)

(B2

2 −B21

)k21(B2

1 +B22)

, (58)

r12 = −r21 k21(B2

1 +B22) + 2νB1B2(πλ+ k1)

k21(B21 +B2

2), (59)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 26 / 49

The Parameterized Form of the mKdV Surfaces

in order to write F in the form F = −i(σ1y1 + σ2y2 + σ3y3).

Hence we obtain a family of surfaces parameterized by

y1 = W14 · sech2(ξ1)[W15

(2Ω2 · sinh(ξ1)− (16/3) cosh(ξ1)

)sin(Ω1)

+W16 · Ω2 · cosh(ξ1) cos(Ω1)

+W8

(2Ω3 · cosh(2ξ1) + 2k21λ sinh(2ξ1) + Ω4

)], (60)

y2 = W14 · sech2(ξ1)[W17

(2Ω2 · sinh(ξ1)− (16/3) cosh(ξ1)

)cos(Ω1)

−W18 · Ω2 · cosh(ξ1) sin(Ω1) +W19

(cosh(2ξ1) + 1

)], (61)

y3 = W14 · sech2(ξ1)[W20

(2Ω2 · sinh(ξ1)− (16/3) cosh(ξ1)

)sin(Ω1)

−W21 · Ω2 · cosh(ξ1) cos(Ω1)

+ (W2/2)(

2Ω3 · cosh(2ξ1) + 2k21λ · sinh(2ξ1) + Ω4

)], (62)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 27 / 49

The Parameterized Form of the mKdV Surfaces

in order to write F in the form F = −i(σ1y1 + σ2y2 + σ3y3).

Hence we obtain a family of surfaces parameterized by

y1 = W14 · sech2(ξ1)[W15

(2Ω2 · sinh(ξ1)− (16/3) cosh(ξ1)

)sin(Ω1)

+W16 · Ω2 · cosh(ξ1) cos(Ω1)

+W8

(2Ω3 · cosh(2ξ1) + 2k21λ sinh(2ξ1) + Ω4

)], (60)

y2 = W14 · sech2(ξ1)[W17

(2Ω2 · sinh(ξ1)− (16/3) cosh(ξ1)

)cos(Ω1)

−W18 · Ω2 · cosh(ξ1) sin(Ω1) +W19

(cosh(2ξ1) + 1

)], (61)

y3 = W14 · sech2(ξ1)[W20

(2Ω2 · sinh(ξ1)− (16/3) cosh(ξ1)

)sin(Ω1)

−W21 · Ω2 · cosh(ξ1) cos(Ω1)

+ (W2/2)(

2Ω3 · cosh(2ξ1) + 2k21λ · sinh(2ξ1) + Ω4

)], (62)

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 27 / 49

The Parameterized Form of the mKdV Surfaces

whereΩ2 = t k31 + 4x k1/3, Ω3 =

(4λ2 + k21

)(k31[λ+ 1]t− 4λ k0

),

Ω4 = tk31

(4λ2[λ+ 1] + k21[7λ+ 1]

)/4 + λ

(k21[2x k1 − k0]− 4λ2 k0

)and

W14, . . . ,W21 are constants.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 28 / 49

The Parameterized Form of the mKdV Surfaces

Graph of Some of the mKdV Surfaces

We obtained the position vector −→y = (y1(x, t), y2(x, t), y3(x, t)) , of themKdV surfaces arising from deformation of parameters.

We plot some of these mKdV surfaces for some special values of theconstants.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 29 / 49

The Parameterized Form of the mKdV Surfaces

Graph of Some of the mKdV Surfaces from k0deformationExample:Taking λ = 0.4, ν = 1, B1 = 1, B2 = 1, k0 = 1.5, k1 = 1.3and r21 = 1 in Eqs. (53)-(55), we get the surface given in Figure 1.

Figure : (x, t) ∈ [−15, 15]× [−15, 15]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 30 / 49

The Parameterized Form of the mKdV Surfaces

Example: Taking λ = 1.2, ν = 1, B1 = 1, B2 = 1, k0 = 0.5, k1 = 1.4and r21 = 1 in Eqs. (53)-(55), we get the surface given in Figure 2.

Figure : (x, t) ∈ [−5, 5]× [−5, 5]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 31 / 49

The Parameterized Form of the mKdV Surfaces

Example: Taking λ = 0.6, ν = 1, B1 = 1, B2 = 1, k2 = 0.2, k3 = 0.4and r21 = 1 in Eqs. (53)-(55), we get the surface given in Figure 3.

a) b)

Figure : (a), (b) (x, t) ∈ [−10, 10]× [−10, 10]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 32 / 49

The Parameterized Form of the mKdV Surfaces

Taking λ = 2.7, ν = 1, B1 = 1, B2 = 1, k0 = 0.3, k1 = 1.5 and r21 = 1in Eqs. (53)-(55), we get the surface given in Figure 4.

Figure : (x, t) ∈ [−5, 5]× [−5, 5]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 33 / 49

The Parameterized Form of the mKdV Surfaces

Graph of Some of the mKdV Surfaces from k1deformationExample: Taking λ = 0.15, ν = 1, B1 = 1, B2 = 1, k0 = 0.1,k1 = −0.5 and r21 = 1 in Eqs. (60)-(62), we get the surface given inFigure 5.

Figure : (x, t) ∈ [−200, 200]× [−200, 200]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 34 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = 0.03, ν = 1, B1 = 1, B2 = 1, k0 = 0, k1 = −0.1 and r21 = 1in Eqs. (60)-(62), we get the surface given in Figure 6.

Figure : (x, t) ∈ [−3000, 3000]× [−3000, 3000]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 35 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = −0.2, ν = 1, B1 = 1, B2 = 1, k0 = 0, k1 = 0.7 and r21 = 1in Eqs. (60)-(62), we get the surface given in Figure 7.

Figure : (x, t) ∈ [−100, 100]× [−100, 100]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 36 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = −1.3, ν = 1, B1 = 1, B2 = 1, k0 = 0, k1 = 4 and r21 = 1 inEqs. (60)-(62), we get the surface given in Figure 8.

Figure : (x, t) ∈ [−5, 5]× [−5, 5]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 37 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = 0.4, ν = 1, B1 = 1, B2 = 1, k0 = 0.6, k1 = 0.7 and r21 = −2in Eqs. (60)-(62), we get the surface given in Figure 9.

Figure : (x, t) ∈ [−50, 50]× [−50, 50]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 38 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = 0, ν = 1, B1 = 1, B2 = 1, k0 = 0, k1 = 0.7 and r21 = 1 inEqs. (60)-(62), we get the surface given in Figure 10.

Figure : (x, t) ∈ [−100, 100]× [−100, 100]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 39 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = −0.8, ν = 1, B1 = 1, B2 = 1, k0 = 0, k1 = −0.2 and r21 = 1in Eqs. (60)-(62), we get the surface given in Figure 11.

Figure : (x, t) ∈ [−20, 20]× [−20, 20]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 40 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = −0.8, ν = 1, B1 = 1, B2 = 1, k0 = 5, k1 = −0.2 and r21 = 1in Eqs. (60)-(62), we get the surface given in Figure 12.

Figure : (x, t) ∈ [−20, 20]× [−20, 20]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 41 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = −0.1, ν = 1, B1 = 1, B2 = 1, k0 = −4, k1 = −0.2 andr21 = 1 in Eqs. (60)-(62), we get the surface given in Figure 13.

Figure : (x, t) ∈ [−500, 500]× [−500, 500]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 42 / 49

The Parameterized Form of the mKdV Surfaces

Example:Taking λ = 0.4, ν = 1, B1 = 1, B2 = 1, k0 = 0, k1 = 0.2 and r21 = 1 inEqs. (60)-(62), we get the surface given in Figure 14.

Figure : (x, t) ∈ [−80, 80]× [−80, 80]

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 43 / 49

Conclusion

• To develop surfaces from integrable equations we used deformation ofparameters of solution of integrable equation

A = µ∂U

∂ki, B = µ

∂V

∂ki.

• Using this method, we construct 2-surfaces from modifiedKorteweg-de Vries (mKdV) equation.

• There are two types of deformations of parametrs. The first one gives2-surfaces on spheres and the second one gives highly complicated2-surfaces in R3.

• We also give the graph of interesting mKdV surfaces arise fromparametric deformations.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 44 / 49

Conclusion

• To develop surfaces from integrable equations we used deformation ofparameters of solution of integrable equation

A = µ∂U

∂ki, B = µ

∂V

∂ki.

• Using this method, we construct 2-surfaces from modifiedKorteweg-de Vries (mKdV) equation.

• There are two types of deformations of parametrs. The first one gives2-surfaces on spheres and the second one gives highly complicated2-surfaces in R3.

• We also give the graph of interesting mKdV surfaces arise fromparametric deformations.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 44 / 49

Conclusion

• To develop surfaces from integrable equations we used deformation ofparameters of solution of integrable equation

A = µ∂U

∂ki, B = µ

∂V

∂ki.

• Using this method, we construct 2-surfaces from modifiedKorteweg-de Vries (mKdV) equation.

• There are two types of deformations of parametrs. The first one gives2-surfaces on spheres and the second one gives highly complicated2-surfaces in R3.

• We also give the graph of interesting mKdV surfaces arise fromparametric deformations.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 44 / 49

Conclusion

• To develop surfaces from integrable equations we used deformation ofparameters of solution of integrable equation

A = µ∂U

∂ki, B = µ

∂V

∂ki.

• Using this method, we construct 2-surfaces from modifiedKorteweg-de Vries (mKdV) equation.

• There are two types of deformations of parametrs. The first one gives2-surfaces on spheres and the second one gives highly complicated2-surfaces in R3.

• We also give the graph of interesting mKdV surfaces arise fromparametric deformations.

S.Tek and M.Gurses Surfaces from Defor. Parameters June 5-10, 2015 44 / 49

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