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BETCHOV-DA RIOS EQUATION BY NULL CARTAN, PSEUDO NULL AND PARTIALLY NULL CURVE IN MINKOWSKI SPACETIME
(Korean Mathematical Soc, 2023) Erdogdu, Melek; Li, Yanlin; Yavuz, Ayse
The aim of this paper is to investigate Betchov-Da Rios equation by using null Cartan, pseudo null and partially null curve in Minkowski spacetime. Time derivative formulas of frame of s parameter null Cartan, pseudo null and partially null curve are examined, respectively. By using the obtained derivative formulas, new results are given about the solution of Betchov-Da Rios equation. The differential geomet-ric properties of these solutions are obtained with respect to Lorentzian causal character of s parameter curve. For a solution of Betchov-Da Rios equation, it is seen that null Cartan s parameter curves are space curves in three-dimensional Minkowski space. Then all points of the soliton sur-face are flat points of the surface for null Cartan and partially null curve. Thus, it is seen from the results obtained that there is no surface corresponding to the solution of Betchov-Da Rios equation by using the pseudo null s parameter curve.
Cayley formula in Minkowski space-time
(World Scientific Publ Co Pte Ltd, 2015) Erdogdu, Melek; Ozdemir, Mustafa
In this paper, Cayley formula is derived for 4 x 4 semi-skew-symmetric real matrices in E-1(4). For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = theta(1)A(1) + theta(2)A(2) by two unique semi-skew-symmetric matrices A(1) and A(2) satisfying the properties A(1)A(2) = 0, A(1)(3) = A(1) and A(2)(3) = -A(2). Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay(A).
Congruence of degenerate surface along pseudo null curve and Landau-Lifshitz equation
(Elsevier, 2022) Yavuz, Ayse; Erdogdu, Melek
This paper is devoted to the geometry of pseudo null curve in terms of anholonomic coordinates in Minkowski space. Firstly, extended Frenet formulas for pseudo null curves are deeply discussed. Then binormal congruence of degenerate surfaces containing the s - lines and n - lines are investigated with the condition mu(b) = 0. This condition represents the necessary and sufficient condition for the existence of a one-parameter family of surfaces containing the s - lines and n - lines. Moreover, normal congruence of surfaces containing the s - lines and b - lines are examined with the condition mu(n) = 0. Again, the condition represents the necessary and sufficient condition for the existence of a one-parameter family of surfaces containing the s - lines and b - lines. Considering the compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this binormal and congruence of surfaces, respectively. Finally, some relations are given for constructing the moving pseudo null curve with using the integrable Landau-Lifshitz equation. (C) 2022 Elsevier B.V. All rights reserved.
Curves Lying on Non-lightlike Surface: Differential Equation for Position Vector
(Univ Punjab, Dept Mathematics, 2022) Erdogdu, Melek
The main purpose of this study is to examine curves lying on a given non-lightlike surface with the help of its position vectors. For this purpose, the darboux frame is used and the position vector of the curve is expressed as a linear combination of the darboux frame with differentiable functions. Then, nonhomogeneous systems of differential equations revealed by the position vector of the curve are obtained for timelike and spacelike surfaces, respectively. For both timelike and spacelike surfaces, the solutions of nonhomogeneous systems of differential equations are obtained depending on the character of the curves and the values kg, kn and tr. The general solutions of the systems of differential equations are obtained separately for each case. Moreover, by considering only the particular solution of the systems of differential equations, new results regarding the differential geometric structure of the curves on the surface are presented with the help of the position vector.
A Different Approach by System of Differential Equations for the Characterization Position Vector of Spacelike Curves
(Univ Punjab, Dept Mathematics, 2021) Yavuz, Ayse; Erdogdu, Melek
The purpose of this study is to obtain a characterization of unit speed spacelike curve with constant curvature and torsion in the Minkowski 3-space. According to this purpose, the position vector of a spacelike curve is expressed by a linear combination of its Serret Frenet Frame with differentiable functions. Since a spacelike curve has different kinds of frames, then we investigate the curve with respect to the Lorentzian casual characterizations of the frame. Hence we examine the results in three different cases including different subcases. Moreover, we illustrate some examples for each case.
DIFFERENTIAL GEOMETRIC ASPECTS OF NONLINEAR SCHRODINGER EQUATION
(Ankara Univ, Fac Sci, 2021) Erdogdu, Melek; Yavuz, Ayse
The main scope of this paper is to examine the smoke ring (or vortex flament) equation which can be viewed as a dynamical system on the space curve in E-3: The differential geometric properties the soliton surface associated with Nonlinear Schrodinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature of Hasimoto surface are found in terms of Darboux curvatures k(n), k(g) and tau(g). Then, we give a different proof of that the s-parameter curves of NLS surface are the geodesics of the soliton surface. As applications we examine two NLS surfaces with Darboux Frame.
Electromagnetic waves along pseudo null curves in Minkowski space
(Elsevier Sci Ltd, 2022) Erdogdu, Melek
Considering the importance of Minkowski space in physics, it is an incomplete approach to deal with EM waves only in Euclidean space. For this reason, this paper deals with EM waves along pseudo null curves in Minkowski space. The main purpose of this study is to examine electromagnetic waves by defining an adapted orthogonal frame along the EM wave which contains both electric and magnetic fields. For this purpose, the extended derivative formulas of pseudo null frame are obtained. Depending on the values of Bishop curvatures, the linear transformations between the pseudo null frame and EM wave vector fields are described in two cases. For all these cases, the relations between these frames are stated, respectively. Moreover, the derivative formulas EM wave vectors are stated by means of geometric phase. Furthermore, the necessary and sufficient conditions provided by the geometric phase are expressed for EM wave vectors to be parallel transportation of the pseudo null frame. Finally, an application is given to investigate the obtained results.
Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane
(Elsevier, 2022) Yang, Zhichao; Li, Yanlin; Erdogdu, Melek; Zhu, Yushu
In this paper, we take advantage of envelope theory and singularity theory to study the evolutoids and pedaloids in Minkowski plane. We illustrate an internal correlation from algebraic and geometric viewpoints, and give the geometric explanation of evolutoids and pedaloids. Then, we generalize the notions of evolutoids and pedaloids to the category of frontals in Minkowski plane. Furthermore, we apply the technique of singularity theory, using the discriminants and versal unfolding tools, to consider evolving evolutoids and give the classifications of singularities of the evolutoids, and explain when cusps and inflexions occur and how evolutoids evolving. Besides, we found that there is a close relationship between the evolutoids and pedaloids and good correspondence between their singularities.(c) 2022 Elsevier B.V. All rights reserved.
A GEOMETRIC APPROACH TO TIMELIKE FLOWS IN TERMS OF ANHOLONOMIC COORDINATES
(Honam Mathematical Soc, 2022) Yavuz, Ayse; Erdogdu, Melek
This paper is devoted to the geometry of vector fields and timelike flows in terms of anholonomic coordinates in three dimensional Lorentzian space. We discuss eight parameters which are related by three partial differential equations. Then, it is seen that the curl of tangent vector field does not include any component in the direction of principal normal vector field. This implies the existence of a surface which contains both s - lines and b - lines. Moreover, we examine a normal congruence of timelike surfaces containing the s - lines and b - lines. Considering the compatibility conditions, we obtain the Gauss-Mainardi-Codazzi equations for this normal congruence of timelike surfaces in the case of the abnormality of normal vector field is zero. Intrinsic geometric properties of these normal congruence of timelike surfaces are obtained. We have dealt with important results on these geometric properties.
Geometry of Hasimoto Surfaces in Minkowski 3-Space
(Springer, 2014) Erdogdu, Melek; Ozdemir, Mustafa
In this paper, we investigate the Hasimoto surfaces in Minkowski 3-space. We discussed the geometric properties of Hasimoto surfaces in M-3 for three cases. The Gaussian and mean curvature of Hasimoto surface are found for each case. Then, we give the characterization of parameter curves of Hasimoto surfaces in M-3
Non-lightlike Bertrand W curves: A new approach by system of differential equations for position vector
(Amer Inst Mathematical Sciences-Aims, 2020) Yavuz, Ayse; Erdogdu, Melek
In this study, the characterization of position vectors belonging to non-lightlike Bertrand W curve mate with constant curvature are obtained depending on differentiable functions. The position vector of Bertrand W curve is stated by a linear combination of its Frenet frame with differentiable functions. There exist also different cases for the curve depending on the value of curvature and torsion. The relationships between Frenet apparatuas of these curves are stated separately for each case. Finally, the timelike and spacelike Bertrand curve mate visualized of given curves as examples, separately.
NONNULL SOLITON SURFACE ASSOCIATED WITH THE BETCHOV-DA RIOS EQUATION
(Pergamon-Elsevier Science Ltd, 2022) Li, Yanlin; Erdogdu, Melek; Yavuz, Ayse
The aim of this paper is to investigate the nonnull soliton surfaces associated with Betchov-Da Rios equation in Minkowski space-time. The differential geometric properties of these kind of nonnull soliton surfaces are examined with respect to the Lorentzian casual characterizations. Moreover, the linear maps of Weingarten type are obtained which are defined on tangent spaces of these soliton surfaces. Some new results are obtained by means of two geometric invariants ?? and h which are generated by linear maps of Weingarten type. Then, the mean curvature vector field and Gaussian curvature of the nonnull soliton surface are obtained. Finally, it is shown that this kind of soliton surface consists of flat points as a numerical example.
On Backlund transformation and motion of null Cartan curves
(World Scientific Publ Co Pte Ltd, 2022) Erdogdu, Melek; Yavuz, Ayse
The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector Omega. In the case of tau(t) = 0, we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve C.
On Complex Split Quaternion Matrices
(Springer Basel Ag, 2013) Erdogdu, Melek; Ozdemir, Mustafa
In this paper, we present some important properties of complex split quaternions and their matrices. We also prove that any complex split quaternion has a 4 x 4 complex matrix representation. On the other hand, we give answers to the following two basic questions If AB = I, is it true that BA = I for complex split quaternion matrices? and How can the inverse of a complex split quaternion matrix be found?. Finally, we give an explicit formula for the inverse of a complex split quaternion matrix by using complex matrices.
On differential analysis of spacelike flows on normal congruence of surfaces
(Amer Inst Mathematical Sciences-Aims, 2022) Erdogdu, Melek; Yavuz, Ayse
The present paper examines the differential analysis of flows on normal congruence of spacelike curves with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters, which are related by three partial differential equations, are discussed. Then, it is seen that the curl of tangent vector field does not include any component with principal normal direction. Thus there exists a surface which contains both s-lines and b - lines. Also, we examine a normal congruence of surfaces containing the s - lines and b - lines. By compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this normal congruence of surface. Intrinsic geometric properties of this normal congruence of surfaces arc given.
On Eigenvalues of Split Quaternion Matrices
(Springer Basel Ag, 2013) Erdogdu, Melek; Ozdemir, Mustafa
The main purpose of this paper is to set a method of finding eigenvalues of split quaternion matrices. In particular, we will give an extension of Gershgorin theorem, which is one of the fundamental theorems of complex matrix theory, for split quaternion matrices.
On exponential of split quaternionic matrices
(Elsevier Science Inc, 2017) Erdogdu, Melek; Ozdemir, Mustafa
The exponential of a matrix plays an important role in the theory of Lie groups. The main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their Lie algebras into the groups. Since the set of split quaternions is a noncommutative algebra, the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers. Therefore, we give a method of finding exponential of a split quaterriion matrix by its complex adjoint matrix. (C) 2017 Elsevier Inc. All rights reserved.
ON REFLECTIONS AND ROTATIONS IN MINKOWSKI 3-SPACE
(Inst Biophysics & Biomedical Engineering, Bulgarian Acad Sciences, 2015) Erdogdu, Melek; Ozdemir, Mustafa
In this paper, we investigate the reflections in Minkowski three-space by three different approaches. Firstly, we define Lorentzian reflections with Lorentzian inner product. Then, we examine Lorentzian reflections in view of Lorentzian Householder matrices. Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we find the matrix representation of Lorentzian reflections and characterize the plane of reflection by using this matrix representation. Moreover, we prove that any Lorentzian orthogonal transformation can be represented by the composition of at most six reflections.
On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions
(Springer Basel Ag, 2014) Ozdemir, Mustafa; Erdogdu, Melek; Simsek, Hakan
In this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Moreover, we find that the casual characters of rotation axis depend only on first component of the corresponding quaternion. Finally, we give the way to generate an orthogonal basis for by using eigenvectors of a rotation matrix.
ON THE ROTATION MATRIX IN MINKOWSKI SPACE-TIME
(Pergamon-Elsevier Science Ltd, 2014) Ozdemir, Mustafa; Erdogdu, Melek
In this paper, a Rodrigues-like formula is derived for 4 x 4 semi skew-symmetric real matrices in E-1(4). For this purpose, we use the decomposition of a semi skew-symmetric matrix A = theta(1)A(1) + theta(2)A(2) by two unique semi skew-symmetric matrices A(1) and A(2) satisfying the properties A(1)A(2) = theta, A(1)(3) = A1 and A(2)(3) = A(2). Then, we find Lorentzian rotation matrices with semi skew-symmetric matrices by Rodrigues-like formula. Furthermore, we give a way to find the semi skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = e(A).

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