Advances in Mathematical Physics

Volume 2015, Article ID 905978, 7 pages

http://dx.doi.org/10.1155/2015/905978

## Geometry of the Solutions of Localized Induction Equation in the Pseudo-Galilean Space

^{1}Department of Mathematics, Faculty of Science, Firat University, 23200 Elazig, Turkey^{2}Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania^{3}Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, 59030 Tekirdağ, Turkey

Received 26 January 2015; Accepted 21 May 2015

Academic Editor: Soheil Salahshour

Copyright © 2015 Muhittin Evren Aydin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the surfaces corresponding to solutions of the localized induction equation in the pseudo-Galilean space . We classify such surfaces with null curvature and characterize some special curves on these surfaces in .

#### 1. Introduction

The* localized induction equation* (LIE), also called the* vortex filament equation* or the* Betchov-Da Rios equation*, is an idealized model of the evolution of the centerline of a thin vortex tube in a three-dimensional inviscid incompressible fluid. The connection of LIE with the theory of solitons was discovered by Hasimoto proving that the solutions of LIE are related to solutions of the cubic nonlinear Schrödinger equation, which is well known to be an equation with soliton solutions. For details, see [1–5].

The soliton surface associated with the nonlinear Schrödinger equation is called a* Hasimoto surface*. The geometric properties of such surfaces are investigated in [4, 6, 7]. By motivating the fact that the study of the Hasimoto surfaces can be interesting in the pseudo-Galilean space which is one of the Cayley-Klein spaces, we are mainly interested in the geometric properties of these surfaces in the pseudo-Galilean space.

Let be a 3-dimensional Riemannian manifold and the Levi-Civita connection with respect to . Note that the cross product of two vector fields on may be defined aswhere is the volume form of the manifold and . Denote by the vortex filament; then LIE or the Betchov-Da Rios filament equation is

On the other hand, the Galilean space which can be defined in three-dimensional projective space is the space of Galilean Relativity. The geometries of Galilean space and pseudo-Galilean space have similarities but, of course, are different. For study of surfaces in the pseudo-Galilean space, we refer to Šipuš and Divjak’s series of works [8–13].

In classical geometry of surfaces, it makes sense to classify the surfaces having null curvature. In particular, a surface is said to be* developable* if it has null Gaussian curvature. In this case the surface can be flattened onto a plane without distortion. We remark that cylinders and cones are examples of developable surfaces, but the spheres are not under any metric.

There exist significant applications of the results obtained on the surfaces of null curvature in different fields, for example, in microeconomics. When the graphs of production functions in microeconomics have null Gaussian curvature, one can realize a “good” analysis of isoquants by projections, without losing essential information about their geometry (see [14, 15]). By targeting that, we study the Hasimoto surfaces in the pseudo-Galilean space .

In this paper, we classify the Hasimoto surfaces having null curvature in . Then we analyze the parameter curves on such surfaces from the point of view to be the geodesics, asymptotic lines, and principal curves in .

#### 2. Preliminaries

The pseudo-Galilean space is one of the Cayley-Klein spaces with absolute figure that consists of the ordered triple , where is the absolute plane in the three-dimensional real projective space , is the absolute line in , and is the fixed hyperbolic involution of points of [8, 9, 13].

Homogenous coordinates of can be given in the following way: The absolute plane is given by , the absolute line by , and the hyperbolic involution by , which is equivalent to the requirement for the conic to be the absolute conic. The metric connections in are introduced with respect to the absolute figure. According to the affine coordinates given by , the* distance* between the points and is defined by (see [16])The* pseudo-Galilean scalar product* of the vectors and is defined byIn this sense, the* pseudo-Galilean norm* of a vector is . A vector is called* isotropic* if ; otherwise it is called* nonisotropic*. All unit nonisotropic vectors are of the form . The isotropic vector is called* spacelike*,* timelike*, and* lightlike* if ,, and , respectively. The* pseudo-Galilean cross product* of and on is given bywhere , .

Let be an admissible curve given by in ; that is, it has no isotropic tangent vectors for . Note that an admissible curve has nonzero curvature. Then its curvature and torsion functions are, respectively, defined bywhere and so forth, . If is an admissible curve parameterized by the arc length , it is of the formThe associated Frenet frame field of is the trihedron such that where . In this sense, the Frenet formulas for the curve are [17, 18]

Let be a surface in the pseudo-Galilean space parameterized byDenote and , and . Then such a surface is admissible if and only if for some . The coefficients of the first fundamental form of areor, in matrix form,where and .

Let us define the function asThus a side tangential vector is defined bywhich can be a spacelike isotropic or a timelike isotropic vector. The unit normal vector field of is given byThe second fundamental form of and its coefficients are defined bywhere . Two types of admissible surfaces can be distinguished: spacelike surfaces having timelike unit normals and timelike ones having spacelike unit normals . The third fundamental form of iswhereThe Gaussian curvature and the mean curvature of are, respectively, defined byA surface in is said to be* minimal* if its mean curvature vanishes.

#### 3. Hasimoto Surfaces with Null Curvature in

Let and be the open intervals of and open domain of .

A* Hasimoto surface * is the surface traced out by a curve in as it evolves over time according to this evolution equation:where is smooth and regular mapping such that .

In order to parameterize by arc length the curve for , the parameterization of the surface in may be chosen as follows:where is a smooth function of one variable on . After using the pseudo-Galilean cross product given by (5) and (21), it can be easily seen that the function is a constant function. In this sense, we get the parameterization of a Hasimoto surface in asfor arbitrary constant . Note that the Hasimoto surface given by (23) is always admissible; that is, it has no pseudo-Euclidean tangent planes, if .

Now, let be a Hasimoto surface given by (23) in . Denote the associated Frenet frame field of the curve for by . Then, the derivative of with respect to has the expressionfor some smooth functions on . By (21) and (24), we getMoreover, applying the compatibility condition yields that In similar ways, we deduce The compatibility conditions and give

Summing up, we have the following result.

Lemma 1. *Let be a Hasimoto surface given by (23) in . Then the following equations hold:*(i)* and ;*(ii)*where is the Frenet frame field of the curve for .*

*Example 2. *Let us consider the spacelike admissible Hasimoto surface in parameterized byIt can be easily seen that We plot the surface as in Figure 1.