1 Introduction

Integrable nonlinear evolution equations occur in many branches of physics, applied mathematics and many fields [3, 4, 6, 14, 21, 24]. Such equations possess a number of interesting properties such as soliton solutions, infinite number of conservation laws, infinite number of symmetries, Bäcklund and Darboux transformations, bi-Hamiltonian structures and so on, see [5, 7, 8, 11, 13, 15]. The study of geometrical flows of curves has a deep connection with integrable nonlinear evolution equations. One of the simplest examples illustrating this connection arises from a vortex filament flow in two and three dimensional inviscid fluid dynamics. In three dimensional space, a vortex filament flow in an inviscid fluid can be described by the dynamical evolution of its vortex filament flow, which is given by the binormal flow \(\gamma _t = \kappa {\mathbf{b}}\) for an arc-length parameterized curve \(\gamma (s,t)\). In the work [12], Hasimoto discovered the relationship between integrable nonlinear evolution equations and vortex filament flows and he showed that the nonlinear Schrödinger (NLS) equation is equivalent to the binormal flow \(\gamma _t = \kappa {\mathbf{b}}\) of the curve \(\gamma (s,t)\) by using a transformation relating the wave function of the NLS equation to the curvature and the torsion of the curves (so-called Hasimoto transformation). Curve flows for the vortex filament have been studied by many experts and geometers [2, 9,10,11, 19], etc. Anco and Asadi [1] studied the general results on parallel frames and Hasimoto variables to extend the geometrical relationships among the NLS equation, the vortex filament equation and the Heisenberg spin model to the general setting of Hermitian symmetric spaces. Arroyo, Garay and Pámpano [2] studied curve motions by the binormal flow with the curvature and the torsion depending velocity and sweeping out immersed surfaces, and they obtained the vortex filaments evolving with constant torsion which arise from extremal curves of the curvature energy functionals. Also, Xu and Cao [23] gave three nonlinear partial differential equations which are associated with binormal flows of constant torsion curves in Minkowski 3-space, and the authors gave Bäcklund transformations for the equations, as well as for surfaces swept out by related moving curves. Mohamed [17] investigated the general description of the binormal flow of a spacelike and a timelike curve in a 3-dimensional de-Sitter space and gave some explicit examples of a binormal flow of the curves. In [22] Wang investigated the nonlinear stability of Hasimoto solitons, in energy space, for a fourth order NLS equation which arises in the context of the vortex filament. The traveling wave solutions play an important role in the long time dynamics of NLS equations at infinity. Ivey [13] discussed travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in Euclidean 3-space that carry mutually orthogonal foliations by geodesics and by helices.

The outline of the paper is organized as follows: In Sect. 2, we give some geometric concepts of nonnull curves in Minkowski 3-space. In Sect. 3, we study nonlinear Schrödinger equations of timelike and spacelike curves and give Hasimoto travelling wave for these curves. In Sect. 4, we discuss the method to find the exact shape of the timelike and spacelike curve from the vortex filament flows by solving the tangent, principal normal and binormal vector in Minkowski 3-space. In the last section, we give some applications to find the position vector of the spacelike and timelike curve from the single soliton solution of the nonlinear Schrödinger equation.

2 Preliminaries

The Minkowski 3-space \(\mathbb R_1^3\) is a real space \(\mathbb R^3\) with the indefinite inner product \(\langle \cdot ~, \cdot \rangle \) defined on each tangent space by

$$\begin{aligned} \langle \mathbf{x}, \mathbf{y} \rangle = -x_1 y_1 + x_2 y_2 + x_3 y_3, \end{aligned}$$

where \({\mathbf{x}}=(x_1, x_2, x_3)\) and \({\mathbf{y}}=(y_1, y_2, y_3)\) are vectors in \(\mathbb R^3_1\).

A nonzero vector \(\mathbf{x}\) in \(\mathbb R^3_1\) is said to be spacelike or timelike if \(\langle \mathbf{x}, \mathbf{x} \rangle >0\) or \(\langle \mathbf{x}, \mathbf{x} \rangle <0\), respectively. Similarly, an arbitrary curve \(\gamma = \gamma (s)\) is spacelike or timelike if all of its tangent vector \(\frac{d \gamma (s) }{ds}={\gamma _s}(s)\) are spacelike or timelike, respectively.

For two vectors \({\mathbf{x}}=(x_1, x_2, x_3)\) and \({\mathbf{y}}=(y_1, y_2, y_3)\) in \(\mathbb R^3_1\), a Lorentz cross product \(\mathbf{x} \times \mathbf{y}\) is defined by

$$\begin{aligned} \mathbf{x}\times \mathbf{y} = (-x_2y_3+x_3y_2,x_3y_1-x_1y_3,x_1y_2-x_2y_1). \end{aligned}$$

Let \({\gamma } : I \rightarrow \mathbb R^3_1\) be a spacelike or timelike curve parametrized by the arc-length s in Minkowski 3-space \(\mathbb R^3_1\) and the vector \(\gamma _s(s)={\mathbf{t}}(s)\) be the unit tangent vector of \(\gamma \) with \(|| {\mathbf{t}}(s)||=\varepsilon _1\). If \(\varepsilon _1=1\), the curve \(\gamma \) is spacelike and if \(\varepsilon _1=-1\), the curve \(\gamma \) is timelike.

Since \( {\gamma _{ss} }\) is perpendicular to \(\mathbf{t}\), we define the principal normal vector \(\mathbf{n}\) as the normalized vector \( {\gamma _{ss} }\) and take the binormal vector \(\mathbf{b}\) which is the unique vector perpendicular to the tangent plane generated by \(\{ {\mathbf{t}}(s), {\mathbf{n}}(s)\}\) at every point \(\gamma (s)\) of \(\gamma \). In this case, \(\{ {\mathbf{t}}, {\mathbf{n}}, {\mathbf{b}}\}\) is the Frenet frame of the curve \(\gamma \) and the Frenet formulas are expressed in matrix notion as (cf. [16]):

$$\begin{aligned} \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) _s = \left( \begin{array}{ccc} 0 &{} \varepsilon _2\kappa &{} 0 \\ -\varepsilon _1 \kappa &{} 0 &{} -\varepsilon _3\tau \\ 0 &{} \varepsilon _2\tau &{} 0 \\ \end{array} \right) \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) , \end{aligned}$$
(1)

where \(\kappa \) and \(\tau \) are the curvature and the torsion of the curve \(\gamma \). Here \(\varepsilon _2\) and \(\varepsilon _3\) are the signs of the vectors \(\mathbf{n}\) and \(\mathbf{b}\), respectively.

Suppose next that we have a fluid in Minkowsi 3-space \(\mathbb R^3_1\) that evolves according to a given one parameter family of diffeomorphism yielding the position of a fluid particle. The corresponding vortex filament flow is assumed to be parametrized by the arc length s and it is expressed as [12]

$$\begin{aligned} \gamma _t = \gamma _s \times \gamma _{ss}. \end{aligned}$$
(2)

Now, we explain the geometric meaning of the evolution equation (2) of the spacelike or timelike curve in \(\mathbb R^3_1\). We known that the flow is binormal, that is, \(\gamma _t = \kappa {\mathbf{b}}\), the time evolution of the moving frames \(\{ {\mathbf{t}}, {\mathbf{n}}, {\mathbf{b}}\}\) of the timelike curve \(\gamma \) is expressed as (cf. [7])

$$\begin{aligned} \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) _t = \left( \begin{array}{ccc} 0 &{} \kappa \tau &{} \kappa _s \\ \kappa \tau &{} 0 &{}\frac{\kappa _{ss}}{\kappa } - \tau ^2\\ \kappa _s &{} \frac{-\kappa _{ss}}{\kappa } + \tau ^2 &{} 0 \\ \end{array} \right) \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) , \end{aligned}$$
(3)

and the time evolution of the spacelike curve \(\gamma \) is given by

$$\begin{aligned} \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) _t = \left( \begin{array}{ccc} 0 &{} \varepsilon _2\kappa \tau &{} -\varepsilon _2\varepsilon _3\kappa _s \\ -\varepsilon _1\kappa \tau &{} 0 &{}-\varepsilon _1\varepsilon _3 \left( \frac{\kappa _{ss}}{\kappa } - \tau ^2\right) \\ -\varepsilon _1\varepsilon _3\kappa _s &{} \varepsilon _1\varepsilon _2\left( \frac{-\kappa _{ss}}{\kappa } + \tau ^2\right) &{} 0 \\ \end{array} \right) \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) . \end{aligned}$$
(4)

It is well-known that equation (2) is equivalent to the NLS equation for the timelike curve \(\gamma \) as follows [10]:

$$\begin{aligned} i \psi _t = - \psi _{ss} + \frac{1}{2} \mid \psi \mid ^2 \psi \end{aligned}$$
(5)

via the Hasimoto transformations

$$\begin{aligned} \psi (s,t) = \kappa (s,t) ~{} \exp \left( -i \int ^s \tau (\rho , t) d \rho \right) . \end{aligned}$$

The NLS equation describes a wide range of physical phenomena and it has many applications.

Next, if \(\gamma \) is the spacelike curve with a timelike principal normal and a timelike binormal vector, equation (2) is equivalent to the nonlinear heat system [10]

$$\begin{aligned} \phi _t= \phi _{ss} + \phi ^2 \varphi , \qquad \varphi _t= -\varphi _{ss} - \varphi ^2 \phi \end{aligned}$$
(6)

in terms of the Hasimoto transformations

$$\begin{aligned} \phi (s,t) = \kappa (s,t) ~{} \exp \left( - i \int ^s \tau (\rho , t) d \rho \right) , \qquad \varphi (s,t) = \kappa (s,t) ~{} \exp \left( i \int ^s \tau (\rho , t) d \rho \right) . \end{aligned}$$
(7)

3 Vortex Filament Equations

In this section, we construct the position vector of the spacelikeor timelike curve from the single soliton solution of the nonlinear Schrödinger equation. It gives a geometrical tool to analyze localized evolution of dynamics in various dynamical systems. To obtain the main results we split it into two cases.

3.1 Timelike Vortex Filament

Theorem 3.1

(cf.[7]) Let \(\gamma \) be a timelike curve parametrized by the arc length s with the curvature \(\kappa \) and the torsion \(\tau \) and satisfies the timelike binormal flow \(\gamma _t = \kappa {\mathbf{b}}\) in Minkowsi 3-space. If we take the Hasimoto transformation

$$\begin{aligned} \psi (s,t) = \kappa (s,t) ~{} \exp \left( -i \int ^s \tau (\bar{s}, t) d \bar{s} \right) , \end{aligned}$$
(8)

then \(\psi \) satisfies the nonlinear Schrödinger equation

$$\begin{aligned} i \psi _t= -\psi _{ss} + \frac{1}{2} \left( \mid \psi \mid ^2 + R(t) \right) \psi \end{aligned}$$
(9)

for some smooth function R(t).

Theorem 3.2

(Timelike Hasimoto travelling wave) If we consider a soliton solution of the NLS equation (9) such that \(\kappa \rightarrow 0\) as \(s \rightarrow \infty \), then it gives a solution of a traveling wave with a kink that becomes a line at infinity.

Proof

To prove the theorem, we take the new variable \(\rho = s +a - ct\) with a constant velocity c and a positive constant a, then this variable will be used in order to obtain the our result with a soliton of the nonlinear Schrödinger equation. The variable implies

$$\begin{aligned} \psi (\rho ) = \kappa (\rho ) ~{} \exp \left( - i \int ^s \tau (\rho ) d s \right) . \end{aligned}$$
(10)

Since (10) is the solution of (9), the real and imaginary parts of (9) lead to

$$\begin{aligned} c \kappa (\rho ) ( \tau (\rho ) - \tau (a - ct) )= & {} \kappa '' (\rho ) - \kappa (\rho ) \tau ^2 (\rho ) - \frac{1}{2} ( \kappa ^2(\rho ) + R(t)) \kappa (\rho ), \end{aligned}$$
(11)
$$\begin{aligned} c \kappa ' (\rho )= & {} - 2 \kappa '(\rho ) \tau (\rho ) - \kappa (\rho ) \tau '(\rho ). \end{aligned}$$
(12)

Then, we easily obtain

$$\begin{aligned} (c + 2 \tau ) \kappa ^2 =0 \end{aligned}$$

from this we have

$$\begin{aligned} \tau = \tau _0 = -\frac{1}{2} c \end{aligned}$$
(13)

assuming that the curvature is not identically zero. So the torsion is constant along the vortex filament. It follows that a solution of the ODE (11) is given by

$$\begin{aligned} \kappa (\rho )= 2 b_1 \mathrm{csch} ( b_1 \rho ), \end{aligned}$$
(14)

and in this case \(R= 2 ( b_1^2 - \tau _0^2 )\) with a nonzero constant \(b_1\). Thus the curvature and the torsion are determined, so we can construct the shape of the timelike binormal flow by using (13) and (14).

Now, we will construct the position vector of \(\gamma \) satisfying the timelike binormal flow. Since \(\gamma \) is the timelike curve, we take \(\varepsilon _1=-1, \varepsilon _2=1\) and \(\varepsilon _3=1\) in (1) and we obtain the following equation:

$$\begin{aligned} \tau _0 ( {\mathbf{t}}' - \kappa {\mathbf{n}}) = \left( \frac{1}{\kappa } ( {\mathbf{b}}'' + \tau _0^2 {\mathbf{b}}) \right) ' - \kappa {\mathbf{b}}' =0. \end{aligned}$$

It is equivalently to

$$\begin{aligned} \frac{d^3 {\mathbf{b}}}{d \xi ^3} + \mathrm{coth} (\xi ) \frac{d^2 {\mathbf{b}}}{d \xi ^2} + \left( Q^2 - 4 \mathrm{csch} ^2 ( \xi ) \right) \frac{d {\mathbf{b}}}{d \xi } + \coth ( \xi ) Q^2 {\mathbf{b}}=0, \end{aligned}$$
(15)

where \(\xi = b_1 \rho \) and \(Q= \frac{\tau _0}{b_1}\).

If we put

$$\begin{aligned} {\mathbf{c}}= \frac{d {\mathbf{b}}}{d \xi } + \coth (\xi ) {\mathbf{b}}, \end{aligned}$$
(16)

then equation (15) is rewritten as

$$\begin{aligned} \frac{d^2 {\mathbf{c}}}{d \xi ^2} +( Q^2 - 2 \mathrm{csch}^2 (\xi ) ) {\mathbf{c}}=0. \end{aligned}$$
(17)

It’s solution is given by

$$\begin{aligned} {\mathbf{c}}= \left( \coth (\xi ) + i Q \right) \exp (-i Q \xi ). \end{aligned}$$
(18)

By combining (16) and (18), we can solve the ODE and the solutions become

$$\begin{aligned} {\mathbf{b}} =\mathrm{csch} (\xi ), \quad {\mathbf{b}}=\left( 1 - Q^2 + 2 i Q \coth (\xi ) \right) \exp (-i Q \xi ). \end{aligned}$$
(19)

We substitute (19) into Frenet formula (1) with \(\varepsilon _1=-1, \varepsilon _2=1\) and \(\varepsilon _3=1\) and determine the coefficients so as to satisfy, without loss generality, the conditions for the vortex filament to be parallel to the x axis at infinity as:

$$\begin{aligned}&{\mathbf{t}}_x \rightarrow 1 \quad {\text{ a }s } \quad \xi \rightarrow \infty \\{\mathbf{n}}_y + i {\mathbf{n}}_z = i ( {\mathbf{b}}_y + {\mathbf{b}}_z)&\quad = \exp (-i ( \tau _0 \xi + A(t))) \quad {\text{ a }s } \quad \xi \rightarrow \infty , \end{aligned}$$

where A(t) is a real function of t and the subscripts denote the x, y,  and z components of the vector, respectively. The above conditions are suggested by asymptotic behaviour of the solution of Frenet formulas. First, we can find the unit spacelike binormal vector \({\mathbf{b}}\) as

$$\begin{aligned} \begin{aligned} {\mathbf{b}}_x&= 2 \mu Q \mathrm{csch}(\xi ), \\ {\mathbf{b}}_y + i {\mathbf{b}}_z&= i \mu ( 1 - Q^2 + 2 i Q \coth (\xi ))\exp (- i ( \tau _0 s + A(t))) \end{aligned} \end{aligned}$$

and Frenet equation \( {\mathbf{b}}_s = \tau _0 {\mathbf{n}}\) implies the unit spacelike principal normal vector \(\mathbf{n}\) as follows:

$$\begin{aligned} \begin{aligned} {\mathbf{n}}_x&= - 2 \mu \mathrm{csch}^2 (\xi ) \cosh (\xi ), \\ {\mathbf{n}}_y + i {\mathbf{n}}_z&= \left[ -1 + 2 \mu \coth (\xi ) ( i Q + \coth (\xi )) \right] \exp (- i ( \tau _0 s + A(t))). \end{aligned} \end{aligned}$$

So, we can compute the unit timelike tangent vector \(\mathbf{t}\) by using the principal and binormal normal vectors and it leads to

$$\begin{aligned} \begin{aligned} {\mathbf{t}}_x&= 2 \mu \coth ^2(\xi ) + \mu \left( \frac{\tau _0^2 - b_1^2}{b_1^2} \right) , \\ {\mathbf{t}}_y + i {\mathbf{t}}_z&= -2 \mu \mathrm{csch} ( \xi ) ( i Q + \coth (\xi ) ) \exp (- i ( \tau _0 s + A(t))), \end{aligned} \end{aligned}$$

where \(\mu = \frac{b_1^2}{\tau _0^2 + b_1^2}\).

By integrating of the tangent vector, the timelike curve \(\gamma \) is expressed as

$$\begin{aligned} \begin{aligned} \gamma _x&= \mu \left( \frac{\tau _0^2 - b_1^2}{b_1^2} \right) s + \frac{2 \mu }{b_1} ( \xi - \coth (\xi )),\\ \gamma _y + i \gamma _z&= \frac{2 \mu }{b_1} \mathrm{csch} (\xi ) \exp (-i (\tau _0 s + ( \tau _0^2 - b_1^2 )t)), \end{aligned} \end{aligned}$$
(20)

and the curve \(\gamma \) satisfies the time evolution (3), in this case we can obtain \( A(t) = (\tau _0^2 - b_1^2) t\). Thus the curve \(\gamma \) determined by (20) satisfies the timelike binormal flow \(\gamma _t = \kappa {\mathbf{b}}\) and gives a soliton solution of the traveling wave for the timelike curve.

3.2 Spacelike Vortex Filament

In Sect. 2, we explain briefly the vortex filament for a spacelike curve in Minkowski 3-space. In this case the corresponding vortex filament \(\gamma _t = \gamma _{s} \times \gamma _{ss} = \kappa {\mathbf{b}}\) is equivalent to the nonlinear heat system (6) by using Hasimoto transformations (7). In this subsection, we want to give the parametrization of the spacelike curve from the single solitary wave solution of the nonlinear Schrödinger equation for a spacelike curve in Minkowski 3-space. In [18] and [15], authors considered the new complex frame in terms of the Frenet frame of a space curve in Euclidean 3-space and studied integrable system for a vortex filament flow.

Now, we consider a complex frame \(\{ {\mathbf{p}}_1 , {\mathbf{p}}_2, {\mathbf{p}}_2^* \}\) for the spacelike curve \(\gamma \) in Minkowski 3-space as follows:

$$\begin{aligned} \begin{aligned} {\mathbf{p}}_1&= {\mathbf{b}},\\ {\mathbf{p}}_2&=( {\mathbf{n}} + i {\mathbf{t}} ) e ^{-i \int \kappa },\\ {\mathbf{p}}_2^*&= ( {\mathbf{n}} - i {\mathbf{t}}) e ^{i \int \kappa }, \end{aligned} \end{aligned}$$
(21)

and the Hasimoto transformation

$$\begin{aligned} \phi (s,t) = \tau (s,t) ~{} \exp \left( -i \int ^s \kappa (\bar{s}, t) d \bar{s} \right) . \end{aligned}$$
(22)

In this case, if we take \(\gamma _s = {\mathbf{p}}_1\), the spacelike vortex filament equation is expressed by

$$\begin{aligned} \gamma _t = \gamma _s \times \gamma _{ss} = \tau {\mathbf{t}}. \end{aligned}$$
(23)

On the other hand, we have the time evolution for Frenet frame of the spacelike curve as follows [10]:

$$\begin{aligned} \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) _t = \left( \begin{array}{ccc} 0 &{} \frac{\tau _{ss}}{\tau } - \kappa ^2 &{} \tau _s \\ - \frac{\tau _{ss}}{\tau } + \kappa ^2 &{} 0 &{} \kappa \tau \\ \tau _s &{} \kappa \tau &{} 0 \\ \end{array} \right) \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) . \end{aligned}$$
(24)

Theorem 3.3

([10]) Let \(\gamma \) be a spacelike curve parametrized by the arc length s with the curvature \(\kappa \) and the torsion \(\tau \) and satisfies the spacelike vortex filament equation (23) in Minkowsi 3-space. Then the Hasimoto transformation \(\phi \) is a solution of the nonlinear Schrödinger equation

$$\begin{aligned} i \phi _t= -\phi _{ss} + \frac{1}{2} \left( \mid \phi \mid ^2 + R(t) \right) \phi \end{aligned}$$
(25)

for some smooth function R(t).

Theorem 3.4

(Spacelike Hasimoto traveling wave) If we consider a soliton solution of the nonlinear Schrödinger equation (25) such that \(\tau \rightarrow 0\) as \(s \rightarrow \infty \), then it gives a solution of a traveling wave with a kink that becomes a line at infinity.

Proof

To prove the theorem, consider the new variable \(\rho = s +a - ct\) with a constant velocity c and a positive constant a, then the variable implies

$$\begin{aligned} \phi (\rho ) = \tau (\rho ) ~{} \exp \left( - i \int ^s \kappa (\rho ) d s \right) . \end{aligned}$$
(26)

Since (26) is the solution of (25), the real and imaginary parts of (25) lead to

$$\begin{aligned}&c \tau (\rho ) ( \kappa (\rho ) - \kappa (a - ct) ) = \tau '' (\rho ) - \tau (\rho ) \kappa ^2 (\rho ) - \frac{1}{2} ( \tau ^2(\rho ) + R(t)) \tau (\rho ), \end{aligned}$$
(27)
$$\begin{aligned}&c \tau ' (\rho ) = - 2 \tau '(\rho ) \kappa (\rho ) - \tau (\rho ) \kappa '(\rho ). \end{aligned}$$
(28)

Look at (11), (12), (27) and (28), they are symmetric with respect to \(\kappa \) and \(\tau \), respectively. So, we have

$$\begin{aligned} \kappa= & {} \kappa _0 = -\frac{1}{2} c, \end{aligned}$$
(29)
$$\begin{aligned} \tau (\rho )= & {} 2 b_1 \mathrm{csch} ( b_1 \rho ), \end{aligned}$$
(30)

where \(R= 2 ( b_1^2 - \kappa _0^2 )\) with a nonzero constant \(b_1\).

Since \(\gamma \) is the spacelike curve, we take \(\varepsilon _1=1, \varepsilon _2=1\) and \(\varepsilon _3=-1\) in (1) and we obtain the following equation:

$$\begin{aligned} \kappa _0 ( {\mathbf{b}}'- \tau {\mathbf{n}}) = \left( \frac{1}{\tau } ( {\mathbf{t}}'' + \kappa _0^2 {\mathbf{t}}) \right) ' - \tau {\mathbf{t}}' =0. \end{aligned}$$

It is equivalently to

$$\begin{aligned} \frac{d^3 {\mathbf{t}}}{d \xi ^3} + \mathrm{coth} (\xi ) \frac{d^2 {\mathbf{t}}}{d \xi ^2} + \left( Q^2 - 4 \mathrm{csch} ^2 ( \xi ) \right) \frac{d {\mathbf{t}}}{d \xi } + \coth ( \xi ) Q^2 {\mathbf{t}}=0, \end{aligned}$$
(31)

where \(\xi = b_1 \rho \) and \(Q= \frac{\kappa _0}{b_1}\).

Applying the similar method of the timelike vortex filament, we have

$$\begin{aligned} \begin{aligned} {\mathbf{t}}_x&= 2 \mu \mathrm{csch} (\xi ),\\ {\mathbf{t}}_y + {\mathbf{t}}_z&= i\mu (1 - Q^2 -2 i Q \coth (\xi ) )e ^{-i(\kappa _0 s + ( \kappa _0^2 - b_1^2)t )},\\ {\mathbf{n}}_x&= - 2 \mu \mathrm{csch}^2 (\xi ) \coth (\xi ),\\ {\mathbf{n}}_y + {\mathbf{n}}_z&= [ 1 + 2 \mu \coth (\xi ) ( i Q - \coth (\xi ) ] e ^{-i(\kappa _0 s + ( \kappa _0^2 - b_1^2)t )},\\ {\mathbf{b}}_x&= 2 \mu \coth ^2(\xi ) + \mu \left( \frac{\kappa _0^2 - b_1^2}{b_1^2} \right) ,\\ {\mathbf{b}}_y + {\mathbf{b}}_z&= 2 \mu \mathrm{csch}(\xi ) ( - i Q + \coth (\xi ) e ^{-i(\kappa _0 s + ( \kappa _0^2 - b_1^2)t )}, \end{aligned} \end{aligned}$$
(32)

where \(\mu = \frac{b_1^2}{\kappa _0^2 + b_1^2}\). Thus, the spacelike curve \(\gamma \) satisfying the spacelike vortex filament \(\gamma _t = \tau {\mathbf{t}}\) is determined by

$$\begin{aligned} \begin{aligned} \gamma _x&= - \frac{2 \mu }{b_1} ( \xi - \coth (\xi ) ) - \mu \left( \frac{\kappa _0^2 - b_1^2}{b_1^2} \right) s,\\ \gamma _y + i \gamma _z&= \frac{2 \mu }{b_1} \mathrm{csch} (\xi ) \exp (i (\kappa _0 s + ( \kappa _0^2 - b_1^2 )t)). \end{aligned} \end{aligned}$$
(33)

This provides the traveling wave soliton solution of the spacelike curve.

4 Position Vectors of Vortex Filament

In this section, we give the method to find the exact shape of the timelike and spacelike curve from the vortex filament by solving the tangent, principal normal and binormal vector in Minkowski 3-space. The method is called an inverse Hasimoto transformation and we use the tool described by Shah [20].

First of all, let \(\gamma \) be a timelike curve with \(\varepsilon _1 =-1, \varepsilon _2 =\varepsilon _3 =1\) in (1). Then these equations can be expressed as the first order ODEs:

$$\begin{aligned} \frac{d}{ds} \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) = \left( \begin{array}{ccc} 0 &{} \kappa (s,t) &{} 0 \\ \kappa (s,t) &{} 0 &{} -\tau (s,t)\\ 0 &{} \tau (s,t)&{} 0 \\ \end{array} \right) \left( \begin{array}{ccc} {\mathbf{t}}\\ {\mathbf{n}}\\ {\mathbf{b}} \end{array} \right) , \end{aligned}$$
(34)

and this can be represented as follows

$$\begin{aligned} \frac{d U}{ds} = \mathcal {A}(s,t) U, \end{aligned}$$
(35)

where \(U= ({\mathbf{t}} \quad {\mathbf{n}} \quad {\mathbf{b}})^T\) and

$$\begin{aligned} \mathcal {A}(s,t)=\left( \begin{array}{ccc} 0 &{} \kappa (s,t) &{} 0 \\ \kappa (s,t) &{} 0 &{} -\tau (s,t)\\ 0 &{} \tau (s,t)&{} 0 \\ \end{array} \right) . \end{aligned}$$

To solve the ODE (35), we use \(\exp ( - \int ^s {\mathcal A} (\sigma , t) d \sigma )\) as an ansatz for the integrating factor and multiples this ansatz both sides of (35), then we have

$$\begin{aligned} \exp \left( - \int ^s {\mathcal A} (\sigma , t) d \sigma \right) \left( \frac{d U}{ds} - \mathcal {A}(s,t) U \right) =0, \end{aligned}$$

that is,

$$\begin{aligned} \frac{d}{ds} \left( \exp \left( - \int ^s {\mathcal A} (\sigma , t) d \sigma \right) U(s,t) \right) =0. \end{aligned}$$

It follows that we get

$$\begin{aligned} \exp \left( - \int ^s {\mathcal A} (\sigma , t) d \sigma \right) U(s,t) = C(t), \end{aligned}$$
(36)

where C(t) is a matrix that is dependent on time t.

If \(\exp ( - \int ^s {\mathcal A} (\sigma , t) d \sigma ) \) is non-singular, then equation (36) leads to

$$\begin{aligned} U(s,t) = \exp ( {\mathcal M}(s,t) ) C(t), \end{aligned}$$
(37)

where

$$\begin{aligned} {\mathcal M}(s,t)= \int ^s {\mathcal A}(\sigma ,t) d \sigma =\left( \begin{array}{ccc} 0 &{} \int ^s \kappa (\sigma ,t)d \sigma &{} 0 \\ \int ^s \kappa (\sigma ,t)d \sigma &{} 0 &{} -\int ^s \tau (\sigma ,t)d \sigma \\ 0 &{} \int ^s \tau (\sigma ,t) d \sigma &{} 0 \\ \end{array} \right) . \end{aligned}$$

Now, we must show how to calculate U(s, t) from (37) to demonstrate the method. Applying matrix exponential \(\exp ({\mathcal M}) = \sum _{k=0}^{\infty } \frac{1}{k !} {\mathcal M}^k\) for the \(3 \times 3\) matrix \(\mathcal {M}\), \(\exp ({\mathcal M})\) is given by

$$\begin{aligned} \exp ({\mathcal M})=\left( \begin{array}{ccc} \frac{1}{w^2} ( q^2 - p^2 \cos w) &{} \frac{p}{w} \sin w &{} \frac{pq}{w^2} ( \cos w -1)\\ \frac{p}{w} \sin w &{} \cos w &{} -\frac{q}{w} \sin w \\ \frac{pq}{w^2} (1 - \cos w) &{} \frac{q}{w} \sin w &{} \frac{1}{w^2} ( q^2 \cos w - p^2) \\ \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} p= \int ^s \kappa (\sigma , t) d \sigma , \quad q= \int ^s \tau (\sigma , t) d \sigma , \quad w^2 = q^2 - p^2. \end{aligned}$$
(38)

Let \(e_{ij}\) and \(c_{ij}\) be the (i, j)-th components of the matrices \(\exp ({\mathcal M})\) and C(t), respectively. Then equation (37) becomes

$$\begin{aligned} U(s,t) =\left( \begin{array}{ccc} \sum _{k=1} ^3 e_{1k}(s,t) c_{k1}(t) &{} \sum _{k=1}^3 e_{1k}(s,t) c_{k2}(t) &{}\sum _{k=1}^3 e_{1k}(s,t) c_{k3}(t) \\ \sum _{k=1}^3 e_{2k}(s,t) c_{k1}(t) &{} \sum _{k=1}^3 e_{2k}(s,t) c_{k2}(t) &{}\sum _{k=1}^3 e_{2k}(s,t) c_{k3}(t) \\ \sum _{k=1}^3 e_{3k}(s,t) c_{k1}(t) &{} \sum _{k=1}^3 e_{3k}(s,t) c_{k2}(t) &{}\sum _{k=1}^3 e_{3k}(s,t) c_{k3}(t) \\ \end{array} \right) . \end{aligned}$$

Since \(U= ({\mathbf{t}} \quad {\mathbf{n}} \quad {\mathbf{b}})^T\), the tangent vector of the timelike curve \(\gamma \) to the vortex filament equation \(\gamma _t = \tau {\mathbf{b}}\) is given by

$$\begin{aligned} {\mathbf{t}}(s,t) = \left( \sum _{k=1} ^3 e_{1k}(s,t) c_{k1}(t), \sum _{k=1}^3 e_{1k}(s,t) c_{k2}(t), \sum _{k=1}^3 e_{1k}(s,t) c_{k3}(t) \right) , \end{aligned}$$

it follows that the position vector \({\mathbf{r}}(s,t)\) of the timelike curve \(\gamma \) can be expressed as

$$\begin{aligned} {\mathbf{r}}(s,t) = \left( \begin{array}{ccc} x_0(t) + \sum _{k=1}^3 c_{k1}(t) \int ^s e_{1k}(\sigma ,t)d \sigma \\ y_0(t) + \sum _{k=1}^3 c_{k2}(t) \int ^s e_{1k}(\sigma ,t)d \sigma \\ z_0(t) + \sum _{k=1}^3 c_{k3}(t) \int ^s e_{1k}(\sigma ,t)d \sigma \\ \end{array} \right) . \end{aligned}$$
(39)

Thus, given the curvature \(\kappa (s,t)\) and the torsion \(\tau (s,t)\) of the timelike curve \(\gamma \) to the vortex filament we can construct the curve \(\gamma \) with the help of (39).

By the similar discussion as above, we give the position vector of the spacelike curve \(\gamma \) to the vortex filament equation \(\gamma _t = \tau {\mathbf{b}}\). In this case,

$$\begin{aligned} {\mathcal M}(s,t)= \left( \begin{array}{ccc} 0 &{} \int ^s \kappa (\sigma ,t)d \sigma &{} 0 \\ -\int ^s \kappa (\sigma ,t)d \sigma &{} 0 &{} \int ^s \tau (\sigma ,t)d \sigma \\ 0 &{} \int ^s \tau (\sigma ,t) d \sigma &{} 0 \\ \end{array} \right) \end{aligned}$$

and \(\exp ({\mathcal M})\) is given by

$$\begin{aligned} \exp ({\mathcal M})=\left( \begin{array}{ccc} \frac{1}{r^2} ( p^2 \cos r - q^2) &{} \frac{p}{r} \sin r &{} \frac{pq}{r^2} (1- \cos r)\\ -\frac{p}{r} \sin w &{} \cos r &{} \frac{q}{r} \sin r \\ \frac{pq}{r^2} (\cos r-1) &{} \frac{q}{r} \sin r &{} \frac{1}{r^2} (p^2- q^2 \cos r ) \\ \end{array} \right) , \end{aligned}$$

where \(r^2 =p^2 - q^2\).

Thus the position vector \({\mathbf{r}}(s,t)\) of the spacelike curve \(\gamma \) to the vortex filament equation with the help of (37) is determined by

$$\begin{aligned} {\mathbf{r}}(s,t) = \left( \begin{array}{ccc} x_0(t) + \sum _{k=1}^3 c_{k1}(t) \int ^s e_{3k}(\sigma ,t)d \sigma \\ y_0(t) + \sum _{k=1}^3 c_{k2}(t) \int ^s e_{3k}(\sigma ,t)d \sigma \\ z_0(t) + \sum _{k=1}^3 c_{k3}(t) \int ^s e_{3k}(\sigma ,t)d \sigma \\ \end{array} \right) . \end{aligned}$$
(40)

5 Applications

Consider the new variable

$$\begin{aligned} \Phi = \varphi _{\pm }\exp \left( \frac{1}{2} i \int ^t R(t) dt \right) , \end{aligned}$$

where \(\varphi _{-}\) and \(\varphi _{+}\) are the Hasimoto transformations of the timelike and spacelike curves to the vortex filament, respectively, that is, \(\varphi _{-}=\psi \) and \(\varphi _{+}=\phi \). Then, the NLS equations (9) and (25) of the timelike curve and the spacelike curve are written as

$$\begin{aligned} i\Phi _t = - \Phi _{ss} + \frac{1}{2} \mid \Phi \mid ^2 \Phi =0. \end{aligned}$$
(41)

In order to obtain the solitary wave solution to the NLS equation (41), the starting ansatz is taken to be

$$\begin{aligned} \Phi (s,t) = u(s,t) \exp (i v(s,t)). \end{aligned}$$

This ansatz is used in the work of inverse Hasimoto transformation. Substituting the last equation into (41) and separating the real and imaginary parts, it is obtained as

$$\begin{aligned} \begin{aligned} u_t(s,t)&= - 2 v_s(s,t) u_s(s,t) - v_{ss}(s,t) u(s,t),\\ v_t (s,t)u (s,t)&= u_{ss}(s,t) - v_s^2(s,t) u(s,t) - \frac{1}{2} u^3(s,t). \end{aligned} \end{aligned}$$
(42)

Example 5.1

If we take

$$\begin{aligned} u(s,t) = 2 \tanh ( s- 2 ct),\quad v(s,t) = cs - ( c^2 + 2 )t, \end{aligned}$$

it is a solution of the PDEs (42) and from (38) we have

$$\begin{aligned} p (s,t) = 2 ~\mathrm{ln} \cosh ( s- 2 ct), \quad q(s,t) = - cs + ( c^2 +2 )t \end{aligned}$$

which are connected to the curvature and the torsion of the timelike and spacelike curve, respectively. Thus, the inverse Hasimoto parametrization can directly be constructed by

$$\begin{aligned} \mathbf{r }(s,t)= ( x(s,t), y(s,t), z(s,t)), \end{aligned}$$
(43)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} x(s,t) = t \int _0^s \frac{2 \mathrm{ln} \cosh (s -2t) \sin \left( {\sqrt{ (-s + 3t)^2 - (2 \mathrm{ln} \cosh (s -2t) )^2 }} \right) }{\sqrt{ (-s + 3t)^2 - (2 \mathrm{ln} \cosh (s -2t) )^2 }} ds,\\ y(s,t) = t + \int _0^s \frac{ (-s + 3t)^2 - (2 \mathrm{ln} \cosh (s -2t) )^2 \cos \left( {\sqrt{ (-s + 3t)^2 - (2 \mathrm{ln} \cosh (s -2t) )^2 }} \right) }{ (-s + 3t)^2 - (2 \mathrm{ln} \cosh (s -2t) )^2 } ds,\\ z(s,t) = t \int _0^s \frac{2 \mathrm{ln} \cosh (s -2t)(-s + 3t) \left( \cos \left( \sqrt{ (-s + 3t)^2 - (2 \mathrm{ln} \cosh (s -2t) )^2 } \right) -1 \right) }{(-s + 3t)^2 - (2 \mathrm{ln} \cosh (s -2t) )^2} ds,\\ \mathrm{if} ~\gamma (s)~ \text { is a timelike curve}, \end{array}\right. } \end{aligned}$$
(44)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} x(s,t) = t \int _0^s \frac{(-s + 3t) \sin \left( {\sqrt{ (2 \mathrm{ln} \cosh (s -2t) )^2 -(-s + 3t)^2 }} \right) }{\sqrt{ (2 \mathrm{ln} \cosh (s -2t) )^2 -(-s + 3t)^2 }} ds,\\ y(s,t) = t + \int _0^s \frac{ 2 \mathrm{ln} \cosh (s -2t) (-s + 3t) \left( \cos \left( {\sqrt{ (2 \mathrm{ln} \cosh (s -2t) )^2-(-s + 3t)^2 }} \right) -1 \right) }{ (2 \mathrm{ln} \cosh (s -2t) )^2 -(-s + 3t)^2 } ds,\\ z(s,t) = t \int _0^s \frac{ (2 \mathrm{ln} \cosh (s -2t))^2 -(-s + 3t)^2 \cos \left( \sqrt{ (2 \mathrm{ln} \cosh (s -2t))^2 -(-s + 3t)^2} \right) }{ (2 \mathrm{ln} \cosh (s -2t) )^2-(-s + 3t)^2} ds,\\ \mathrm{if} ~\gamma (s)~ \text { is a spacelike curve}, \end{array}\right. } \end{aligned}$$
(45)

in this case we choose a matrix C(t) given by

$$\begin{aligned} C(t)=\left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ t &{} 0 &{} 0 \\ 0 &{} 0 &{} t \\ \end{array} \right) , \end{aligned}$$
(46)

and \(x_0(t)=0, y_0(t) = t, z_0(t)=0, c=1.\)

Example 5.2

We consider that the torsion of the timelike curve \(\gamma \) vanishes, that is, \(\tau =0\). In this case, \(v=v(s,t)\) is constant in (42) and we have

$$\begin{aligned} u_t=0, \quad u_{ss}- \frac{1}{2} u^3=0. \end{aligned}$$

Its solution is given by \( u(s,t)= - \frac{2}{s}, \) which implies

$$\begin{aligned} p= - 2 \ln s, \quad q= q_0. \end{aligned}$$
(47)

If we choose a matrix C(t) given by (46), then from (39) we have the inverse Hasimoto parametrization \(\mathbf{r }(s,t)\) as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} x(s,t) = t\int ^se_{12}ds = t \int ^s \frac{p \sin w}{w} ds = t \int ^s \frac{-2 \ln s \sin ({\sqrt{q_0^2 - 4 (\ln s)^2} } ) }{\sqrt{ q_0^2 - 4 (\ln s)^2 }} ds,\\ y(s,t) = t + \int ^s e_{11} ds = t + \int ^s \frac{q_0^2 - p^2 \cos w}{w^2} =t + \int ^s \frac{q_0^2 - 4 (\ln s)^2 \cos ({\sqrt{q_0^2 - 4 (\ln s)^2} } )}{ q_0^2 - 4 (\ln s)^2 } ds ,\\ z(s,t) = t \int ^s e_{13} ds = t \int ^s \frac{pq_0 (\cos w -1)}{w^2}ds= t \int ^s \frac{ -2 q_0 \ln s \left( \cos ( {\sqrt{q_0^2 - 4 (\ln s)^2} } ) -1 \right) }{ q_0^2 - 4 (\ln s)^2 } ds . \end{array}\right. } \end{aligned}$$
(48)

6 Conclusion

One of classical nonlinear differential equations by through inverse scattering transform is the vortex filament equation \(\gamma _t = \gamma _s \times \gamma _{ss}\) and this equation becomes the binormal flow \(\gamma _t = \kappa {\mathbf{b}}\) when the parameter s of the curve \(\gamma (s)\) is arc-length. We know that the binormal flow of the timelike curve or the spacelike curve is equivalent to the NLS equation or the heat equation, respectively. However, in [10] authors studied a spacelike curve with the new complex frame (21), in this case they showed that the vortex filament equation is equivalent to the nonlinear Schrödinger equation (25).

In this work, we construct the parametrizations of the timelike curve and the spacelike curve from the traveling wave soliton solution of the nonlinear Schrödinger equation. Also, we give the method to find the inverse Hasimoto transformation of the timelike and spacelike curve for the vortex filament by solving the Frenet vectors in Minkowski 3-space and provide applications to illustrate the inverse Hasimoto transformation.