Framed Curve Families Induced by Real and Complex Coupled Dispersionless-Type Equations

Abstract

In this study, we investigate coupled real and complex dispersionless equations for curve families, even if they have singular points. Even though the connections with the differential equations and regular curves were considered in various ways in the past, since each curve does not need to be regular, we establish the connections for framed base curves, which generalize regular curves with linear independent conditions. Also, we give the Lax pairs of the real and complex coupled dispersionless equations from the motions of any framed curve. These give us significant conditions based on the framed curvatures and associated curvatures of the framed curves for integrability since it is well known that the Lax pair provides the integrability of differential equations.

1. Introduction

Nonlinear partial differential equation (PDE) systems have been utilized to model a wide range of issues in recent years. Since they offer comprehensive discrimination of the dynamics of complex systems and aid in visualizing and controlling their behavior, it is important to understand how these systems behave in order to advance numerous fields of science and engineering. These systems are an active area of research, and many mathematical and computational techniques are continuously being developed to contribute to the revelation of vital phenomena, diseases, populations, and many other physical problems. Real coupled dispersionless (CD) or complex coupled dispersionless (CCD) equations are types of PDE systems that describe the dynamics of multiple variables or fields without dispersion. New coupled integrable dispersionless equations were introduced, and the inverse scattering method was used to solve them [1]. The inverse scattering method solves the general form of the integrable dispersionless equation. Soliton solutions consist of growing solitons, decaying solitons, and stationary solitons, and their interaction and properties were considered in [2]. In addition to the CD and CCD equations, real short pulse (SP) and complex short pulse (CSP) equations were investigated by Shen et al. in [3]. This study demonstrated that the motions of the space curves are first linked to CD and CCD equations and then to SP and CSP equations via hodograph transformations, and in addition, it was expressed that providing these equations’ Lax pairs guarantees their integrability. Kakuhata and Konno suggested a generalized inverse technique for the coupled integrable dispersionless system from a group theoretic perspective [4]. Moreover, the authors of ref. [5] derived the Hamiltonian starting from the Lagrangian of the unified integrable and distributionless equations and obtained the conserved quantities associated with Lagrangian symmetries, which the inverse scattering approach cannot obtain. In [6], the defocusing CSP equations were investigated with geometric and algebraic approaches, and a link of the CCD system with the motion of space curves was stated in Minkowski space. Recently, in [7], the CCD equations for non-null curves were interpreted in Minkowski space, which has spatial and temporal dimensions and forces us to investigate curves with a physical insight via the non-Euclidean metric. Besides the gauge equivalence between the Zhaidary-III and CD equations, the families of involute–evolute curves generated by the CD equations were derived by Eren et al. [8]. In a similar vein, but this time according to the Bishop frame, the CD equations were constructed by Eren [9] for pairs of curves such as Mannheim curves in [10] and Bertrand curves in [11]. In addition to all of these, since differential equations are mathematical tools used for capturing some natural phenomena, various approaches have been used to study differential equations; for instance, Ref. [12] presented the definition of the finite volume method’s mechanism and applied the suggested comparisons of the numerical methods described in the paper using an inviscid Burgers equation. Further, the major findings of multiple simulations for the Burgers equation and one for the Buckley–Leverett equation were presented. The fourth-order nonlinear damped delayed differential equation class oscillation with distributed deviation arguments was the subject of the research [13]. A new explanation of quaternary equation oscillation in terms of undamped oscillation of a similar well-studied quadratic linear differential equation was proposed. The extended Riccati transform, integral averaging approach, and comparison principles were used to procure some additional oscillation criteria. Bazighifan et al. investigated the asymptotic behavior of fourth-order advanced differential equations in a certain form, and new results were obtained in the form of Philos-type and Hille–Nehari oscillation criteria for the oscillation behavior of these equations [14]. Vikas et al. carried out an analytical study of coupled convective heat and mass transfer with volumetric heating data describing the sublimation of a porous body under the most sensitive temperature inputs as well as a numerical study of a nonlinear porous sublimation issue with temperature-dependent thermal conductivity and concentration-dependent mass diffusivity [15,16]. As it is known, modeling via PDEs takes into account possible configurations of the dynamic systems. Therefore, PDE-based controllers are more precise and reliable for implementation. For instance, some implementations of complex and coupled dispersionless-type equations yielding dynamic controllers can be seen in [17,18,19]. Additionally, the physical applications or geometric interpretations of special nonlinear PDEs are attractive research areas, and its importance is increasing in line with the developments in physical and engineering sciences. In the majority of investigations that include geometrical interpretations, the Frenet frames of the moving smooth space curves are taken into consideration. However, the Frenet frame of a moving space curve cannot be defined at the singular points. An alternative frame that can be accepted as relatively new, a Frenet-type frame, was defined along a curve under certain conditions to solve this problem. Also, the basic concepts of the Frenet-type framed base curve and the existence and uniqueness conditions of the framed curve introduced in [20] have affected the following investigations widely. For instance, the Bertrand and Mannheim curve pairs of framed curves were considered by [21], and evolutes and focal surfaces of framed immersions were constructed in Euclidean 3-space in [22]. In addition, framed slant helices [23], generalized oscillating type ruled surfaces of singular curves in Euclidean 3-space [24], and framed normal curves in Euclidean 4-space [25] are recent studies using this frame.

The novelty of the paper arises from the benefit of the Frenet-type frame at the singular points. Herein, we discuss the relationships of the CD and CCD equations with the motions of the space curve, even if they are not regular at each point. Starting from this idea, we find the Lax pairs providing the integration of the CD and CCD equations to be obtained via the framed curvatures and associated curvatures of the framed curves with some singular points. We conclude by demonstrating that the conserved quantities in the CD and CCD equations are constants. The novelty allows us to incorporate the results of the link of the regular curves’ motions to the CD and CCD equations that are given [3].

2. Preliminaries

2.1. Coupled Dispersionless-Type Equations

Konno et al. introduced the real CD equation in [1] as

$$\left\{\begin{array}{c}{\omega }_{us}=\rho \omega ,\hfill \\ {\rho }_s+\omega {\omega }_u=0.\hfill \end{array}\right.$$

The Lax pair of this real CD equation is

$$\left\{\begin{array}{c}{\psi }_u=C\psi ,\hfill \\ {\psi }_s=D\psi ,\hfill \end{array}\right.$$

where

$$C=-i\lambda \left(\begin{array}{cc}\rho & {\omega }_u\\ {\omega }_u& -\rho \end{array}\right),D=\left(\begin{array}{cc}\frac{i}{4\lambda }& -\frac{1}{2}\omega \\ \frac{1}{2}\omega & -\frac{i}{4\lambda }\end{array}\right)$$

in [1,2]. Here, $\lambda \ne 0$ and $C_s-D_u+\left[C,D\right]=0$ are also satisfied [3].

The CCD equation is defined as

$$\left\{\begin{array}{c}{\omega }_{us}=\rho \omega ,\hfill \\ {\rho }_s+\frac{1}{2}{\left({\left|\omega \right|}^2\right)}_u=0,\hfill \end{array}\right.$$

where $\omega$ is a real-valued function, and the subscripts u and s denote partial differentiations [4,5]. The Lax pair, which procures the integrability of this CCD equation, is given by

$${\psi }_u=A\psi ,{\psi }_s=B\psi ,$$

where

$$A=-i\lambda \left(\begin{array}{cc}\rho & {\omega }_u\\ {\omega }_u^{*}& -\rho \end{array}\right),B=\left(\begin{array}{cc}\frac{i}{4\lambda }& -\frac{1}{2}\omega \\ \frac{1}{2}{\omega }^{*}& -\frac{i}{4\lambda }\end{array}\right),$$

and the function ${\omega }^{*}$ indicates the complex conjugate of $\omega$ [4,5].

2.2. Framed Curves in Euclidean 3-Space

Let $\gamma :I\to {\mathbb{R}}^3$ be a smooth space curve that does not need to be regular in Euclidean 3-space and let the set ${\Delta }_2$ be a three-dimensional smooth manifold defined by

$${\Delta }_2=\left\{\mu =\left({\mu }_1,{\mu }_2\right)\in {\mathbb{R}}^3\times {\mathbb{R}}^3:〈{\mu }_i,{\mu }_j〉={\delta }_{ij},i,j=1,2\right\}.$$

Let $\mu =\left({\mu }_1,{\mu }_2\right)\in {\Delta }_2$ and let $\upsilon$ be a unit vector such that $\upsilon ={\mu }_1\times {\mu }_2$ [20].

Definition 1. 

$\left(\gamma ,\mu \right):I\to {\mathbb{R}}^3\times {\Delta }_2$ is called a framed curve provided that $〈{\gamma }^{\prime },{\mu }_1〉=〈{\gamma }^{\prime },{\mu }_2〉=0$ for each $u\in I$. Also, $\gamma :I\to {\mathbb{R}}^3$ is called a framed base curve if $\mu :I\to {\Delta }_2$, where $\left(\gamma ,\mu \right)$ is a framed curve [20].

Let $\left(\gamma ,{\mu }_1,{\mu }_2\right):I\to {\mathbb{R}}^3\times {\Delta }_2$ be a framed curve and $\upsilon ={\mu }_1\times {\mu }_2$. Then, the Frenet-type formula of the framed curve is written as follows:

$$\left[\begin{array}{c}{{\mu }_1}^{\prime }\left(u\right)\\ {{\mu }_2}^{\prime }\left(u\right)\\ {\upsilon }^{\prime }\left(u\right)\end{array}\right]=\left[\begin{array}{ccc}0& l\left(u\right)& m\left(u\right)\\ -l\left(u\right)& 0& n\left(u\right)\\ -m\left(u\right)& -n\left(u\right)& 0\end{array}\right]\left[\begin{array}{c}{\mu }_1\left(u\right)\\ {\mu }_2\left(u\right)\\ \upsilon \left(u\right)\end{array}\right].$$

(1)

Here, $l\left(u\right)=〈{{\mu }^{\prime }}_1\left(u\right),{\mu }_2\left(u\right)〉$, $m\left(u\right)=〈{{\mu }^{\prime }}_1\left(u\right),\upsilon \left(u\right)〉$, and $n\left(u\right)=〈{{\mu }^{\prime }}_2\left(u\right),\upsilon \left(u\right)〉$. $\left(l\left(u\right),m\left(u\right),n\left(u\right),\alpha \left(u\right)\right)$ is called the curvature of the framed curve $\gamma$ where the smooth mapping $\alpha :I\to \mathbb{R}$ is given by

$${\gamma }^{\prime }\left(u\right)=\alpha \left(u\right)\upsilon \left(u\right).$$

$\alpha \left(u_0\right)=0$ only if $u_0$ is a singular point of the curve $\gamma$. If $m\left(u\right)=n\left(u\right)=0$, then ${\upsilon }^{\prime }\left(u\right)=0$.

In this article, we make the assumption that ${\upsilon }^{\prime }\left(u\right)\ne 0$ as in [20]. The sequential two basic theorems on the existence and uniqueness of framed curves admit a foundation for understanding the behavior of framed curves and ensuring the validity of solutions to framed curve equations.

Theorem 1 

(The Existence Theorem). Let $\left(l,m,n,\alpha \right):I\to {\mathbb{R}}^4$ be a smooth mapping. There exists a framed curve, $\left(\gamma ,\mu \right):I\to {\mathbb{R}}^3\times {\Delta }_2$, whose associated curvatures are $\left\{l,m,n,\alpha \right\}$ [20].

Theorem 2 

(The Uniqueness Theorem). Let $\left(\gamma ,\mu \right)$ and $\left(\overline{\gamma },\overline{\mu }\right)$ be framed curves whose curvatures $\left\{l,m,n,\alpha \right\}$ and $\left\{\overline{l},\overline{m},\overline{n},\overline{\alpha }\right\}$ are coincided. Then, the curves $\left(\gamma ,\mu \right)$ and $\left(\overline{\gamma },\overline{\mu }\right)$ are congruent as framed curves [20].

Let $\left(\gamma ,{\mu }_1,{\mu }_2\right):I\to {\mathbb{R}}^3\times {\Delta }_2$ be a framed curve with the curvatures $\left\{l,m,n,\alpha \right\}$. The normal plane of the curve $\gamma \left(u\right)$ is spanned by ${\mu }_1$ and ${\mu }_2$. Let $\phi \left(u\right)$ be a smooth function. If the relationship

$$\left[\begin{array}{c}{\overline{\mu }}_1\left(u\right)\\ {\overline{\mu }}_2\left(u\right)\end{array}\right]\left[\begin{array}{cc}cos\phi \left(u\right)& -sin\phi \left(u\right)\\ sin\phi \left(u\right)& cos\phi \left(u\right)\end{array}\right]\left[\begin{array}{c}{\mu }_1\left(u\right)\\ {\mu }_2\left(u\right)\end{array}\right]$$

is expressed, then $\left(\gamma ,{\overline{\mu }}_1,{\overline{\mu }}_2\right):I\to {\mathbb{R}}^3\times {\Delta }_2$ is also a framed curve. Thus, it is evident that

$$\overline{\upsilon }\left(u\right)={\mu }_1\left(u\right)\times {\mu }_2\left(u\right)={\overline{\mu }}_1\left(u\right)\times {\overline{\mu }}_2\left(u\right)=\upsilon \left(u\right).$$

Using straightforward calculations, there are the relations

$$\begin{array}{c}{\upsilon }^{\prime }\left(u\right)=p\left(u\right){\overline{\mu }}_1\left(u\right),\hfill \\ {{\overline{\mu }}^{\prime }}_1\left(u\right)=-p\left(u\right)\upsilon \left(u\right)+\left(l\left(u\right)-{\phi }^{\prime }\left(u\right)\right){\overline{\mu }}_2\left(u\right),\hfill \\ {{\overline{\mu }}^{\prime }}_2\left(u\right)=-\left(l\left(u\right)-{\phi }^{\prime }\left(u\right)\right){\overline{\mu }}_1\left(u\right),\hfill \end{array}$$

where $\phi :I\to \mathbb{R}$ is a smooth function, and there exist the equalities

$m\left(u\right)sin\phi \left(u\right)=-n\left(u\right)cos\phi \left(u\right)$, $m\left(u\right)=-p\left(u\right)cos\phi \left(u\right)$, and $n\left(u\right)=p\left(u\right)sin\phi \left(u\right)$.

The vectors $\upsilon \left(u\right)$, ${\overline{\mu }}_1\left(u\right)$, and ${\overline{\mu }}_2\left(u\right)$ construct an adapted frame along the framed curve $\gamma \left(u\right)$, so the following Frenet frame formula emerges:

$$\left[\begin{array}{c}{\upsilon }^{\prime }\left(u\right)\\ {{\overline{\mu }}^{\prime }}_1\left(u\right)\\ {{\overline{\mu }}^{\prime }}_2\left(u\right)\end{array}\right]=\left[\begin{array}{ccc}0& p\left(u\right)& 0\\ -p\left(u\right)& 0& q\left(u\right)\\ 0& -q\left(u\right)& 0\end{array}\right]\left[\begin{array}{c}\upsilon \left(u\right)\\ {\overline{\mu }}_1\left(u\right)\\ {\overline{\mu }}_2\left(u\right)\end{array}\right],$$

(2)

where

$$p\left(u\right)=∥{\upsilon }^{\prime }\left(u\right)∥>0\mathrm{and}q\left(u\right)=l\left(u\right)-{\phi }^{\prime }\left(u\right).$$

The vectors $\upsilon \left(u\right)$, ${\overline{\mu }}_1\left(u\right)$, and ${\overline{\mu }}_2\left(u\right)$ are called the generalized tangent vector, the generalized principal normal vector, and the generalized binormal vector of $\gamma \left(u\right)$, respectively. Additionally, $\left(p\left(u\right),q\left(u\right),\alpha \left(u\right)\right)$ is called the framed curvature of the curve $\gamma \left(u\right)$ [20].

3. Framed Curve Families Induced by the Real and Complex Coupled Dispersionless Equations

In this section of the study, the CCD and CD equations will be linked to a framed curve based on its curvatures, respectively, in Euclidean 3-space. Let us consider the following family of framed curves:

$$\gamma :\left[0,l_1\right]\times \left[0,l_2\right]\to {\mathbb{R}}^3$$

where $s\in \left[0,l_2\right]$ represents the time and $u\in \left[0,l_1\right]$ is the arc-length parameter of each framed curve $\gamma (u,s_0)$ at any moment $s_0$. Considering Equation (1), the time evolution of the orthonormal Frenet-type frame $\left\{{\mu }_1,{\mu }_2,\upsilon \right\}$ in matrix form for the framed curve is given by

$${\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\\ \upsilon \end{array}\right]}_s=\left[\begin{array}{ccc}0& f_1& f_2\\ -f_1& 0& f_3\\ -f_2& -f_3& 0\end{array}\right]\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\\ \upsilon \end{array}\right],$$

(3)

where $f_1,f_2$, and $f_3$ are functions of u and s. The partial differentiations of the equations in (1) according to the parameter s and the equations in (3) according to the parameter u are

$$\begin{array}{c}{{\mu }_1}_{us}=\left(l_s{\mu }_2+l{{\mu }_2}_s\right)+\left(m_s\upsilon +m{\upsilon }_s\right),\hfill \\ {{\mu }_2}_{us}=-\left(l_s{\mu }_1+l{{\mu }_1}_s\right)+\left(n_s\upsilon +n{\upsilon }_s\right),\hfill \\ {\upsilon }_{us}=-\left(m_s{\mu }_1+m{{\mu }_1}_s\right)-\left(n_s{\mu }_2+n{{\mu }_2}_s\right),\hfill \end{array}$$

and

$$\begin{array}{c}{{\mu }_1}_{su}=\left({f_1}_u{\mu }_2+f_1{{\mu }_2}_u\right)+\left({f_2}_u\upsilon +f_2{\upsilon }_u\right),\hfill \\ {{\mu }_2}_{su}=-\left({f_1}_u{\mu }_1+f_1{{\mu }_1}_u\right)+\left({f_3}_u\upsilon +f_3{\upsilon }_u\right),\hfill \\ {\upsilon }_{su}=-\left({f_2}_u{\mu }_1+f_2{{\mu }_1}_u\right)-\left({f_3}_u{\mu }_2+f_3{{\mu }_2}_u\right),\hfill \end{array}$$

respectively. Thus, using the equations ${{\mu }_1}_{su}={{\mu }_1}_{us},{{\mu }_2}_{su}={{\mu }_2}_{us}$, and ${\upsilon }_{su}={\upsilon }_{us}$ gives us the relations

$$l_s={f_1}_u-n{f}_2+m{f}_3,$$

(4)

$$m_s=n{f}_1+{f_2}_u-l{f}_3,$$

(5)

$$n_s=-m{f}_1+l{f}_2+{f_3}_u.$$

(6)

Theorem 3. 

Let $\left(\gamma ,{\mu }_1,{\mu }_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ be a framed curve family and $\upsilon ={\mu }_1\times {\mu }_2$; then, the following relations provide the CCD equations:

$${\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\\ \upsilon \end{array}\right]}_u=\left[\begin{array}{ccc}0& c\rho & 0\\ -c\rho & 0& c{\omega }_u\\ 0& -c{\omega }_u& 0\end{array}\right]\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\\ \upsilon \end{array}\right]and{\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\\ \upsilon \end{array}\right]}_s=\left[\begin{array}{ccc}0& -c^{-1}& \omega \\ c^{-1}& 0& 0\\ -\omega & 0& 0\end{array}\right]\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\\ \upsilon \end{array}\right],$$

where $c\ne 0$ is constant. The ω and ρ are smooth functions of u and s.

Proof. 

If $\left(\gamma (u,s),{\mu }_1,{\mu }_2\right):I\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family by associating Equations (5) and (6), the following equation is immediate:

$${\left(n+im\right)}_s={\left(f_3+i{f}_2\right)}_u+i{f}_1\left(n+im\right)-il\left(f_3+i{f}_2\right).$$

(7)

Based on the hypothesis, $l=c\rho ,m=0,n=c{\omega }_u,f_1=-c^{-1},f_2=\omega ,$ and $f_3=0$ are satisfied and then Equation (7) gives us the first equation of the CCD equations as

$${\omega }_{us}=\rho \omega$$

(8)

such that $n+im=c{\omega }_u$, $l=c\rho$, $f_3+i{f}_2=i\omega$, and $f_1=-c^{-1}.$ Therefore, it is found that

$$\frac{1}{2}{\left({\left|\omega \right|}^2\right)}_u=\frac{1}{2}\left(\omega {\omega }_u^{*}+{\omega }_u{\omega }^{*}\right)=\frac{n{f}_2-m{f}_3}{c}$$

(9)

is satified, where ${\omega }^{*}$ denotes the conjugate of $\omega$. By substituting the equalities $l=c\rho$ and $f_1=-c^{-1}$ into Equation (4), we find that

$$n{f}_2-f_{3}m=-c{\rho }_s.$$

(10)

Consequently, considering Equations (9) and (10) together gives us the second equation of CCD equations as

$${\rho }_s+\frac{1}{2}{\left({\left|\omega \right|}^2\right)}_u=0.$$

(11)

So, Equations (8) and (11) correspond to the CCD equations for a framed curve family. □

Corollary 1. 

Let $\left(\gamma ,{\mu }_1,{\mu }_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ be a framed curve family and $\upsilon ={\mu }_1\times {\mu }_2$; then, the CCD equations correspond to the set $\left\{l,m,n,f_1,f_2,f_3\right\}=\left\{c\rho ,0,c{\omega }_u,-c^{-1},\omega ,0\right\}$.

Now, we investigate the correlation between the CD equations and a framed curve according to its adapted frames. The time evolution of the orthonormal Frenet frame $\left\{\upsilon ,{\overline{\mu }}_1,{\overline{\mu }}_2\right\}$ of a framed curve is given in matrix form as follows:

$${\left[\begin{array}{c}\upsilon \\ {\overline{\mu }}_1\\ {\overline{\mu }}_2\end{array}\right]}_s=\left[\begin{array}{ccc}0& g_1& g_2\\ -g_1& 0& g_3\\ -g_2& -g_3& 0\end{array}\right]\left[\begin{array}{c}\upsilon \\ {\overline{\mu }}_1\\ {\overline{\mu }}_2\end{array}\right],$$

(12)

where $g_1,g_2$, and $g_3$ are smooth functions of u and s. The partial differentiations of the equations in (2) according to the parameter s and the equations in (12) according to the parameter u are

$$\begin{array}{c}{\upsilon }_{us}=\left(p_s{\overline{\mu }}_1+p{\overline{\mu }}_{1s}\right),\hfill \\ {\overline{\mu }}_{1us}=-\left(p_s\upsilon +p{\upsilon }_s\right)+\left(q_s{\overline{\mu }}_2+q{\overline{\mu }}_{2s}\right),\hfill \\ {\overline{\mu }}_{2us}=-\left(q_s{\overline{\mu }}_1+q{\overline{\mu }}_{1s}\right),\hfill \end{array}$$

and

$$\begin{array}{c}{\upsilon }_{su}=\left({g_1}_u{\overline{\mu }}_1+g_1{\overline{\mu }}_{1u}\right)+\left({g_2}_u{\overline{\mu }}_2+g_2{\overline{\mu }}_{2u}\right),\hfill \\ {\overline{\mu }}_{1su}=-\left({g_1}_u\upsilon +g_1{\upsilon }_u\right)+\left({g_3}_u{\overline{\mu }}_2+g_3{\overline{\mu }}_{2u}\right),\hfill \\ {\overline{\mu }}_{2su}=-\left({g_2}_u\upsilon +g_2{\upsilon }_u\right)-\left({g_3}_u{\overline{\mu }}_1+g_3{\overline{\mu }}_{1u}\right),\hfill \end{array}$$

respectively. Thus, considering the equalities ${\upsilon }_{su}={\upsilon }_{us},{\overline{\mu }}_{1su}={\overline{\mu }}_{1us}$, and ${\overline{\mu }}_{2su}={\overline{\mu }}_{2us}$, which imply the inextensibility of the framed curve, we have

$${g_1}_u=p_s+g_{2}q,$$

(13)

$${g_2}_u=-q{g}_1+p{g}_3,$$

(14)

$${g_3}_u=q_s-p{g}_2.$$

(15)

Theorem 4. 

Let $\left(\gamma ,{\overline{\mu }}_1,{\overline{\mu }}_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ be a framed curve family with non-zero framed curvatures p and q for $u\in I$; then, the following relations ensure the CD equations:

$${\left[\begin{array}{c}\upsilon \\ {\overline{\mu }}_1\\ {\overline{\mu }}_2\end{array}\right]}_u=\left[\begin{array}{ccc}0& c\rho & 0\\ -c\rho & 0& c{\omega }_u\\ 0& -c{\omega }_u& 0\end{array}\right]\left[\begin{array}{c}\upsilon \\ {\overline{\mu }}_1\\ {\overline{\mu }}_2\end{array}\right]and{\left[\begin{array}{c}\upsilon \\ {\overline{\mu }}_1\\ {\overline{\mu }}_2\end{array}\right]}_s=\left[\begin{array}{ccc}0& -c^{-1}& \omega \\ c^{-1}& 0& 0\\ -\omega & 0& 0\end{array}\right]\left[\begin{array}{c}\upsilon \\ {\overline{\mu }}_1\\ {\overline{\mu }}_2\end{array}\right]$$

where c is a non-zero constant. The ω and ρ are smooth functions of u and s.

Proof. 

Let $\left(\gamma (u,s),{\overline{\mu }}_1,{\overline{\mu }}_2\right):I\to {\mathbb{R}}^3\times {\Delta }_2$ be a framed curve family with the framed curvatures p and q that do not vanish for all $u\in I$. Based on the hypothesis $g_1=-c^{-1},g_2=\omega$ and $g_3=0$ are satisfied, and Equations (13)–(15) are obtained as

$$0=p_s+\omega q,$$

(16)

$${\omega }_u=q{c}^{-1},$$

(17)

$$0=q_s-p\omega ,$$

(18)

respectively. In addition, by substituting the curvature functions $p=c\rho$ and $q=c{\omega }_u$ into Equations (16) and (18), respectively, the CD equations can be found as desired, i.e.,

$${\rho }_s+\omega {\omega }_u=0 \mathrm{and} \rho \omega ={\omega }_{us}.$$

□

Corollary 2. 

Let $\left(\gamma ,{\overline{\mu }}_1,{\overline{\mu }}_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ be a framed curve family with the framed curvatures p and q that do not vanish for all $u\in I$; then, the CD equations correspond to the set $\left\{g_1,g_2,g_3,p,q\right\}=\left\{-c^{-1},\omega ,0,c\rho ,c{\omega }_u\right\}$.

Example 1. 

Let $\gamma :\left[0,\pi \right]\times \left[0,1\right]\to {\mathbb{R}}^3$ denote a family of astroids given by

$$\gamma \left(u,s\right)=\left\{{cos}^3\left(u-s\right)+s,{sin}^3\left(u-s\right),cos2\left(u-s\right)\right\}.$$

At any moments, $s_0=u$ and $s_0=u+\frac{\pi }{2}$, so it is impossible to define the Frenet frames of the curve $\gamma \left(u,s_0\right)$ at the cuspidal singular points. Additionally, Frenet frames of the astroid $\gamma \left(u,0\right)$ can be constructed at regular points such as $\left(\frac{\pi }{4},0\right)$ and $\left(\frac{3\pi }{4},0\right)$ (see Figure 1).

On the other hand, $\left(\gamma ,{\mu }_1,{\mu }_2\right):\left[0,\pi \right]\times \left[0,1\right]\to {\mathbb{R}}^3\times {\Delta }_2$ is a family of framed curves with

$$\begin{array}{c}\gamma \left(u,s\right)=\left\{{cos}^3\left(u-s\right)+s,{sin}^3\left(u-s\right),cos2\left(u-s\right)\right\},\hfill \\ {\mu }_1\left(u,s\right):=\left\{\frac{4}{5}cos\left(u-s\right),-\frac{4}{5}sin\left(u-s\right),-\frac{3}{5}\right\},\hfill \\ {\mu }_2\left(u,s\right):=\left\{sin\left(u-s\right),cos\left(u-s\right),0\right\}.\hfill \end{array}$$

Obviously, the generalized tangent of $\left(\gamma \left(u,s\right),{\mu }_1,{\mu }_2\right)$ is $\upsilon \left(u,s\right)=\{\frac{3}{5}cos\left(s-u\right),\frac{3}{5}sin\left(s-u\right),$$\frac{4}{5}\}$, where $\alpha \left(u,s\right)=\frac{5}{2}sin\left(s-u\right)$. The Frenet-type frames of the framed curve can be constituted at singular points as well as at regular ones; see Figure 2.

Here, the found curvatures $\left\{l,m,n\right\}=\left\{-\frac{4}{5},0,\frac{3}{5}\right\}$ and $\left\{f_1,f_2,f_3\right\}=\left\{\frac{4}{5},0,-\frac{3}{5}\right\}$ show that the framed curve $\left(\gamma \left(u,0\right),{\mu }_1,{\mu }_2\right)$ does not satisfy the CCD equation. Moreover, the framed curvatures $\left\{p,q\right\}=\left\{\frac{3}{5},\frac{4}{5}\right\}$ and $\left\{g_1,g_2,g_3\right\}=\left\{\frac{3}{5},-\frac{4}{5},0\right\}$ indicate that $\left(\gamma \left(u,0\right),{\mu }_1,{\mu }_2\right)$ does not satisfy the CD equation.

Example 2. 

Let $\gamma :\left[-1,1\right]\times \left[0,1\right]\to {\mathbb{R}}^3$ denote a family of curves given by $\gamma \left(u,s\right)=\left\{s,\frac{{\mathrm{e}}^s{u}^3}{3},\frac{{\mathrm{e}}^s{u}^2}{2}\right\}$. $u=0$ is a singular point of the curve $\gamma \left(u,s_0\right)$ at each moment $s_0$. The Frenet frames of this curve are drawn at its regular points $\left(-1,0\right)$ and $\left(1,0\right)$ as shown in Figure 3.

$\left(\gamma ,{\mu }_1,{\mu }_2\right):\left[-1,1\right]\times \left[0,1\right]\to {\mathbb{R}}^3\times {\Delta }_2$ is a family of framed curves with

$$\begin{array}{c}\gamma \left(u,s\right)=\left\{s,\frac{{\mathrm{e}}^s{u}^3}{3},\frac{{\mathrm{e}}^s{u}^2}{2}\right\},\hfill \\ {\mu }_1\left(u,s\right):=\left\{-1,0,0\right\},\hfill \\ {\mu }_2\left(u,s\right):=\left\{0,-\frac{u}{\sqrt{1+u^2}},\frac{1}{\sqrt{1+u^2}}\right\}.\hfill \end{array}$$

Then, $\upsilon \left(u,s\right)={\mu }_1\left(u,s\right)\times {\mu }_2\left(u,s\right)=\left\{0,\frac{1}{\sqrt{1+u^2}},\frac{u}{\sqrt{1+u^2}}\right\}$ is the generalized tangent vector, where $\alpha \left(u,s\right)={\mathrm{e}}^{s}u\sqrt{1+u^2}$; see Figure 4.

Using direct calculations, the curvatures $l=c\rho =0$, $m=0$, $n=c{\omega }_u=-\frac{1}{\left(1+u^2\right)}$, and $f_2=\omega =0$ are found, and it is seen that they satisfy the CCD equation ${\omega }_{us}=\rho \omega$, ${\rho }_s+\frac{1}{2}{\left({\left|\omega \right|}^2\right)}_u=0$. Furthermore, the framed curvatures $p=c\rho =\frac{1}{1+u^2}$, $q=c{\omega }_u=0$, and $g_2=\omega =0$ require the CD equation ${\omega }_{us}=\rho \omega$, ${\rho }_s+\omega {\omega }_u=0$.

Let us recall the basic information on the isomorphism between the real vector space of dimension 3 of the traceless anti-Hermitian $2\times 2$ complex matrices denoted by $\mathrm{su}\left(2\right)$ and the vector space of the antisymmetric real $3\times 3$ matrices denoted by $\mathrm{so}\left(3\right)$. The Lie algebra of $\mathrm{so}\left(3\right)$ is represented by the real vector space with a basis of $\left\{D_1,D_2,D_3\right\}$ such that

$$D_1=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right),D_2=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right),D_3=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right).$$

The basis matrices $D_k$$\left(k=1,2,3\right)$ satisfy the following commutation relations:

$$\left[D_1,D_2\right]=D_3,\left[D_2,D_3\right]=D_1,\left[D_3,D_1\right]=D_2.$$

Similarly, the basis of $\mathrm{su}\left(2\right)$ is $\left\{d_1,d_2,d_3\right\}$, where

$$d_1=\frac{1}{2i}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),d_2=\frac{1}{2i}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),d_3=\frac{1}{2i}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

are generated under the commutation relations $\left[d_1,d_2\right]=d_3,\left[d_2,d_3\right]=d_1,\left[d_3,d_1\right]=d_2$. The relation between the basis matrices $\mathrm{su}\left(2\right)$ and the Pauli matrices is

$$d_k=\frac{{\sigma }_k}{2i}$$

such that

$${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

It is well known that the Lie algebra $\mathrm{su}\left(2\right)$ is isomorphic to $\mathrm{so}\left(3\right)$ under the correspondence $D_k\leftrightarrow d_k$. This allows us to construct a $2\times 2$ linear representation from the $3\times 3$ linear representation as explained in [3].

Using this correspondence, it is possible to give the Lax pairs introduced by P.D Lax in 1968 [3]. The Lax pairs satisfying the integrability conditions of the CCD and CD equations obtained for the framed curve families are stated in the theorem below.

Theorem 5. 

Let $\gamma :I_1\times I_2\to {\mathbb{R}}^3$ be a family of curves that do not need to be regular in Euclidean 3-space.

i. 

If $\left(\gamma ,{\mu }_1,{\mu }_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family satisfying the CCD equation, then the Lax pair of this equation is

$${\psi }_u=A\psi ,{\psi }_s=B\psi$$

such that

$$A=l{d}_3+m{d}_2-n{d}_1,B=-f_1{e}_3+f_2{e}_2-f_3{e}_1.$$

where the associated curvatures are $\left\{l,m,n,\alpha \right\}$ and the Frenet-type frame formulae are given by Equation (1).

ii. 

If $\left(\gamma ,{\overline{\mu }}_1,{\overline{\mu }}_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family satisfying the CD equation, then the Lax pair of this equation is

$${\psi }_u=C\psi ,{\psi }_s=D\psi$$

such that

$$C=p{d}_3-q{d}_1,D=-g_1{d}_3+g_2{d}_2-g_3{d}_1.$$

where $\left\{p,q,\alpha \right\}$ are the framed curvatures and the Frenet frame formulae are given by Equation (2).

Proof. 

Let the framed curve family be defined as follows:

i.

If $\left(\gamma ,{\mu }_1,{\mu }_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family, whose associated curvatures are $\left\{l,m,n,\alpha \right\}$ and whose Frenet-type frame formulae are given by Equation (1), then the functions A and B, which construct a Lax pair of the CCD equations for framed curve families, are found with the help of the isomorphism $D_k\leftrightarrow d_k$ and Theorem 1 as follows:

$$\begin{array}{ccc}\hfill A=l{d}_3+m{d}_2-n{d}_1& =& \frac{i}{2}\left(\begin{array}{cc}-l& n+mi\\ n-mi& l\end{array}\right)\hfill \\ & =& \frac{ci}{2}\left(\begin{array}{cc}\rho & {\omega }_u\\ {\omega }_u^{*}& -\rho \end{array}\right)\hfill \\ & =& -\lambda i\left(\begin{array}{cc}\rho & {\omega }_u\\ {\omega }_u^{*}& -\rho \end{array}\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill B=-f_1{d}_3+f_2{d}_2-f_3{d}_1& =& \frac{i}{2}\left(\begin{array}{cc}{f}_1& f_3+i{f}_2\\ f_3-i{f}_2& f_1\end{array}\right)\hfill \\ & =& \frac{i}{2}\left(\begin{array}{cc}-c^{-1}& i\omega \\ -i{\omega }^{*}& c^{-1}\end{array}\right)\hfill \\ & =& \left(\begin{array}{cc}\frac{i}{4\lambda }& -\frac{\omega }{2}\\ \frac{{\omega }^{*}}{2}& \frac{i}{4\lambda }\end{array}\right),\hfill \end{array}$$

where $c=-2\lambda$. Also, the compatibility condition $A_u-B_s+AB-BA=0$ satisfies the CCD equations.

ii.

If $\left(\gamma ,{\overline{\mu }}_1,{\overline{\mu }}_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family whose associated curvatures are $\left\{p,q,\alpha \right\}$ and whose Frenet formulae are given by Equation (2), then the functions C and D, which construct a Lax pair of the CD equations for framed curve families, are found with the help of the isomorphism $D_k\leftrightarrow d_k$ as follows:

$$\begin{array}{ccc}\hfill C=-p{d}_3-q{d}_1& =& \frac{i}{2}\left(\begin{array}{cc}p& q\\ q& -p\end{array}\right)\hfill \\ & =& \frac{ic}{2}\left(\begin{array}{cc}\rho & {\omega }_u\\ {\omega }_u& -\rho \end{array}\right)\hfill \\ & =& -\lambda i\left(\begin{array}{cc}\rho & {\omega }_u\\ {\omega }_u& -\rho \end{array}\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill D=-g_1{d}_3+g_2{d}_2& =& \frac{i}{2}\left(\begin{array}{cc}{g}_1& g_{2}i\\ -g_{2}i& -g_1\end{array}\right)\hfill \\ & =& \left(\begin{array}{cc}\frac{-i}{2c}& \frac{-\omega }{2}\\ \frac{\omega }{2}& \frac{i}{2c}\end{array}\right)\hfill \\ & =& \left(\begin{array}{cc}\frac{i}{4\lambda }& \frac{-\omega }{2}\\ \frac{\omega }{2}& \frac{-i}{4\lambda }\end{array}\right),\hfill \end{array}$$

where $c=-2\lambda$. Also, the compatibility condition $C_u-D_s+CD-DC=0$ satisfies the CD equations.

□

Let us give the geometric interpretation of the conserved quantity of the CCD and CD equations for the framed curve family by following Theorem 3.

Theorem 6. 

Let $\gamma :I_1\times I_2\to {\mathbb{R}}^3$ be a family of curves that do not need to be regular in Euclidean 3-space.

i. 

If $\left(\gamma ,{\mu }_1,{\mu }_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family whose associated curvatures are $\left\{l,m,n,\alpha \right\}$, satisfying the CCD equation, then the conserved quantity of the CCD equations is the constant

$$I={\rho }^2+{\left|{\omega }_u\right|}^2,$$

where $\rho =\frac{l}{c}$, ${\omega }_u=\frac{n}{c}$.

ii. 

If $\left(\gamma ,{\overline{\mu }}_1,{\overline{\mu }}_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family whose framed curvatures are $\left\{p,q,\alpha \right\}$, satisfying the CD equation, then the conserved quantity of the CD equations is the constant

$$I={\rho }^2+{{\omega }_u}^2,$$

where $\rho =\frac{p}{c}$, ${\omega }_u=\frac{q}{c}$.

Proof. 

Assume that the framed curve family is expressed as below.

i.

If $\left(\gamma ,{\mu }_1,{\mu }_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family, then under consideration of the equations $\left\{l,m,n,f_1,f_2,f_3\right\}=\left\{c\rho ,0,c{\omega }_u,-c^{-1},\omega ,0\right\}$, we find that

$$\begin{array}{ccc}\hfill \frac{d}{ds}\left({\rho }^2+{\left|{\omega }_u\right|}^2\right)& =& \frac{d}{ds}\left({\rho }^2+{\omega }_u{\omega }_u^{*}\right)\hfill \\ & =& \frac{d}{ds}\left(\frac{l^2}{c^2}+\frac{\left(n+im\right)\left(n-im\right)}{c^2}\right)\hfill \\ & =& \frac{1}{c^2}\frac{d}{ds}\left(l^2+m^2+n^2\right).\hfill \end{array}$$

On the other hand, by using Equations (4)–(6), we obtain

$$\begin{array}{ccc}\hfill \frac{d}{ds}\left(l^2+m^2+n^2\right)& =& 2l{l}_s+2m{m}_s+2n{n}_s\hfill \\ & =& -2cl\omega {\omega }_u+2cn\rho \omega \hfill \\ & =& 0.\hfill \end{array}$$

By comparing these last equations, it is proven that that ${\rho }^2+{\left|{\omega }_u\right|}^2$ is constant.

ii.

If $\left(\gamma ,{\overline{\mu }}_1,{\overline{\mu }}_2\right):I_1\times I_2\to {\mathbb{R}}^3\times {\Delta }_2$ is a framed curve family, then considering the equations $\left\{g_1,g_2,g_3,p,q\right\}=\left\{-c^{-1},\omega ,0,c\rho ,c{\omega }_u\right\}$ gives us

$$\begin{array}{ccc}\hfill \frac{d}{ds}\left(p^2+q^2\right)& =& \frac{d}{ds}\left({\left(c\rho \right)}^2+{\left(c{\omega }_u\right)}^2\right)\hfill \\ & =& c^2\frac{d}{ds}\left({\rho }^2+{{\omega }_u}^2\right).\hfill \end{array}$$

On the other hand, by using Equations (16) and (18), it is easily seen that

$$\begin{array}{ccc}\hfill \frac{d}{ds}\left(p^2+q^2\right)& =& 2p{p}_s+2q{q}_s\hfill \\ & =& 2p\left(-q\omega \right)+2q\left(p\omega \right)\hfill \\ & =& 0.\hfill \end{array}$$

Consequently, this is proof that ${\rho }^2+{{\omega }_u}^2$ is constant.

□

4. Conclusions

The following conclusions can be drawn from this study:

• The connections between the CD and CCD equations and the curve families, even if they have singular points, were presented.
• The new relations based on the framed curvatures and the associated curvatures of the framed curves extended the results of [3] since the framed base curves generalize the regular curves with linear independent conditions.
• The Lax pairs of the obtained CD and CCD equations based on the motions of any framed curve were given.
• The conditions’ integrability of these differential equations was based on the curvatures of any framed curve, which allows us to study integrability in a broader context.

Since the relationship between nonlinear partial differential equations and curve families with singular points has not yet been discussed from this perspective, this paper is expected to be a source for studies of different types of nonlinear partial differential equations or framed curves in non-Euclidean spaces.