1 Introduction

In [1], an intrinsic geometric decomposition procedure was developed in a kinematic analysis of certain hydrodynamic motions. It has subsequently been applied in diverse areas such as, inter alia, nonlinear elasticity, gasdynamics, magnetohydrostatics, magnetohydrodynamics and modern soliton theory. Thus, in magnetohydrodynamics it has been applied to analyse motions with the admitted geometric characteristic that the Maxwellian surfaces contain the streamlines as geodesics and the magnetic lines as parallels on a normal congruence [2]. Viable helical geometries were thereby isolated. In soliton theory, this kind of anholonomic representation has been applied, in particular, in the derivation of both an auto-Bäcklund transformation and Lax pair in a purely geometric manner for the canonical nonlinear Schrödinger (NLS) equation [3]. NLS soliton surfaces may be thereby constructed systematically in a geometric way [4]. In [5], this intrinsic geometric approach uncovered a remarkable connection between a previously well-studied class of hydrodynamic motions [6,7,8,9] and the integrable solitonic Heisenberg spin equation subject to a geometric constraint. The nature of this Heisenberg spin link has subsequently been elaborated upon in [10,11,12,13] and most recently extended to spatial relativistic gasdynamics [14]. The corresponding solitonic connection in magnetohydrostatics to the S-integrable homogeneous Heisenberg spin equation has been employed in [15] to derive exact magnetic flux configurations which incorporate, in particular, nested toroidal geometries as originally described in [16] with relevance to tokamak design. An extension to magnetohydrodynamics leading to the solitonic Pohlmeyer-Lund-Regge model subject to a volume-preserving constraint was presented in [17]. This geometric procedure was again employed in soliton theory in [18] in an analysis of the spatial binormal motion of inextensible curves of either constant curvature or torsion. Novel integrable extensions with geometric origin of both the Dym equation [19] and classical sine-Gordon equation of Bour [20] together with associated Bäcklund transformations [21] and soliton surfaces were thereby constructed.

The important connection between certain motions of inextensible curves and the generation of associated soliton equations is well-established (see, e.g., [4, 22] and literature cited therein). The study of the motion of fibre-reinforced fluids on the other hand, as described in the authoritative monograph of Spencer [23] likewise requires the analysis of viable motions of inextensible curves, in that physical context, embedded in a viscous liquid. This commonality with soliton theory involving the constrained motion of inextensible curves indicated potential integrable structure underlying certain admitted deformations in the motion of fibre-reinforced materials in formation processes. Such integrable connections have indeed been established in [24,25,26].

The basic mathematical theory of the deformation of fibre-reinforced materials is contained in [27]. However, it is remarked that certain aspects of the deformation of elastic materials reinforced by inextensible chords had been treated in the classic work by Green and Adkins [28]. In [24, 25], it was established by geometric means that kinematic constraints as set down in [23] on the steady two-dimensional motion of an ideal fibre-reinforced fluid can be consolidated in a single 3rd order nonlinear equation. Importantly, the latter admits a particular solitonic reduction compatible with the classical sine-Gordon equation. The kinematic conditions in that case admit both a Bäcklund transformation and a novel duality property. In [24], kinematically admissible spatial motions were obtained in which the fibres constitute geodesic windings on nested toroidal surfaces. The analysis of the kinematics of 2+1-dimensional motions of a fibre-reinforced fluid was subsequently undertaken in [26]. Under a constant divergence constraint, remarkably, the evolution was shown to be linked to the inverse scattering problem for the solitonic modified Korteweg-de Vries hierarchy which, in turn, may be generated in a purely geometric manner [4]. Motions were isolated in [26] wherein the fibres which are convected by the viscous fluid constitute generalized tractrices.

Two-dimensional steady magnetohydrodynamics has previously been the subject of analysis when the magnetic and velocity fields are everywhere orthogonal [29] or, more generally, inclined at a constant angle [30]. These cases were subsequently subsumed in non-aligned planar magnetohydrodynamics in [31] via a novel representation involving a quartet of Bernoulli integrals admitted by the governing nonlinear system. The present work serves to complement that of [31] and applies an intrinsic geometric formulation. It emerges here that, remarkably, a key 3rd order nonlinear equation of a geometric nature as originally derived in the fibre-reinforced viscous liquid context of [24, 25] naturally arises in the present magnetohydrodynamic analysis. In most recent work [32] a time-dependent variant of this canonical equation has been derived in the description of the 2+1-dimensional motion of fibre-reinforced fluids. Importantly, this development indicates the potential to extend the present geometric analysis to 2+1-dimensional magnetodydrodynamics and magnetogasdynamics. It is remarked that, in the latter connection, novel integrable pulsrodon [33] and Ermakov-Ray-Reid structure [34, 35] has been previously isolated in [36].

2 The magnetohydrodynamic system

Here, a geometric decomposition is undertaken of the two-dimensional reduction of the time-independent nonlinear magnetohydrodynamic system

$$\begin{aligned} \begin{aligned} {\text {div}}\varvec{q}=&\, 0\\ (\varvec{q}\cdot \nabla )\varvec{q}- {\text {curl}}\varvec{H}\times \varvec{H}+ \nabla p =&\, \varvec{0}\\ {\text {div}}\varvec{H}=&\, 0\\ {\text {curl}}(\varvec{q}\times \varvec{H}) =&\, \varvec{0}, \end{aligned} \end{aligned}$$
(1)

where \(\varvec{H}\), \(\varvec{q}\) and p denote, in turn, the magnetic field, velocity and pressure respectively. The density and magnetic permeability are assumed to be constant and have been scaled to unity. It is remarked that the system (1) has been previously shown in [37] to be analogous mutatis mutandis to that descriptive of the plane elastostatic deformation of neo-Hookean materials [38]. In the current case of a translational symmetry in the direction of a unit vector \(\varvec{b}\), the magnetic induction equation (1)\(_3\) admits a representation

$$\begin{aligned} \varvec{H}= \nabla A\times \varvec{b}, \end{aligned}$$

where A is the magnetic flux. Faraday’s law (1)\(_4\) then shows that

$$\begin{aligned} \varvec{q}\cdot \nabla A = \tilde{c} = \text{ const }. \end{aligned}$$
(2)

The general solution of the latter may be decomposed into a particular solution \(A_0\) and a function of any non-constant quantity \(\rho\) which is convected by \(\varvec{q}\), that is,

$$\begin{aligned} A = A_0 + f(\rho ),\quad \varvec{q}\cdot \nabla \rho = 0. \end{aligned}$$

It is this observation which, in part, motivates the following alignment of magnetohydrodynamic motions with those of fibre-reinforced fluids in which quantities which are convected by the fluid flow arise naturally.

3 Fibre-reinforced fluid flows

In order to make a connection with the steady planar motion of ideal fibre-reinforced fluids, we now assume that there exist notional inextensible “fibres” embedded in the fluid which are convected by \(\varvec{q}\). If \(\varvec{t}\) denotes the unit tangent to the fibre lines then the existence of these fibre lines is encoded in the commutativity of the vector fields \(\varvec{q}\cdot \nabla\) and \(\varvec{t}\cdot \nabla\), that is,

$$\begin{aligned} (\varvec{q}\cdot \nabla )\varvec{t}= (\varvec{t}\cdot \nabla )\varvec{q}. \end{aligned}$$
(3)

Accordingly, the compatibility condition which guarantees the existence of a function \(\rho\) obeying the pair of equations

$$\begin{aligned} \varvec{t}\cdot \nabla \rho = 1,\quad \varvec{q}\cdot \nabla \rho = 0 \end{aligned}$$

is satisfied. Geometrically, this means that \(\rho\) constitutes arc length along the fibres which is convected by the fluid flow. It is noted that this is consistent with the assumption that the fibres are inextensible.

In order to proceed, it is convenient to introduce the directional derivatives

$$\begin{aligned} D_s = \varvec{t}\cdot \nabla ,\quad D_n = \varvec{n}\cdot \nabla , \end{aligned}$$

where \(\varvec{n}\) denotes the principal (unit) normal to the fibres. If \(\kappa\) is the curvature of the fibres then the orthonormal pair \((\varvec{t},\varvec{n})\) obeys the Serret-Frenet equations [39]

$$\begin{aligned} D_s\begin{pmatrix} \varvec{t}\\ \varvec{n}\end{pmatrix} = \begin{pmatrix} 0 &{} \kappa \\ -\kappa &{} 0\end{pmatrix}\begin{pmatrix} \varvec{t}\\ \varvec{n}\end{pmatrix} \end{aligned}$$
(4)

and, analogously,

$$\begin{aligned} D_n\begin{pmatrix} \varvec{t}\\ \varvec{n}\end{pmatrix} = \begin{pmatrix} 0 &{} \theta \\ -\theta &{} 0\end{pmatrix}\begin{pmatrix} \varvec{t}\\ \varvec{n}\end{pmatrix} \end{aligned}$$
(5)

with \(\theta ={\text {div}}\varvec{t}\) being the negative of the curvature of the trajectories which are orthogonal to the fibres. Furthermore, \(\varvec{t}\) and \(\varvec{n}\) are represented by

$$\begin{aligned} \varvec{t}= \begin{pmatrix}\cos \varphi \\ \sin \varphi \end{pmatrix},\quad \varvec{n}= \begin{pmatrix}-\sin \varphi \\ \cos \varphi \end{pmatrix} \end{aligned}$$
(6)

and the fibres and orthogonal trajectories are parametrised by s and n respectively so that a canonical orthogonal coordinate system (x(s, n), y(s, n)) on the plane is obtained. In terms of this parametrisation, it has been shown that the kinematic equation (3) subject to the continuity equation (1)\(_1\) may be brought into the following compact form [24, 25].

Theorem 1

Given a solution \((\rho ,\psi (\rho ))\) of the partial differential equation

$$\begin{aligned} \left( \frac{\rho _{sn}}{\psi }\right) _n + \psi ''\rho _s = 0, \end{aligned}$$
(7)

a solution of the kinematic pair (1)\(_1\), (3) governing steady planar motions of fibre-reinforced fluids is given by (6) and

$$\begin{aligned} \varvec{q}= c(-\rho _n\varvec{t}+ \psi \varvec{n}), \end{aligned}$$
(8)

where c is an arbitrary constant and the angle \(\varphi\) is determined by the compatible system

$$\begin{aligned} \varphi _s = -\frac{\rho _{sn}}{\psi },\quad \varphi _n = \psi '. \end{aligned}$$
(9)

The parametrisation of the Cartesian coordinates x and y is obtained via integration of the compatible system

$$\begin{aligned} \begin{aligned} x_s&= \rho _s\cos \varphi ,\quad&x_n&= -\psi \sin \varphi \\ y_s&= \rho _s\sin \varphi ,\quad&y_n&= \psi \cos \varphi . \end{aligned} \end{aligned}$$
(10)

The one-parameter family of fibres is given by \((x,y)(s,n=\text{ const})\). Conversely, any generic motion is encoded in this manner.

Remark 1

The relations (10) show that the metric of the plane and the gradient are given by

$$\begin{aligned} dx^2 + dy^2 = \rho _s^2ds^2 + \psi ^2dn^2 \end{aligned}$$

and

$$\begin{aligned} \nabla = \varvec{t}D_s + \varvec{n}D_n = \frac{\varvec{t}}{\rho _s}\partial _s + \frac{\varvec{n}}{\psi }\partial _n \end{aligned}$$

so that combination of the Serret-Frenet-type equations (4), (5) and (9) reveals that

$$\begin{aligned} \kappa = -\frac{\rho _{sn}}{\rho _s\psi },\quad \theta = \frac{\psi '}{\psi }. \end{aligned}$$
(11)

It is also observed that

$$\begin{aligned} \varvec{q}\cdot \nabla = c\left( -\frac{\rho _n}{\rho _s}\partial _s + \partial _n\right) . \end{aligned}$$
(12)

by virtue of the representation (8) of the velocity \(\varvec{q}\).

4 The momentum equation

In order to evaluate the compatibility condition for the momentum equation (1)\(_2\) which guarantees the existence of the pressure p, we first derive an explicit expression for the magnetic field \(\varvec{H}\). Inspection of the directional derivative (12) reveals that \(A_0 = n\tilde{c}/c\) is a particular solution of (2) so that the magnetic flux is given by

$$\begin{aligned} A = f(\rho ) + \frac{\tilde{c}}{c}n. \end{aligned}$$

If \(\varvec{b}\) is chosen as the binormal to the fibres defined by \(\varvec{b}= \varvec{t}\times \varvec{n}\) then

$$\begin{aligned} \begin{aligned} \varvec{H}&= (\varvec{t}D_s + \varvec{n}D_n) \left( f(\rho ) + \frac{\tilde{c}}{c}n\right) \times \varvec{b}\\&= \left[ f'\left( \varvec{t}+ \frac{\rho _n}{\psi }\varvec{n}\right) + \frac{\tilde{c}}{c\psi }\varvec{n}\right] \times \varvec{b}\\&= g\left( \frac{\rho _n}{\psi }\varvec{t}- \varvec{n}\right) + \frac{\tilde{c}}{c\psi }\varvec{t}, \end{aligned} \end{aligned}$$

where \(g(\rho ) = f'(\rho )\). Accordingly, we arrive at the decomposition

$$\begin{aligned} \varvec{H}= \frac{1}{c\psi }(-g\varvec{q}+ \tilde{c}\varvec{t}) =: a(\rho )\varvec{q}+ \tilde{c}b(\rho )\varvec{t}\end{aligned}$$
(13)

which turns out to be convenient in the following discussion.

The momentum equation (1)\(_2\) may be formulated as

$$\begin{aligned} {\text {curl}}\varvec{q}\times \varvec{q}- {\text {curl}}\varvec{H}\times \varvec{H}+ \nabla \left( p + \frac{\varvec{q}^2}{2}\right) = \varvec{0}\end{aligned}$$

so that the pressure p is well defined if and only if the associated compatibility condition

$$\begin{aligned} {\text {curl}}({\text {curl}}\varvec{q}\times \varvec{q}- {\text {curl}}\varvec{H}\times \varvec{H}) = \varvec{0}\end{aligned}$$
(14)

holds. Since the two terms in the compatibility condition are of the form \({\text {curl}}(\varvec{A}\times \varvec{B})\) with

$$\begin{aligned} {\text {div}}\varvec{A}= 0,\quad {\text {div}}\varvec{B}= 0,\quad \varvec{A}= \bar{A}\varvec{b},\quad \varvec{b}\cdot \nabla \equiv 0 \end{aligned}$$

so that the vector identity

$$\begin{aligned} \begin{aligned} {\text {curl}}(\varvec{A}\times \varvec{B})&= \varvec{A}{\text {div}}\varvec{B}- \varvec{B}{\text {div}}\varvec{A}\\&+ (\varvec{B}\cdot \nabla )\varvec{A}- (\varvec{A}\cdot \nabla )\varvec{B}\end{aligned} \end{aligned}$$

reduces to

$$\begin{aligned} {\text {curl}}(\varvec{A}\times \varvec{B}) = (\varvec{B}\cdot \nabla \bar{A})\,\varvec{b}, \end{aligned}$$

the compatibility condition (14) assumes the compact form

$$\begin{aligned} \begin{aligned} \varvec{q}\cdot \nabla \alpha = \varvec{H}\cdot \nabla \beta \\ \alpha = \varvec{b}\cdot {\text {curl}}\varvec{q},\quad \beta = \varvec{b}\cdot {\text {curl}}\varvec{H}. \end{aligned} \end{aligned}$$

On use of the expansion (13), this may be formulated as

$$\begin{aligned} \varvec{q}\cdot \nabla \left( \frac{\alpha - a\beta }{b}\right) = \varvec{t}\cdot \nabla (\tilde{c}\beta ) \end{aligned}$$
(15)

and, hence,

$$\begin{aligned} (\rho _s\partial _n - \rho _n\partial _s)\left( \frac{\alpha - a\beta }{b}\right) = \partial _s(\tilde{c}\beta ). \end{aligned}$$

The vorticity \(\alpha\) may now be calculated. Thus, since \(\varvec{b}\) is constant, we conclude that

$$\begin{aligned} \begin{aligned} \alpha /c&= \nabla \cdot (\varvec{q}\times \varvec{b})/c\\&= (\varvec{t}D_s + \varvec{n}D_n)\cdot (\rho _n\varvec{n}+ \psi \varvec{t})\\&= -\rho _n\kappa + D_s\psi + D_n\rho _n + \psi \theta \\&= \frac{\rho _{sn}\rho _n}{\rho _s\psi } + \frac{\rho _{nn}}{\psi } + 2\psi ', \end{aligned} \end{aligned}$$

wherein the Serret-Frenet-type equations (4), (5) and the expressions (11) for the curvatures \(\kappa\) and \(\theta\) have been used. Two of the above terms may be combined to obtain

$$\begin{aligned} \alpha = c\left[ \frac{{(\rho _s\rho _n)}_n}{\rho _s\psi } + 2\psi '\right] . \end{aligned}$$

Moreover, since \(\beta\) is linear in \(\tilde{c}\), we may write it as \(\beta = \beta _0 + \tilde{c}\beta _1\) so that, on the one hand,

$$\begin{aligned} \begin{aligned} \beta _0&= \nabla \cdot (a\varvec{q}\times \varvec{b})\\&= a\alpha + \nabla a\cdot (\varvec{q}\times \varvec{b})\\&= a\alpha + ca'(\varvec{t}D_s\rho + \varvec{n}D_n\rho )\cdot (\rho _n\varvec{n}+ \psi \varvec{t})\\&= a\alpha + ca'\left( \psi + \frac{\rho _n^2}{\psi }\right) \\&= -\frac{g}{\psi }\frac{{(\rho _s\rho _n)}_n}{\rho _s\psi } - \left( \frac{g}{\psi }\right) '\left( \psi + \frac{\rho _n^2}{\psi }\right) - 2g\frac{\psi '}{\psi }\\&= -\frac{1}{\rho _s\psi }\left( \rho _s\rho _n\frac{g}{\psi }\right) _n - \frac{(g\psi )'}{\psi }. \end{aligned} \end{aligned}$$

On the other hand,

$$\begin{aligned} \begin{aligned} \beta _1&= \nabla \cdot (b\varvec{t}\times \varvec{b})\\&= - (\varvec{t}D_s + \varvec{n}D_n)\cdot (b\varvec{n})\\&= -b'D_n\rho + b\kappa \\&= - b'\frac{\rho _n}{\psi } - b\frac{\rho _{sn}}{\rho _s\psi }\\&= - \frac{{(b\rho _n)}_s}{\rho _s\psi }\\&= - \frac{1}{c\rho _s\psi }\left( \frac{\rho _n}{\psi }\right) _s. \end{aligned} \end{aligned}$$

Accordingly, the compatibility condition (15) is explicitly given by

$$\begin{aligned} \varvec{q}\cdot \nabla U = \varvec{t}\cdot \nabla V, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} U&= c^2\frac{{(\rho _s\rho _n)}_n}{\rho _s} - \frac{g}{\rho _s\psi }\left( \rho _s\rho _n\frac{g}{\psi }\right) _n - \frac{\tilde{c}g}{c\rho _s\psi }\left( \frac{\rho _n}{\psi }\right) _s\\ V&= -\frac{\tilde{c}}{\rho _s\psi }\left( \rho _s\rho _n\frac{g}{\psi }\right) _n - \frac{\tilde{c}}{\psi }(g\psi )' - \frac{\tilde{c}^2}{c\rho _s\psi }\left( \frac{\rho _n}{\psi }\right) _s. \end{aligned} \end{aligned}$$

We therefore conclude that any solution of the overdetermined system of the third-order partial differential equations (7) and

$$\begin{aligned} (\rho _s\partial _n - \rho _n\partial _s)U = \partial _sV \end{aligned}$$
(16)

gives rise to an essentially unique magnetohydrodynamic motion.

5 Classes of solutions

Even though the preceding analysis demonstrates that considering cognate magnetohydrodynamic and fibre-reinforced fluid motions is natural, it is by no means evident that the overdetermined system (7), (16) admits non-trivial solutions.

5.1 Rotationally symmetric motions

A reduction to ordinary differential equations is obtained via the ansatz

$$\begin{aligned} \rho = \rho (s+n). \end{aligned}$$
(17)

Here, it should be observed that even though it is evident that the ansatz (17) is compatible with the constraint (16), it is, a priori, not obvious that any other symmetry reduction of (7) is consistent with (16). In the case (17), the third-order equation (7) may be integrated once to obtain the nonlinear pendulum-type equation

$$\begin{aligned} \ddot{\rho } + \psi (\rho )\psi '(\rho ) = \hat{c}\psi (\rho ), \end{aligned}$$

where \(\hat{c}\) is the constant of integration. Moreover, the left-hand side of (16) vanishes identically, while the right-hand constitutes a differential equation for \(\hat{g}(s+n) = g(\rho (s+n))\) if \(\tilde{c}\ne 0\). If the velocity and the magnetic field are aligned (\(\tilde{c}=0\)) then the function g is arbitrary.

Fibre-reinforced fluid motions generated by the ansatz (17) have been discussed in detail in [24]. In particular, it is readily verified that the Cartesian coordinate vector \(\varvec{r}= (x\,\, y)^T\) obeying the system (10) is given by

$$\begin{aligned} \varvec{r}= \frac{1}{\hat{c}}(\psi \varvec{t}+ \dot{\rho }\varvec{n}). \end{aligned}$$

The vector-valued additive constant of integration which has been suppressed is irrelevant unless \(\hat{c}=0\) in which case the constant has to be chosen in such a manner that the limit \(\hat{c}\rightarrow 0\) exists. In the case \(\hat{c}\ne 0\), we conclude that \(\varvec{r}^2\) is a function of \(s+n\) and, hence, (8) and (13) reveal that the velocity and magnetic field are of the form

$$\begin{aligned} \begin{aligned} \varvec{q}&= c^*\begin{pmatrix} -y\\ x\end{pmatrix}\\ \varvec{H}&= H_0(x^2+y^2)\begin{pmatrix} -y\\ x\end{pmatrix} + H_1(x^2+y^2)\begin{pmatrix} x\\ y\end{pmatrix}. \end{aligned} \end{aligned}$$
(18)

If we set aside the genesis of the latter then direct evaluation of the original magnetohydrodynamic system (1) leads to

$$\begin{aligned} H_1 = \frac{H_{10}}{x^2+y^2} \end{aligned}$$

and

$$\begin{aligned} H_0 = \frac{H_{00}}{x^2 +y^2} + H_{01}\quad \text{ for } H_{10}\ne 0. \end{aligned}$$

In the aligned case associated with \(H_{10}=0\), the function \(H_0\) is arbitrary. This corresponds to the function g being arbitrary. In fact, one may show that the magnetohydrodynamic motions associated with the current ansatz \(\rho =\rho (s+n)\) and \(\hat{c}\ne 0\) completely represent the rotationally symmetric class (18). The latter is characterised by concentric circular streamlines and magnetic field lines of spiral type. In the case \(H_{01}=0\), one retrieves a sub-class of magnetohydrodnamic motions for which the magnetic field lines and streamlines meet at a constant angle. The entire class of this nature has been classified in [30].

5.2 Compatibility

We now present a route which may be taken to reveal to which extent the differential equations (7) and (16) share non-trivial solutions. The classification of all common solutions will be presented elsewhere. However, here, we will demonstrate that there exist common solutions which may be expressed in terms of Whittaker, Bessel and Airy functions.

We first introduce a function R according to

$$\begin{aligned} \rho _n = R(\rho ,n). \end{aligned}$$
(19)

It is observed that if \(R = R(\rho )\) then, locally, \(\rho = \rho (S(s) + n)\) which is, in turn, equivalent to the ansatz (17) since a suitable reparametrisation of the fibre lines leads to \(S=s\). If we take into account that

$$\begin{aligned} \rho _{sn} = R_\rho \rho _s \end{aligned}$$

then it is readily seen that the third-order equation (7) becomes the second-order equation

$$\begin{aligned} R_{\rho n} + \frac{1}{2}(R^2)_{\rho \rho } - \frac{\psi '}{2\psi }(R^2)_\rho + \psi \psi '' = 0. \end{aligned}$$
(20)

Secondly, if, for simplicity, we confine ourselves to the aligned case then (16) reduces to \(U = l(\rho )\) so that

$$\begin{aligned} \left( c^2 - \frac{g^2}{\psi ^2}\right) {(\rho _s\rho _n)}_n - \frac{\rho _s\rho _n}{2}\left( \frac{g^2}{\psi ^2}\right) _n = l(\rho )\rho _s. \end{aligned}$$
(21)

Once again, the parametrisation (19) reduces the order of this differential equation to obtain

$$\begin{aligned} R_n + (R^2)_\rho = h(\rho )R^2 + k(\rho ), \end{aligned}$$
(22)

where

$$\begin{aligned} h = \frac{(g^2/\psi ^2)'}{2(c^2 - g^2/\psi ^2)},\quad k = \frac{l}{c^2-g^2/\psi ^2}. \end{aligned}$$

Elimination of \(R_n\) between (20) and (22) then produces the ordinary differential equation

$$\begin{aligned} (R^2)_{\rho \rho } + \left( \frac{\psi '}{\psi }-2h\right) (R^2)_\rho - 2h'R^2 = 2k' + 2\psi \psi '' \end{aligned}$$
(23)

which is linear in \(R^2\).

The general solution of the second-order equation (23) is of the form

$$\begin{aligned} R^2 = N_1(n)W_1(\rho ) + N_2(n)W_2(\rho ) + W_0(\rho ). \end{aligned}$$
(24)

In principle, for any given triple \((\psi ,h,k)\), the functions \(W_i\) are determined by (23). Alternatively, we may regard the functions \(W_i\) as being given so that insertion of (24) into (23) leads to a system of ordinary differential equations for \((\psi ,h,k)\). Now, “taking the square” of (22) leads to a functional equation of the form

$$\begin{aligned} \sum _i\mathcal {N}_i(n)\mathcal {R}_i(\rho ) = 0, \end{aligned}$$
(25)

where the functions \(\mathcal {N}_k\) and \(\mathcal {R}_k\) depend on \(N_i\) and \(h,k,W_i\) respectively. A classification of the classes of solutions of this functional equation may be obtained by standard techniques [40], leading to a system of constraints on the functions \(N_i,W_i\) and h, k.

5.3 Separation of variables

As an illustration of the procedure outlined in the preceding, we focus on the case \(W_0=W_2=0\) so that R is a product of a function of n and a function of \(\rho\). In this case, the functional equation (25) admits only one class of solutions. Instead of deriving this class explicitly, we observe that R being separable is equivalent to the ansatz

$$\begin{aligned} \rho = F(z),\quad z = s + N(n) \end{aligned}$$
(26)

and, hence, we may directly deal with the pair of differential equations (7) and (21). Thus, insertion of (26) into (7) produces

$$\begin{aligned} F_{zz}N'' + \left( F_{zzz} - F_{zz}F_z\frac{\psi '}{\psi }\right) N'^2 + \psi \psi ''F_z = 0, \end{aligned}$$
(27)

while (21) becomes

$$\begin{aligned} \begin{aligned} \left( c^2 - \frac{g^2}{\psi ^2}\right) (F_zN''&+ 2F_{zz}N'^2)\\&- \frac{F_z^2}{2}\left( \frac{g^2}{\psi ^2}\right) 'N'^2 = l. \end{aligned} \end{aligned}$$

The latter implies that

$$\begin{aligned} N'' + c_1N'^2 = c_2, \end{aligned}$$
(28)

where \(c_1\) and \(c_2\) are arbitrary constants, so that l and g may be determined by

$$\begin{aligned} l = c_2\left( c^2 - \frac{g^2}{\psi ^2}\right) F_z \end{aligned}$$

and

$$\begin{aligned} \frac{F_z^2}{2}\left( \frac{g^2}{\psi ^2}\right) ' = \left( c^2 - \frac{g^2}{\psi ^2}\right) (2F_{zz} - c_1F_z), \end{aligned}$$

once F is known.

It remains to separate the variables in the differential equation (27). Comparison with (28) shows that

$$\begin{aligned} \begin{aligned} F_{zzz} - F_{zz}F_z\frac{\psi '}{\psi } = c_1F_{zz}\\ \psi \psi ''F_z = -c_2F_{zz}. \end{aligned} \end{aligned}$$
(29)

Since \(\psi _z = \psi 'F_z\), (29)\(_1\) may be formulated as

$$\begin{aligned} F_{zzz} = \left( c_1 + \frac{\psi _z}{\psi }\right) F_{zz} \end{aligned}$$

and, hence,

$$\begin{aligned} F_{zz} = c_3e^{c_1z}\psi . \end{aligned}$$

The second equation (29)\(_2\) then becomes

$$\begin{aligned} (\psi ')_z = -c_2c_3e^{c_1z} \end{aligned}$$

which may be integrated to obtain

$$\begin{aligned} \psi ' = c_4 - \frac{c_2c_3}{c_1}e^{c_1z} \end{aligned}$$

or, equivalently,

$$\begin{aligned} \psi _z = \left( c_4 - \frac{c_2c_3}{c_1}e^{c_1z}\right) F_z. \end{aligned}$$

It is noted that \(c_4\) may be appropriately parametrised so that the above remains valid in the limit \(c_1\rightarrow 0\). We therefore conclude that the functions F and \(\psi\) are obtained via the linear system

$$\begin{aligned} G_z = c_3e^{c_1z}\psi ,\quad \psi _z = \left( c_4 - \frac{c_2c_3}{c_1}e^{c_1z}\right) G \end{aligned}$$
(30)

with

$$\begin{aligned} F(z) = \int G(z) dz. \end{aligned}$$

5.4 Motions involving special functions

The solution of the differential equation (28) may be expressed in terms of elementary functions. The case \(c_1=c_2=0\) essentially corresponds to the ansatz (17). If \(c_1\ne 0\) then, for instance,

$$\begin{aligned} N = \frac{\ln \cosh (\sqrt{c_1c_2}\,n)}{c_1} \end{aligned}$$

is a solution of (28). In connection with the system (30), it is convenient to make the change of independent variable

$$\begin{aligned} t = \frac{1}{c_1}e^{c_1z}, \end{aligned}$$

leading to

$$\begin{aligned} G_t = c_3\psi ,\quad \psi _t = \frac{1}{c_1}\left( \frac{c_4}{t} - c_2c_3\right) G \end{aligned}$$

so that

$$\begin{aligned} G_{tt} = \left( c_5 + \frac{c_6}{t}\right) G,\quad c_5 = -\frac{c_2c_3^2}{c_1},\quad c_6 = \frac{c_3c_4}{c_1} \end{aligned}$$

which is a special case of Whittaker’s equation [41]. In particular, if \(c_2 = 0\), its solution is given in terms of Bessel functions according to

$$\begin{aligned} G = c_7\sqrt{t}\,J_1(2\sqrt{-c_6t}) + c_8\sqrt{t}\,Y_1(2\sqrt{-c_6t}). \end{aligned}$$

If \(c_4=0\) then G is a linear combination of (hyperbolic) sine and cosine functions.

In order to deal with the case \(c_1=0\), we first set \(c_4 = c_9 + c_2c_3/c_1\) so that

$$\begin{aligned} G_z = c_3\psi ,\quad \psi _z = (c_9 - c_2c_3z)G \end{aligned}$$

in the limit \(c_1\rightarrow 0\). Hence, the Airy equation

$$\begin{aligned} G_{zz} = (c_3c_9 - c_2c_3^2z)G \end{aligned}$$

is obtained. A detailed examination of the solutions of the overdetermined system (7), (21) together with the corresponding magnetohydrodynamic motions is presented elsewhere.