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On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

Dmitry K. Demskoi1 and Wolfgang K. Schief2 1 School of Computing and Mathematics, Charles Sturt University, NSW 2678, Australia 2 School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia ddemskoy@csu.edu.au w.schief@unsw.edu.au
Abstract

It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitzéica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitzéica equation by an analogue of the Miura transformation for the (modified) Korteweg-de Vries equation. Finally, the formalism developed in this paper is illustrated by focussing on the simplest “fluid sheets” of constant Gaussian curvature, namely the plane, sphere and pseudosphere.

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Keywords: fibre-reinforced fluid, geometry, integrability

1 Introduction

In a series of papers [1, 2, 3, 4], the geometry and integrable structure residing in the differential equations governing the motion of ideal fibre-reinforced fluids have been investigated in great detail. Recently, this investigation has culminated in an overarching description of the general non-steady (planar) two-dimensional case which captures and extends the results obtained previously [5]. The mathematical theory of the deformation of fibre-reinforced materials has been set down in a monograph by Spencer [6]. Hull et al. [7] adopted an ideal fibre-reinforced fluid model to describe resin matrix fibre-reinforced materials in the important formation state. This model consists of an incompressible fluid which is inextensible along “fibre” lines. The latter occupy the volume of the fluid by which they are convected. The presence of these privileged fibre orientations in the fluid imposes strong kinematic constraints on its admissible motions.

Fully three-dimensional motions of fibre-reinforced fluids have not yet been amenable to the techniques employed in the above-mentioned series of papers. In fact, Spencer [8] has observed that two-dimensional flows of an ideal fibre-reinforced fluid are privileged in that these are essentially determined by kinematic considerations. Therefore, pressure pp and tension TT in the fibre direction can always be determined such that the equations of motion are satisfied. Against this background, the results presented in this paper have had its origin in the quest to determine to what extent the methods applied in the planar two-dimensional case may be transferred to the case of fibre-reinforced “fluid sheets”, that is, motions of fibre-reinforced fluids in thin curved strata which are represented by curved surfaces [9]. It turns out that, remarkably, the results of this investigation may not only be of relevance to mathematical physicists who are interested in the mathematical description of such motions but also differential geometers and researchers whose expertise is integrable systems theory.

Here, we show that the kinematic differential equations governing the steady motion of an ideal fibre-reinforced fluid in a curved stratum may be reduced to the third-order partial differential equation

(unsψ)n+(ψ′′+Kψ)us=0\left(\frac{u_{ns}}{\psi}\right)_{n}+(\psi^{\prime\prime}+K\psi)u_{s}=0 (1)

for a function uu, wherein ψ=ψ(u)\psi=\psi(u) is an arbitrary function depending on uu only. In fact, uu represents arc length along the fibre lines on the surface M2M^{2} on which the motion takes place. Here, KK denotes the Gaussian curvature of M2M^{2}. The metric of M2M^{2} is entirely parametrised in terms of uu and ψ\psi if one chooses the fibre lines (ss-lines) and their orthogonal trajectories (nn-lines) as the coordinate lines on M2M^{2} so that the above third-order differential equation is nothing but the Gauss equation [10] associated with the metric. The geometric nature of the kinematic equations of fibre-reinforced fluids is, in fact, already encoded in their original formulation since, in purely mathematical terms, these capture the commutativity of two tangent vector fields with one of them being divergence free and the other one being normalised to unity.

It turns out that large classes of motions on surfaces M2M^{2} of constant Gaussian curvature KK are governed by a Gauss equation which is integrable in the sense of soliton theory [11, 12]. In particular, the integrable structure discovered in [1, 2] in the planar steady case may be retrieved in the current setting by considering the case K=0K=0. These integrable cases are isolated by exploiting an observation made by Adler and Shabat [13] who have noticed that integrable hyperbolic equations of third-order frequently admit compatible third-order differential equations. Even though the existence of a compatible equation does not necessarily imply integrability, we prove that, in the current context, each constraint on the arbitrary function ψ(u)\psi(u) generated by this approach renders the Gauss equation (1) integrable. In the simplest case, the Gauss equation may be integrated once to obtain the sin(h)-Gordon equation

uns=sinu(K>0),uns=sinhu(K<0).u_{ns}=\sin u\quad(K>0),\qquad u_{ns}=\sinh u\quad(K<0). (2)

It is important to point out that the classical theory of surfaces of constant Gaussian curvature K=±1K=\pm 1 identifies a sin(h)-Gordon equation of the form [12]

ωxx+ωyy+sinhω=0(K>0),ωxxωyy=sinω(K<0)\omega_{xx}+\omega_{yy}+\sinh\omega=0\quad(K>0),\qquad\omega_{xx}-\omega_{yy}=\sin\omega\quad(K<0) (3)

as the Gauss equation, where xx and yy constitute curvature coordinates [10]. Hence, as a by-product of the current investigation, it has been established that there exist natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature.

Remarkably, there exists another class of “integrable” motions of fibre-reinforced fluids on surfaces of constant negative Gaussian curvature which is related to an integrable differential equation arising in an entirely different classical geometric context. Thus, the discovery by Tzitzéica [14, 12] of a class of surfaces, the properties of which are preserved by affine transformations, is widely regarded as the beginning of the theory of affine differential geometry. Tzitzéica demonstrated that these “affine spheres” are governed by the partial differential equation

φxy=eφe2φ,\varphi_{xy}=e^{\varphi}-e^{-2\varphi}, (4)

where xx and yy constitute asymptotic coordinates [10]. We demonstrate that the Gauss equation descriptive of the afore-mentioned class of motions is related to the Tzitzéica equation (4) by an analogue of the celebrated Miura transformation [15] connecting the Korteweg-de Vries (KdV) equation and its modified analogue, the modified KdV (mKdV) equation [11]. Accordingly, the “modified Tzitzéica equation” obtained here gives rise to a canonical orthogonal coordinate system on surfaces of constant negative Gaussian curvature. In this connection, it is observed that a systematic examination of coordinate systems on such pseudospherical surfaces which are defined in terms of integrable systems may be found in [16, 17].

Due to the intrinsic nature of the Gauss equation, the determination of the associated motions on any concrete surface M2M^{2} embedded in 3\mathbb{R}^{3} requires the derivation of the Frobenius system which encapsulates the coordinate transformation between the adapted orthogonal coordinates (s,n)(s,n) and the coordinates which parametrise the given surface M2M^{2}. Here, we solve this “embedding problem” for the simplest surfaces of curvature K=0K=0 and K=±1K=\pm 1, namely, the plane, sphere and pseudosphere. It turns out that the corresponding coordinate transformations are given in terms of quadratures and linear Frobenius systems which are compatible modulo the Gauss equation.

2 The governing equations

2.1 The mathematical setup

Mathematically, the steady motion of a fibre-reinforced fluid in a curved stratum may be described in the following manner [6, 9]. The shape of the fluid stratum is encapsulated in a surface M2M^{2} embedded in Euclidean space 3\mathbb{R}^{3} on which the motion takes place. Let

g=gijdxidxjg=g_{ij}dx^{i}dx^{j} (5)

be the metric tensor [18] on M2M^{2} in terms of local coordinates x1x^{1} and x2x^{2}. Here, the usual rule of summation over repeated indices is adopted. The motion is governed by a system of differential equations for two vector fields 𝒒q and 𝒕t on M2M^{2}. In tensor notation, the vector field qiq^{i} represents the velocity of the fluid, while tit^{i} is a unit vector field, that is,

𝒕2=gijtitj=1,\|{\mbox{\boldmath$t$}}\|^{2}=g_{ij}t^{i}t^{j}=1, (6)

the integral curves of which represent the fibres embedded in the fluid. The fluid velocity obeys the continuity equation which, in the current context, reduces to the vanishing divergence condition

div𝒒=iqi=0\mbox{div}\,{\mbox{\boldmath$q$}}=\nabla_{i}q^{i}=0 (7)

with i\nabla_{i} denoting the covariant derivative with respect to xix^{i}. Finally, the requirement that the fibres be convected with the fluid translates into the commutativity condition

qiitjtiiqj=0.q^{i}\nabla_{i}t^{j}-t^{i}\nabla_{i}q^{j}=0. (8)

In summary, the governing equations for the steady motion of a fibre-reinforced fluid are given by the differential equations (7) and (8) subject to the algebraic constraint (6).

It turns out that the fibre divergence

θ=div𝒕=iti\theta=\mbox{div}\,\mbox{\boldmath$t$}=\nabla_{i}t^{i} (9)

constitutes an important quantity in the subsequent discussion. Indeed, if we calculate the divergence of (8) then we obtain

0=j(qiitjtiiqj)=(jqi)itj+qijitj(jti)iqjtijiqj.\eqalign{0=\nabla_{j}(q^{i}\nabla_{i}t^{j}-t^{i}\nabla_{i}q^{j})\\ \phantom{0}=\left(\nabla_{j}q^{i}\right)\nabla_{i}t^{j}+q^{i}\nabla_{j}\nabla_{i}t^{j}-\left(\nabla_{j}t^{i}\right)\nabla_{i}q^{j}-t^{i}\nabla_{j}\nabla_{i}q^{j}.} (10)

Application of the standard identities [18]

jitj=ijtjRkitkjiqj=Rkiqk,\eqalign{\nabla_{j}\nabla_{i}t^{j}&=\nabla_{i}\nabla_{j}t^{j}-R_{ki}t^{k}\\ \nabla_{j}\nabla_{i}q^{j}&=-R_{ki}q^{k},} (11)

where Rik=RkiR_{ik}=R_{ki} is the symmetric Ricci tensor, leads to

0=qiijtj=qiiθ0=q^{i}\nabla_{i}\nabla_{j}t^{j}=q^{i}\nabla_{i}\theta (12)

so that the fibre divergence is convected with the fluid.

2.2 An adapted orthogonal coordinate system

We now choose a coordinate system which is adapted to the geometry of the fibres. Thus, if x1=sx^{1}=s and x2=nx^{2}=n parametrise the fibre lines and their orthogonal trajectories respectively then the metric on M2M^{2} adopts the diagonal form

g=α2ds2+β2dn2g=\alpha^{2}ds^{2}+\beta^{2}dn^{2} (13)

and the components of the unit vector 𝒕t are given by

t1=1α,t2=0.t^{1}=\frac{1}{\alpha},\quad t^{2}=0. (14)

On use of the general expression [18]

div𝒘=1|g|i(|g|wi)\mbox{div}\,\mbox{\boldmath$w$}=\frac{1}{\sqrt{|g|}}\partial_{i}(\sqrt{|g|}w^{i}) (15)

for the divergence of a vector field 𝒘w, where |g|=α2β2|g|=\alpha^{2}\beta^{2} is the determinant of the metric tensor gg, the divergence condition (7) takes the form

(αβq1)s+(αβq2)n=0{(\alpha\beta q^{1})}_{s}+{(\alpha\beta q^{2})}_{n}=0 (16)

with subscripts denoting partial derivatives. We also observe that (15) applied to the vector field 𝒕t leads to

θ=1αslnβ.\theta=\frac{1}{\alpha}\partial_{s}\ln\beta. (17)

The commutativity condition (8) guarantees the existence of a function uu defined by the pair of equations

tiiu=1,qiiu=0.t^{i}\nabla_{i}u=1,\quad q^{i}\nabla_{i}u=0. (18)

Thus, uu represents arc length along the fibres which is convected with the fluid. Since uu cannot be constant, any other function which is convected with the fluid and uu must be functionally dependent so that (12) implies that

θ=θ(u).\theta=\theta(u). (19)

Accordingly, the arc length condition (18)1 becomes

α=us\alpha=u_{s} (20)

and (17) suggests introducing the parametrisation

θ=ψ(u)ψ(u)\theta=\frac{\psi^{\prime}(u)}{\psi(u)} (21)

so that integration results in

β=N(n)ψ(u),\beta=N(n)\psi(u), (22)

wherein the function of integration N(n)N(n) may be scaled to unity if one re-parametrises the nn-lines appropriately. Here, the prime denotes differentiation with respect to uu. Thus, the metric on M2M^{2} adopts the form

g=us2ds2+ψ2dn2.g=u_{s}^{2}ds^{2}+\psi^{2}dn^{2}. (23)

2.3 The commutativity condition

In order to evaluate the remaining constraint on the vector fields 𝒒q and 𝒕t, that is, the commutativity condition (8), it is recalled that the covariant derivative of a vector field is given by [18]

iwj=iwj+Γjikwk,\nabla_{i}w^{j}=\partial_{i}w^{j}+{\Gamma^{j}}_{ik}w^{k}, (24)

wherein the Christoffel symbols are defined by

Γjik=12gjm(kgmi+igmkmgik),{\Gamma^{j}}_{ik}=\frac{1}{2}g^{jm}(\partial_{k}g_{mi}+\partial_{i}g_{mk}-\partial_{m}g_{ik}), (25)

with gjmg^{jm} being the inverse of the metric tensor. Since the only non-vanishing components of the metric tensor and its inverse are given by

g11=us2,g22=ψ2,g11=us2,g22=ψ2,g_{11}=u_{s}^{2},\quad g_{22}=\psi^{2},\quad g^{11}=u_{s}^{-2},\quad g^{22}=\psi^{-2}, (26)

the Christoffel symbols are calculated to be

Γ111=ussus,Γ112=usnus,Γ122=ψψusΓ211=ususnψ2,Γ212=ψusψ,Γ222=ψunψ.\eqalign{{\Gamma^{1}}_{11}=\frac{u_{ss}}{u_{s}},\quad{\Gamma^{1}}_{12}=\frac{u_{sn}}{u_{s}},\quad{\Gamma^{1}}_{22}=-\frac{\psi\psi^{\prime}}{u_{s}}\\ {\Gamma^{2}}_{11}=-\frac{u_{s}u_{sn}}{\psi^{2}},\quad{\Gamma^{2}}_{12}=\frac{\psi^{\prime}u_{s}}{\psi},\quad{\Gamma^{2}}_{22}=\frac{\psi^{\prime}u_{n}}{\psi}.} (27)

Hence, we obtain

1t1=0,2t1=0,1t2=Γ211t1,2t2=Γ212t1.\nabla_{1}t^{1}=0,\quad\nabla_{2}t^{1}=0,\quad\nabla_{1}t^{2}={\Gamma^{2}}_{11}t^{1},\quad\nabla_{2}t^{2}={\Gamma^{2}}_{12}t^{1}. (28)

Now, the commutativity condition (8)j=2 reduces to

qs2=0,q^{2}_{s}=0, (29)

while (8)j=1 may be formulated as

(usq1)s+usnq2=0.{(u_{s}q^{1})}_{s}+u_{sn}q^{2}=0. (30)

The latter is a differential consequence of the convection condition (18)1 given by

usq1+unq2=0.u_{s}q^{1}+u_{n}q^{2}=0. (31)

Finally, elimination of q1q^{1} between (31) and the divergence condition (16), which reads

(usψq1)s+(usψq2)n=0,{(u_{s}\psi q^{1})}_{s}+{(u_{s}\psi q^{2})}_{n}=0, (32)

results in

qn2=0.q^{2}_{n}=0. (33)

We therefore conclude that

q1=cunus,q2=c=const.q^{1}=-c\frac{u_{n}}{u_{s}},\quad q^{2}=c=\mbox{const}. (34)

It is observed that the constant cc reflects the fact that the original system of governing equations is linear and homogenous in the fluid velocity 𝒒q so that it merely corresponds to a scaling of the fluid flow.

2.4 Motions on surfaces of constant Gaussian curvature

Motions on surfaces of constant Gaussian curvature KK naturally generalise planar motions for which K=0K=0. The geometry and integrability of the latter have been discussed in detail in [2, 3]. In general, the Gaussian curvature is related to the metric coefficients by Gauss’ Theorema Egregium which is given by [10]

(αnβ)n+(βsα)s+Kαβ=0\left(\frac{\alpha_{n}}{\beta}\right)_{n}+\left(\frac{\beta_{s}}{\alpha}\right)_{s}+K\alpha\beta=0 (35)

in the case of a metric of the form (13). The preceding analysis with α=us\alpha=u_{s} and β=ψ(u)\beta=\psi(u) therefore gives rise to the main result of this section.

Theorem 2.1.

The ss-lines of an orthogonal coordinate system (x1,x2)=(s,n)(x^{1},x^{2})=(s,n) on a surface M2M^{2} of constant Gaussian curvature KK may be regarded as the fibre lines in a steady motion of a fibre-reinforced fluid if and only if the metric on M2M^{2} is of the form

g=us2ds2+ψ2dn2,g=u_{s}^{2}ds^{2}+\psi^{2}dn^{2}, (36)

where ψ(u)\psi(u) is an arbitrary function of uu only, and uu obeys the nonlinear third-order differential equation

(unsψ)n+(ψ′′+Kψ)us=0.\left(\frac{u_{ns}}{\psi}\right)_{n}+(\psi^{\prime\prime}+K\psi)u_{s}=0. (37)

The velocity of the fluid and the unit tangent to the fibres are given respectively by

𝒒=c(unus,1),𝒕=(1us,0),\mbox{\boldmath$q$}=c\left(-\frac{u_{n}}{u_{s}},1\right),\quad\mbox{\boldmath$t$}=\left(\frac{1}{u_{s}},0\right), (38)

where cc is an arbitrary constant. Any generic motion of a fibre-reinforced fluid on a surface of constant Gaussian curvature is captured in this manner.

3 Isolation of integrable cases via compatible constraints

In general, finding non-trivial classes of solutions of the nonlinear governing equation (37) constitutes a highly non-trivial task. However, it turns out that the case of constant Gaussian curvature as considered in the above theorem is privileged. A first confirmation of this assertion is the observation that if we demand that the function ψ\psi be a solution of the second-order equation ψ′′+Kψ=0\psi^{\prime\prime}+K\psi=0 then the third-order equation may be integrated to obtain essentially

usn=G(u),G(u){0,u,expu,sinu,sinhu,coshu},u_{sn}=G(u),\quad G(u)\in\{0,u,\exp u,\sin u,\sinh u,\cosh u\}, (39)

depending on the sign of KK. Accordingly, the linear wave and Klein-Gordon equations, the explicitly solvable Liouville equation and the integrable sine-, sinh-, and cosh-Gordon equations [11, 12] govern particular classes of motions on surfaces of constant Gaussian curvature. This observation is the motivation for the classification scheme presented in this section. Thus, the governing equation (37) is a particular case of a third-order hyperbolic equation and, even though the class of third-order hyperbolic equations is not well studied (in comparison to second-order equations), it is known that it contains many integrable examples (see, e.g., [20]-[23]). In fact, Adler and Shabat [13] pointed out that an integrable hyperbolic equation of third order often admits a compatible equation of the form unss=F^u_{nss}=\hat{F} with F^\hat{F} depending on lower-order derivatives of uu. Hence, even though, in general, compatibility does not imply integrability, we demonstrate below that, in the current setting, this observation is very practical in that it results in all specialisations of (37) found in this manner being integrable.

It is easily verified that if we assume that (37) admits a compatible equation of the form unss=F^u_{nss}=\hat{F}, where F^\hat{F} may depend on all derivatives of uu up to second order, then the associated compatibility condition immediately implies that F^\hat{F} cannot depend on unu_{n} and unnu_{nn}. Thus, for convenience, we may bring the compatible equation to be considered into the form

(unsψ)s=F(u,us,uss,uns).\left(\frac{u_{ns}}{\psi}\right)_{s}=F(u,u_{s},u_{ss},u_{ns}). (40)

The compatibility condition

(unsψ)ns=(unsψ)sn\left(\frac{u_{ns}}{\psi}\right)_{ns}=\left(\frac{u_{ns}}{\psi}\right)_{sn} (41)

of (37) and (40) produces the overdetermined system

Fusuns=Fuss(unsusψψ+ψF)+(Funsusussψ)(φ+Kψ2+ψ2)us2ψ2[ψ3+(φψ)+Kψ2ψ],Fu=unsψψFuns\eqalign{\frac{\partial F}{\partial u_{s}}u_{ns}=-\frac{\partial F}{\partial u_{ss}}\left(u_{ns}u_{s}\frac{\psi^{\prime}}{\psi}+\psi F\right)\\ \phantom{\frac{\partial F}{\partial u_{s}}u_{ns}}+\left(\frac{\partial F}{\partial u_{ns}}u_{s}-\frac{u_{ss}}{\psi}\right)(\varphi+K\psi^{2}+\psi^{\prime 2})\\ \phantom{\frac{\partial F}{\partial u_{s}}u_{ns}}-\frac{u_{s}^{2}}{\psi^{2}}[\psi^{\prime 3}+(\varphi\psi)^{\prime}+K\psi^{2}\psi^{\prime}],\\ \frac{\partial F}{\partial u}=-u_{ns}\frac{\psi^{\prime}}{\psi}\frac{\partial F}{\partial u_{ns}}} (42)

for the function FF, where

φ=ψ′′ψψ2.\varphi=\psi^{\prime\prime}\psi-\psi^{\prime 2}. (43)

Here, we exploit the fact that FF does not depend on unu_{n}, which allows us to split (41) into the pair of conditions (42)1(\ref{twocomp})_{1} and (42)2(\ref{twocomp})_{2}. The second equation is easy to integrate, but we postpone this until we go through the compatibility conditions which will enable us to determine the function ψ\psi. Thus, the compatibility condition may be written as

0=2Fusu2Fuus=usuns[φunsψ2FussφFuns+ussφusψ+usψ3(ωψψφ+Kφψ2)],\eqalign{0&=\frac{\partial^{2}F}{\partial u_{s}\partial u}-\frac{\partial^{2}F}{\partial u\partial u_{s}}\cr&=\frac{u_{s}}{u_{ns}}\left[\frac{\varphi u_{ns}}{\psi^{2}}\frac{\partial F}{\partial u_{ss}}-\varphi^{\prime}\frac{\partial F}{\partial u_{ns}}+\frac{u_{ss}\varphi^{\prime}}{u_{s}\psi}+\frac{u_{s}}{\psi^{3}}\big{(}\omega-\psi\psi^{\prime}\varphi^{\prime}+K\varphi\psi^{2}\big{)}\right],} (44)

where

ω=ψ2φ′′+φψ2+φ2.\omega=\psi^{2}\varphi^{\prime\prime}+\varphi\psi^{\prime 2}+\varphi^{2}. (45)

It is then natural to examine the following three cases.

(1) The compatibility condition (44) is evidently satisfied if φ=0,\varphi=0, in which case the function ψ\psi is determined by

ψ=ϰψ.\psi^{\prime}=\varkappa\psi. (46)

(2) If we assume that φ=0\varphi^{\prime}=0, but φ0\varphi\neq 0, then (43) is equivalent to

ψ′′=kψ.\psi^{\prime\prime}=k\psi. (47)

In this case, we can solve (44) for F/uss\partial F/\partial u_{ss} to obtain

Fuss=usψuns(ψ2+Kψ2+φ).\frac{\partial F}{\partial u_{ss}}=-\frac{u_{s}}{\psi u_{ns}}(\psi^{\prime 2}+K\psi^{2}+\varphi). (48)

One may now verify that the system (42), (48) is compatible without any further constraints.

(3) In the case φ0\varphi^{\prime}\neq 0, the compatibility condition (44) may be solved for F/uns\partial F/\partial u_{ns} so that it is required to examine two additional compatibility conditions. One of these conditions is given by

0=2Funsu2Fuuns=ϱunsφ2ψ2Fuss+usψ3φφ2[ϱωψ2(ϱφKφϱ)],\eqalign{0&=\frac{\partial^{2}F}{\partial u_{ns}\partial u}-\frac{\partial^{2}F}{\partial u\partial u_{ns}}\cr&=\frac{\varrho u_{ns}}{\varphi^{\prime 2}\psi^{2}}\frac{\partial F}{\partial u_{ss}}+\frac{u_{s}}{\psi^{3}\varphi\varphi^{\prime 2}}\left[\varrho\omega-\psi^{2}(\varrho^{\prime}\varphi^{\prime}-K\varphi\varrho)\right],} (49)

where

ϱ=φ′′φφ2.\varrho=\varphi^{\prime\prime}\varphi-\varphi^{\prime 2}. (50)

If we assume that ϱ0\varrho\neq 0 then (49) determines F/uss\partial F/\partial u_{ss} which is required to be compatible with the expression for F/uns\partial F/\partial u_{ns}. It turns out that

2Funsuss2Fussuns=usuns2ψφ(ω+Kψ2φψ2ϱφϱ)1usψ\frac{\partial^{2}F}{\partial u_{ns}\partial u_{ss}}-\frac{\partial^{2}F}{\partial u_{ss}\partial u_{ns}}=\frac{u_{s}}{u_{ns}^{2}\psi\varphi}\left(\omega+K\psi^{2}\varphi-\frac{\psi^{2}\varrho^{\prime}\varphi^{\prime}}{\varrho}\right)-\frac{1}{u_{s}\psi} (51)

which cannot vanish for any ψ\psi. Hence, ϱ=0\varrho=0 so that

φ′′=φ2φ\varphi^{\prime\prime}=\frac{\varphi^{\prime 2}}{\varphi} (52)

and (49) is identically satisfied.

The second compatibility condition is given by

0=2Fusuns2Funsus=μunsFuss+Fusunsussψus2+us2φφψ3uns2σ+σ+φφψ,\eqalign{0=\displaystyle\frac{\partial^{2}F}{\partial u_{s}\partial u_{ns}}-\frac{\partial^{2}F}{\partial u_{ns}\partial u_{s}}\\ \phantom{0}=\frac{\mu}{u_{ns}}\frac{\partial F}{\partial u_{ss}}+\frac{F}{u_{s}u_{ns}}-\frac{u_{ss}}{\psi u_{s}^{2}}+\frac{u_{s}^{2}}{\varphi\varphi^{\prime}\psi^{3}u_{ns}^{2}}\sigma_{+}\sigma_{-}+\frac{\varphi^{\prime}}{\varphi\psi},} (53)

where

μ=ussusψusψ+us(ω+Kφψ2)ψ2φψunsF,σ±=ψψφφψ2φ2Kφψ2±Kφψ2.\eqalign{\displaystyle\mu=\frac{u_{ss}}{u_{s}}-\frac{\psi^{\prime}u_{s}}{\psi}+\frac{u_{s}(\omega+K\varphi\psi^{2})}{\psi^{2}\varphi^{\prime}}-\frac{\psi}{u_{ns}}F,\\ \displaystyle\sigma_{\pm}=\psi\psi^{\prime}\varphi^{\prime}-\varphi\psi^{\prime 2}-\varphi^{2}-K\varphi\psi^{2}\pm\sqrt{-K}\varphi^{\prime}\psi^{2}.} (54)

It is noted that, a priori, KK could be positive so that σ+\sigma_{+} and σ\sigma_{-} are complex conjugates. However, the analysis presented below demonstrates that the reality of the function ψ\psi excludes this possibility. Now, once again, the multiplier μ\mu in (53) requires the distinction between μ=0\mu=0 and μ0\mu\neq 0. Thus, if μ=0\mu=0 then one can verify that the expression for FF provided by (54)1(\ref{muF})_{1} does not satisfy (53). Therefore, we now assume that μ0\mu\neq 0 and may solve (53) for F/uss\partial F/\partial u_{ss}. The latter is required to be compatible with F/uns\partial F/\partial u_{ns} obtained from (44), leading to

0=2Funsuss2Fussuns=1ψ5φ2μ2[(12φ(σ++σ)2ψ2φ2)(uns2us+2ψ2φμφ)usσ+σ].\eqalign{0=\displaystyle\frac{\partial^{2}F}{\partial u_{ns}\partial u_{ss}}-\frac{\partial^{2}F}{\partial u_{ss}\partial u_{ns}}\\ \phantom{0}=\frac{1}{\psi^{5}\varphi^{\prime 2}\mu^{2}}\left[\left(\frac{1}{2}\varphi(\sigma_{+}+\sigma_{-})-2\psi^{2}\varphi^{\prime 2}\right)\left(\frac{u_{ns}^{2}}{u_{s}}+2\frac{\psi^{2}\varphi^{\prime}\mu}{\varphi}\right)-u_{s}\sigma_{+}\sigma_{-}\right].} (55)

Since the right-hand side of the above compatibility condition contains the function FF via the expression (54)1(\ref{muF})_{1} for μ\mu, there exist two possibilities. We first consider the case of not being able to solve (55) for FF so that

σ+σ=0,φ(σ++σ)4ψ2φ2=0.\sigma_{+}\sigma_{-}=0,\quad\varphi(\sigma_{+}+\sigma_{-})-4\psi^{2}\varphi^{\prime 2}=0. (56)

If we integrate (52) to obtain

φ=λφ,\varphi^{\prime}=\lambda\varphi, (57)

where λ\lambda is a constant of integration, then we can re-write (54)2 as

σ±=φψ(λψψ′′Kψ±Kλψ)\sigma_{\pm}=\varphi\psi(\lambda\psi^{\prime}-\psi^{\prime\prime}-K\psi\pm\sqrt{-K}\lambda\psi) (58)

and, hence, condition (56)2(\ref{conds})_{2} becomes

λψψ′′Kψ2λ2ψ=0.\lambda\psi^{\prime}-\psi^{\prime\prime}-K\psi-2\lambda^{2}\psi=0. (59)

Comparison with (58) shows that, in order to satisfy (59) and (56)1(\ref{conds})_{1}, we must set

λ=K2\lambda=\mp\frac{\sqrt{-K}}{2} (60)

so that (59) becomes

ψ′′=K2(ψKψ)\psi^{\prime\prime}=\mp\frac{\sqrt{-K}}{2}(\psi^{\prime}\mp\sqrt{-K}\psi) (61)

and it has been verified that K0K\leq 0 since ψ\psi is required to be real. We therefore conclude that the function ψ\psi is constrained by the differential equation

ψ′′=κ2(ψκψ),\psi^{\prime\prime}=-\frac{\kappa}{2}(\psi^{\prime}-\kappa\psi), (62)

where

κ=±K.\kappa=\pm\sqrt{-K}. (63)

In the other case, that is, when (56) is not satisfied, we may solve (55) for FF to obtain

F=φ2φψ3uns3us+unsussψus+unsus(ω+Kφψ2ψ3φφσ+σφψ3[φ(σ++σ)4ψ2φ2]ψψ2).\eqalign{F=\frac{\varphi}{2\varphi^{\prime}\psi^{3}}\frac{u_{ns}^{3}}{u_{s}}+\frac{u_{ns}u_{ss}}{\psi u_{s}}\\ \phantom{F}+u_{ns}u_{s}\left(\frac{\omega+K\varphi\psi^{2}}{\psi^{3}\varphi^{\prime}}-\frac{\varphi\sigma_{+}\sigma_{-}}{\varphi^{\prime}\psi^{3}[\varphi(\sigma_{+}+\sigma_{-})-4\psi^{2}\varphi^{\prime 2}]}-\frac{\psi^{\prime}}{\psi^{2}}\right).} (64)

However, one may verify that this expression makes the system of equations for FF inconsistent. For instance, if (64) is substituted into (42)1(\ref{twocomp})_{1} then the terms containing various powers of unsu_{ns} may not be cancelled regardless of the choice of ψ\psi. Accordingly, the above analysis may be summarised as follows.

Theorem 3.1.

The Gauss equation (37) admits a compatible equation of the form (40) if the function ψ\psi satisfies one of the following constraints.

(1)ψ=ϰψψ=aeϰu(2)ψ′′=kψψ=aeku+beku(3)ψ′′=κ2(ψκψ)ψ=aeκu/2+beκu\eqalign{(1)\quad\psi^{\prime}=\varkappa\psi\quad\Rightarrow\quad\psi=ae^{\varkappa u}\\ (2)\quad\psi^{\prime\prime}=k\psi\quad\Rightarrow\quad\psi=ae^{\sqrt{k}u}+be^{-\sqrt{k}u}\\ (3)\quad\psi^{\prime\prime}=-\frac{\kappa}{2}(\psi^{\prime}-\kappa\psi)\quad\Rightarrow\quad\psi=ae^{\kappa u/2}+be^{-\kappa u}} (65)

Here, the constants a,ba,b and ϰ,k,κ\varkappa,k,\kappa are arbitrary. In Case (2), aa and bb are complex conjugates if k<0k<0 and, in Case (3), K=κ2<0K=-\kappa^{2}<0.

4 Integrability and symmetries

We now address the significance and, in fact, the applicability of the analysis carried out in the previous section.

4.1 Sine-Gordon and modified KdV connections

We begin by assuming that, as in Case (2),

ψ′′=kψ,\psi^{\prime\prime}=k\psi, (66)

which, in fact, includes Case (1) with k=ϰ2k=\varkappa^{2}. The Gauss equation (37) may then be formulated as the coupled system of equations

uns=vψ,vn+(K+k)ψus=0,u_{ns}=v\psi,\quad v_{n}+(K+k)\psi u_{s}=0, (67)

leading to the first integral

(K+k)us2+v2=f(s).(K+k)u_{s}^{2}+v^{2}=f(s). (68)

If k=Kk=-K then one reproduces the initial observation made at the beginning of Section 3, leading to the linearisable/solvable/integrable cases (39). If K+k0K+k\neq 0 then one has to distinguish between two cases. Here, we focus on the case K+k>0K+k>0. The case K+k<0K+k<0 may be dealt with in an analogous manner. Thus, the parametrisation

K+kus=S(s)sinρ,v=S(s)cosρ,f(s)=[S(s)]2\sqrt{K+k}\,u_{s}=S(s)\sin\rho,\quad v=S(s)\cos\rho,\quad f(s)=[S(s)]^{2} (69)

of the first integral results in the coupled system

us=S(s)K+ksinρ,ρn=K+kψ.u_{s}=\frac{S(s)}{\sqrt{K+k}}\sin\rho,\quad\rho_{n}=\sqrt{K+k}\,\psi. (70)

It is noted that an appropriate reparametrisation of the ss-lines and scaling of ψ\psi may be used to remove the function SS and the constant K+k\sqrt{K+k} so that, without loss of generality, we proceed with S=1S=1 and K+k=1K+k=1 to obtain the canonical form

us=sinρ,ρn=ψu_{s}=\sin\rho,\quad\rho_{n}=\psi (71)

which now naturally leads to the following cases.

(A) If k=0k=0 then either ψ=const\psi=\mbox{const}, in which case (71) may be trivially integrated, or ψ=u\psi=u without loss of generality so that (71) becomes the sine-Gordon equation

ρsn=sinρ.\rho_{sn}=\sin\rho. (72)

The latter admits large classes of explicit solutions which may be obtained by means of the powerful techniques of integrable systems theory [12].

(B) If ψ=ϰψ\psi^{\prime}=\varkappa\psi for non-vanishing ϰ\varkappa then we may set ϰ=1\varkappa=1 and ψ=eu\psi=e^{u} for illustrative purposes. In this case, (71) becomes

ρsn=ρnsinρ,\rho_{sn}=\rho_{n}\sin\rho, (73)

which may be integrated to obtain

ρs+cosρ=U(s).\rho_{s}+\cos\rho=U(s). (74)

The substitution

ρ=2arctanχπ2\rho=2\arctan\chi-\frac{\pi}{2} (75)

then leads to the Riccati equation

χs=U2(1+χ2)χ\chi_{s}=\frac{U}{2}(1+\chi^{2})-\chi (76)

which is well-known to be linearisable. Indeed, the linearising transformation is given by

χ=y2y1,\chi=\frac{y_{2}}{y_{1}}, (77)

where y1y_{1} and y2y_{2} obey the linear system

(y1y2)s=12(1UU1)(y1y2).\left(\begin{array}[]{c}y_{1}\\ y_{2}\end{array}\right)_{s}=\frac{1}{2}\left(\begin{array}[]{cc}1&-U\\ U&-1\end{array}\right)\left(\begin{array}[]{c}y_{1}\\ y_{2}\end{array}\right). (78)

The latter constitutes the scattering problem [11] for the modified Korteweg-de Vries (mKdV) equation, wherein the spectral parameter is 11.

(C) If ψ′′=kψ\psi^{\prime\prime}=k\psi for non-vanishing kk and ψ≁ψ\psi^{\prime}\not\sim\psi, we may choose k=1k=-1 and ψ=sinu\psi=\sin u for simplicity. Other choices of kk and ψ\psi may be treated in a similar manner. In the current situation, system (71) reads

us=sinρ,ρn=sinu.u_{s}=\sin\rho,\quad\rho_{n}=\sin u. (79)

The latter constitutes a well-known pair of integrable equations, namely the classical Bäcklund equations for the sine-Gordon equation [12] (with the Bäcklund parameter being 11). In fact, it is easy to see that this system implies that

(u±ρ)sn=sin(u±ρ),{(u\pm\rho)}_{sn}=\sin(u\pm\rho), (80)

which indeed shows that the system (79) provides a link between the two solutions u±ρu\pm\rho of the sine-Gordon equation (80).

The integrability of Cases (A) and (C) may also be revealed in an alternative manner. Thus, if we assume that uu depends on an additional variable τ\tau then one may directly verify that the evolution equation

uτ=unnn+3(K+k)2ψ2unk2un3u_{\tau}=u_{nnn}+\frac{3(K+k)}{2}\psi^{2}u_{n}-\frac{k}{2}u_{n}^{3} (81)

constitutes a higher symmetry of the Gauss equation (37). The latter is indeed compatible with the above third-order equation. In Case (A), (81) becomes the mKdV equation

uτ=unnn+32u2unu_{\tau}=u_{nnn}+\frac{3}{2}u^{2}u_{n} (82)

so that the compatibility condition usτ=uτsu_{s\tau}=u_{\tau s} related to (71)1 produces the potential mKdV (pmKdV) equation

ρτ=ρnnn+12ρn3,\rho_{\tau}=\rho_{nnn}+\frac{1}{2}\rho_{n}^{3}, (83)

which is known to be compatible with the sine-Gordon equation (72) [12].

In Case (C), we obtain the modified mKdV (m2KdV) equation [24, 25]

uτ=unnn+32unsin2u+12un3u_{\tau}=u_{nnn}+\frac{3}{2}u_{n}\sin^{2}u+\frac{1}{2}u_{n}^{3} (84)

which generates the τ\tau-evolution

ρτ=unncosu+12un2sin2u+12un3\rho_{\tau}=u_{nn}\cos u+\frac{1}{2}u_{n}^{2}\sin^{2}u+\frac{1}{2}u_{n}^{3} (85)

via compatibility. By construction, the latter is compatible with (79)2. Moreover, the evolution equations (84) and (85) imply that

(u±ρ)τ=(u±ρ)nnn+12(u±ρ)n3{(u\pm\rho)}_{\tau}={(u\pm\rho)}_{nnn}+\frac{1}{2}{(u\pm\rho)}_{n}^{3} (86)

if one takes into account (79)2. This is consistent with the compatibility of the pmKdV equations (86) with the sine-Gordon equations (80).

4.2 Tzitzéica connections

Case (3) is characterised by the constraint

ψ′′=κ2(ψκψ),κ=±K,K<0.\psi^{\prime\prime}=-\frac{\kappa}{2}(\psi^{\prime}-\kappa\psi),\quad\kappa=\pm\sqrt{-K},\quad K<0. (87)

Standard (computer algebra) algorithms (see, e.g., [13]) may be used to verify the generalised symmetries

uξ=us2unssusunsussψ+23κusuns3ψ312(κψ+2ψ)unsus3ψ2u_{\xi}=\frac{u_{s}^{2}u_{nss}-u_{s}u_{ns}u_{ss}}{\psi}+\frac{2}{3\kappa}\frac{u_{s}u_{ns}^{3}}{\psi^{3}}-\frac{1}{2}\frac{(\kappa\psi+2\psi^{\prime})u_{ns}u_{s}^{3}}{\psi^{2}} (88)

and

uτ=unnnnn52κunnn(κ2un2unn+ψ(κψ+ψ))52κununn(κ2unn+(κψ+ψ)2)+un(κ416un4+54κ2ψ2(κψ+ψ)2).\eqalign{u_{\tau}=u_{nnnnn}-\frac{5}{2}\kappa\,u_{nnn}\left(\frac{\kappa}{2}u_{n}^{2}-u_{nn}+\psi(\kappa\psi+\psi^{\prime})\right)\\ \phantom{u_{\tau}=u_{nnnnn}}-\frac{5}{2}\kappa\,u_{n}u_{nn}\left(\frac{\kappa}{2}u_{nn}+(\kappa\psi+\psi^{\prime})^{2}\right)\\ \phantom{u_{\tau}=u_{nnnnn}}+u_{n}\left(\frac{\kappa^{4}}{16}u_{n}^{4}+\frac{5}{4}\kappa^{2}\psi^{2}(\kappa\psi+\psi^{\prime})^{2}\right).} (89)

This assertion may be made good by verifying compatibility with the Gauss equation (37). Remarkably, the stationary reduction uξ=0u_{\xi}=0 of (88) produces a compatible third-order equation of the type (40) with

F=κ2unsusψ+unsussψus23κuns3usψ3,F=\frac{\kappa}{2}\frac{u_{ns}u_{s}}{\psi}+\frac{u_{ns}u_{ss}}{\psi u_{s}}-\frac{2}{3\kappa}\frac{u_{ns}^{3}}{u_{s}\psi^{3}}, (90)

while (89) may be found in a classification of integrable fifth-order evolution equations [19]. A connection with the integrable Tzitzéica equation of affine differential geometry is obtained by applying the Miura-type transformation

3ew=(ψ+κψ)(unsψ±32κus)3e^{w}=(\psi^{\prime}+\kappa\psi)\left(\frac{u_{ns}}{\psi}\pm\frac{\sqrt{3}}{2}\kappa u_{s}\right) (91)

which provides a link to the third-order equation

wnns=wn(3ew2wns).w_{nns}=w_{n}(3e^{w}-2w_{ns}). (92)

In fact, the latter is independent of the constant κ\kappa so that one may regard κ\kappa as a spectral parameter for this third-order equation as in the original Miura transformation [15, 11] for the KdV equation. Thus, if uu is a solution of the Gauss equation (37) with ψ\psi being a solution of (87) then ww as defined by (91) is a solution of (92).

The differential equation (92) may be integrated once to obtain the Tzitzéica equation [14, 12] in the form

wns=ew+h(s)e2w.w_{ns}=e^{w}+h(s)e^{-2w}. (93)

Its standard form given by h(s)=±1h(s)=\pm 1 may be obtained by applying suitable transformations of the type sS1(s)s\rightarrow S_{1}(s) and ww+S2(s)w\rightarrow w+S_{2}(s). Hence, we may regard the Gauss equation considered in this subsection as a “modified Tzitzéica equation” in the terminology of soliton theory. Its Lax pair

Ψn=UΨ,Ψs=VΨ\Psi_{n}=U\Psi,\quad\Psi_{s}=V\Psi (94)

may be obtained by applying a gauge transformation to the standard Lax pair of the Tzitzéica equation (see, e.g., [12]). One obtains the matrix-valued functions

U=(0ψ+κψψ+κψλ(ψ+κψ)033(ψ+κψ)λ(ψ+κψ)32κψ0)V=(036κλus13λunsψ13unsψκ2us33unsψ33unsψ36κusκus+33unsψκ2us+33unsψ),\eqalign{U=\left(\begin{array}[]{ccc}0&\quad\psi^{\prime}+\kappa\psi&\psi^{\prime}+\kappa\psi\\[2.84526pt] \lambda(\psi^{\prime}+\kappa\psi)&0&\frac{\sqrt{3}}{3}(\psi^{\prime}+\kappa\psi)\\[2.84526pt] -\lambda(\psi^{\prime}+\kappa\psi)&\frac{\sqrt{3}}{2}\kappa\psi&0\\ \end{array}\right)\\ V=\left(\begin{array}[]{ccc}0&\frac{\sqrt{3}}{6}\frac{\kappa}{\lambda}u_{s}&-\frac{1}{3\lambda}\frac{u_{ns}}{\psi}\\[5.69054pt] \frac{1}{3}\frac{u_{ns}}{\psi}&\frac{\kappa}{2}u_{s}-\frac{\sqrt{3}}{3}\frac{u_{ns}}{\psi}&-\frac{\sqrt{3}}{3}\frac{u_{ns}}{\psi}\\[5.69054pt] \frac{\sqrt{3}}{6}\kappa u_{s}&\quad-\kappa u_{s}+\frac{\sqrt{3}}{3}\frac{u_{ns}}{\psi}&-\frac{\kappa}{2}u_{s}+\frac{\sqrt{3}}{3}\frac{u_{ns}}{\psi}\end{array}\right),} (95)

wherein λ\lambda is the spectral parameter. One may directly verify that the linear system (94) for the vector-valued function Ψ\Psi is indeed compatible if uu is a solution of the Gauss equation subject to (87).

As pointed out in the introduction, the above analysis has led to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature. In particular, the discovery of orthogonal coordinate systems on pseudospherical surfaces which are related to the modified Tzitzéica equation is somewhat surprising since the underlying Lie algebraic structure of the Lax pairs associated with surfaces of constant non-vanishing Gaussian curvature is the Kac-Moody algebra A1(1)A_{1}^{(1)}, while the graded loop algebra A2(2)A_{2}^{(2)} is usually associated with the Tzitzéica equation (see [12] and references therein). It is also observed that (apparently different) modified Tzitzéica equations have been recorded elsewhere in the literature. For instance, a modified Tzitzéica equation and the associated Miura-type transformation containing a particular Weierstrass \wp-function have been presented in [26].

5 Determination of the fibres and fluid flow

So far, we have treated the surface M2M^{2} as a Riemannian manifold [18] without considering its embedding in Euclidean space 3\mathbb{R}^{3} which is determined by the second fundamental form [10]

II=eds2+2fdsdn+gdn2,\mbox{II}=e\,ds^{2}+2f\,dsdn+g\,dn^{2}, (96)

the coefficients of which obey the Mainardi-Codazzi equations

enfs=eunsus+f(ψusψussus)+gusunsψ2fngs=eψψus+f(ψunψunsus)gψusψ.\eqalign{e_{n}-f_{s}=e\frac{u_{ns}}{u_{s}}+f\left(\frac{\psi^{\prime}u_{s}}{\psi}-\frac{u_{ss}}{u_{s}}\right)+g\frac{u_{s}u_{ns}}{\psi^{2}}\\ f_{n}-g_{s}=-e\frac{\psi\psi^{\prime}}{u_{s}}+f\left(\frac{\psi^{\prime}u_{n}}{\psi}-\frac{u_{ns}}{u_{s}}\right)-g\frac{\psi^{\prime}u_{s}}{\psi}.} (97)

In the case of a surface of constant Gaussian curvature KK, this system is completed by adding the defining expression for the Gaussian curvature [10]

K=egf2us2ψ2.K=\frac{eg-f^{2}}{u_{s}^{2}\psi^{2}}. (98)

Thus, for a given solution (u,ψ)(u,\psi) of the third-order equation (37), any solution (e,f,g)(e,f,g) of the system (97), (98) uniquely determines a surface M2M^{2} parametrised by the position vector 𝒓(s,n)=(x(s,n),y(s,n),z(s,n))\mbox{\boldmath$r$}(s,n)=(x(s,n),y(s,n),z(s,n)) up to a Euclidean motion. The fibres are then represented by the curves 𝒓(s,n=const)\mbox{\boldmath$r$}(s,n=\mbox{const}) on M2M^{2}.

Conversely, if a parametrisation 𝒓(σ,ν)\mbox{\boldmath$r$}(\sigma,\nu) of a particular surface M2M^{2} of constant Gaussian curvature KK on which the motion takes place is given then it is required to determine the change of coordinates σ=σ(s,n)\sigma=\sigma(s,n) and ν=ν(s,n)\nu=\nu(s,n) which transforms the given metric

I=𝒓σ2dσ2+2𝒓σ𝒓νdσdν+𝒓ν2dν2\mbox{I}=\mbox{\boldmath$r$}_{\sigma}^{2}\,d\sigma^{2}+2\mbox{\boldmath$r$}_{\sigma}\cdot\mbox{\boldmath$r$}_{\nu}\,d\sigma d\nu+\mbox{\boldmath$r$}_{\nu}^{2}\,d\nu^{2} (99)

into gg as given by (36). In order to illustrate this procedure, we solve this problem in the case of the three simplest examples of surfaces of constant Gaussian curvature, namely the plane, the sphere and the pseudosphere.

5.1 Motions on the plane

The simplest surface of Gaussian curvature K=0K=0 (developable surface) is the plane which we parametrise by

x=σ,y=ν,z=0x=\sigma,\quad y=\nu,\quad z=0 (100)

without loss of generality. Hence, the coordinate transformation σ=σ(s,n)\sigma=\sigma(s,n), ν=ν(s,n)\nu=\nu(s,n) is determined by

σs2+νs2=us2,σsσn+νsνn=0,σn2+νn2=ψ2.\sigma_{s}^{2}+\nu_{s}^{2}=u_{s}^{2},\quad\sigma_{s}\sigma_{n}+\nu_{s}\nu_{n}=0,\quad\sigma_{n}^{2}+\nu_{n}^{2}=\psi^{2}. (101)

The latter may be solved by algebraically parametrising the derivatives of σ\sigma and ν\nu in terms of an a priori arbitrary function ϕ(s,n)\phi(s,n) according to

σs=uscosϕ,σn=ψsinϕ,νs=ussinϕ,νn=ψcosϕ.\sigma_{s}=u_{s}\cos\phi,\quad\sigma_{n}=-\psi\sin\phi,\quad\nu_{s}=u_{s}\sin\phi,\quad\nu_{n}=\psi\cos\phi. (102)

However, the associated compatibility conditions σsn=σns\sigma_{sn}=\sigma_{ns} and νsn=νns\nu_{sn}=\nu_{ns} determine ϕ\phi up to a constant of integration. Indeed, one obtains the pair

ϕs=unsψ,ϕn=ψ,\phi_{s}=-\frac{u_{ns}}{\psi},\quad\phi_{n}=\psi^{\prime}, (103)

which is compatible modulo the Gauss equation (37) for K=0K=0. It is observed that the relations (102) and (103) coincide with those found in [1, 2] for steady planar motions of fibre-reinforced fluids.

5.2 Motions on the sphere

The unit sphere is the simplest surface of Gaussian curvature K=1K=1. If we use spherical polar coordinates

x=sinσcosν,y=sinσsinν,z=cosσx=\sin\sigma\cos\nu,\quad y=\sin\sigma\sin\nu,\quad z=\cos\sigma (104)

to parametrise the unit sphere then the re-parametrisation σ=σ(s,n)\sigma=\sigma(s,n), ν=ν(s,n)\nu=\nu(s,n) yields the relations

σs2+νs2sin2σ=us2,σsσn+νsνnsin2σ=0,σn2+νn2sin2σ=ψ2.\sigma_{s}^{2}+\nu_{s}^{2}\sin^{2}\sigma=u_{s}^{2},\quad\sigma_{s}\sigma_{n}+\nu_{s}\nu_{n}\sin^{2}\sigma=0,\quad\sigma_{n}^{2}+\nu_{n}^{2}\sin^{2}\sigma=\psi^{2}. (105)

As in the flat case, it is convenient to introduce a new quantity ϕ\phi, leading to the system

σs=ussinϕ,σn=ψcosϕ,νs=uscosϕcscσ,νn=ψsinϕcscσ,\eqalign{\sigma_{s}=u_{s}\sin\phi,\quad\sigma_{n}=\psi\cos\phi,\\ \nu_{s}=u_{s}\cos\phi\csc\sigma,\quad\nu_{n}=-\psi\sin\phi\csc\sigma,} (106)

the compatibility of which results in the pair

ϕs=uscosϕcotσunsψ,ϕn=ψψsinϕcotσ.\phi_{s}=u_{s}\cos\phi\cot\sigma-\frac{u_{ns}}{\psi},\quad\phi_{n}=\psi^{\prime}-\psi\sin\phi\cot\sigma. (107)

It is observed that the system (106)1,2, (107) is decoupled from (106)3,4 which may be integrated once σ\sigma and ϕ\phi are known. One may also readily verify that the pair (107) is compatible modulo the Gauss equation (37) for K=1K=1.

The vector fields underlying the Frobenius system (106)1,2, (107) may be shown to generate the so(3)so(3) Lie algebra. In fact, this observation gives rise to the linearising transformation

ϕ=arctanϕ2ϕ3,σ=arctan[ϕ3ϕ11+ϕ22ϕ32],ϕ=(ϕ1ϕ2ϕ3),\phi=\arctan\frac{\phi_{2}}{\phi_{3}},\quad\sigma=\arctan\left[\frac{\phi_{3}}{\phi_{1}}\sqrt{1+\frac{\phi_{2}^{2}}{\phi_{3}^{2}}}\,\right],\quad\boldsymbol{\phi}=\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\end{array}\right), (108)

where ϕ\boldsymbol{\phi} obeys the linear system

ϕs=(0us0us0uns/ψ0uns/ψ0)ϕ,ϕn=(00ψ00ψψψ0)ϕ.\boldsymbol{\phi}_{s}=\left(\begin{array}[]{ccc}0&-u_{s}&0\\ u_{s}&0&-u_{ns}/\psi\\ 0&u_{ns}/\psi&0\end{array}\right)\boldsymbol{\phi},\quad\boldsymbol{\phi}_{n}=\left(\begin{array}[]{ccc}0&0&-\psi\\ 0&0&\psi^{\prime}\\ \psi&-\psi^{\prime}&0\end{array}\right)\boldsymbol{\phi}. (109)

It is observed that the above linear system admits the first integral

ϕ12+ϕ22+ϕ32=κ^2=const.\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}=\hat{\kappa}^{2}=\mbox{const}. (110)

Thus, any solution (ϕ1,ϕ2,ϕ3)(\phi_{1},\phi_{2},\phi_{3}) of the linear system (109) is mapped to a solution (ϕ,σ)(\phi,\sigma) of the nonlinear system (106)1,2, (107). Conversely, for any given solution of the nonlinear system (106)1,2, (107), the functions ϕ1\phi_{1}, ϕ2\phi_{2} and ϕ3\phi_{3} defined by (108)1,2 and (110) obey the linear system (109). Finally, in terms of ϕ\boldsymbol{\phi}, the remaining quantity ν\nu is given by

ν=κ^sgn(ϕ1)(ϕ3ϕ22+ϕ32usdsϕ2ϕ22+ϕ32ψdn).\nu=\hat{\kappa}\int\mbox{sgn}(\phi_{1})\left(\frac{\phi_{3}}{\phi_{2}^{2}+\phi_{3}^{2}}u_{s}\,ds-\frac{\phi_{2}}{\phi_{2}^{2}+\phi_{3}^{2}}\psi\,dn\right). (111)

5.3 Motion on the pseudosphere

The classical pseudosphere is the simplest surface of Gaussian curvature K=1K=-1. It admits the parametrisation [12]

x=sinσcosν,y=sinσsinν,z=cosσ+ln(tanσ2)x=\sin\sigma\cos\nu,\quad y=\sin\sigma\sin\nu,\quad z=\cos\sigma+\ln\left(\tan\frac{\sigma}{2}\right) (112)

with 0<σ<π0<\sigma<\pi, in which case the connection between the two sets of coordinates is given by

σs2cot2σ+νs2sin2σ=us2,σsσncot2σ+νsνnsin2σ=0σn2cot2σ+νn2sin2σ=ψ2.\eqalign{\sigma_{s}^{2}\cot^{2}\sigma+\nu_{s}^{2}\sin^{2}\sigma=u_{s}^{2},\quad\sigma_{s}\sigma_{n}\cot^{2}\sigma+\nu_{s}\nu_{n}\sin^{2}\sigma=0\\ \sigma_{n}^{2}\cot^{2}\sigma+\nu_{n}^{2}\sin^{2}\sigma=\psi^{2}.} (113)

The compatibility of the associated parametrisation

σs=ussinϕtanσ,σn=ψcosϕtanσνs=uscosϕcscσ,νn=ψsinϕcscσ\eqalign{\sigma_{s}=u_{s}\sin\phi\tan\sigma,\quad\sigma_{n}=\psi\cos\phi\tan\sigma\\ \nu_{s}=u_{s}\cos\phi\csc\sigma,\quad\nu_{n}=-\psi\sin\phi\csc\sigma} (114)

leads to the pair

ϕs=uscosϕunsψ,ϕn=ψψsinϕ\phi_{s}=u_{s}\cos\phi-\frac{u_{ns}}{\psi},\quad\phi_{n}=\psi^{\prime}-\psi\sin\phi (115)

which is, once again, compatible modulo the Gauss equation (37) for K=1K=-1. In the current case, once the differential equations (115) have been solved, the remaining functions σ\sigma and ν\nu may be found iteratively via quadratures.

It is observed that the structure of the system (115) implies that it may be transformed into a system of Riccati equations and, hence, be linearised. In fact, it turns out that (114)1,2 plays an important role in this connection. Thus, if we consider, for instance, the upper half of the pseudosphere represented by π/2σ<π\pi/2\leq\sigma<\pi then one may directly verify that the change of variables

ϕ=2arctanϕ1ϕ2,σ=πarcsin(ϕ12+ϕ22),ϕ=(ϕ1ϕ2)\phi=2\arctan\frac{\phi_{1}}{\phi_{2}},\quad\sigma=\pi-\arcsin(\phi_{1}^{2}+\phi_{2}^{2}),\quad\boldsymbol{\phi}=\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\end{array}\right) (116)

linearises the coupled system (114)1,2, (115) to obtain

ϕs=12(0usuns/ψus+uns/ψ0)ϕ,ϕn=12(ψψψψ)ϕ\boldsymbol{\phi}_{s}=\frac{1}{2}\left(\begin{array}[]{cc}0&u_{s}-u_{ns}/\psi\\ u_{s}+u_{ns}/\psi&0\end{array}\right)\boldsymbol{\phi},\quad\boldsymbol{\phi}_{n}=\frac{1}{2}\left(\begin{array}[]{cc}-\psi&\psi^{\prime}\\ -\psi^{\prime}&\psi\end{array}\right)\boldsymbol{\phi} (117)

and the remaining function ν\nu is given by

ν=(ϕ12ϕ22(ϕ12+ϕ22)2usds+2ϕ1ϕ2(ϕ12+ϕ22)2ψdn).\nu=-\int\left(\frac{\phi_{1}^{2}-\phi_{2}^{2}}{{(\phi_{1}^{2}+\phi_{2}^{2})}^{2}}u_{s}\,ds+\frac{2\phi_{1}\phi_{2}}{{(\phi_{1}^{2}+\phi_{2}^{2})}^{2}}\psi\,dn\right). (118)

Conclusion

We have demonstrated that large classes of steady motions of ideal fibre-reinforced fluids on surfaces of constant Gaussian curvature are governed by various integrable specialisations of the Gauss equation which constitutes a third-order partial differential equation. Integrable systems theory may now be applied to construct explicitly such motions. On the one hand, one can apply Bäcklund transformations [12] to generate sequences of fibre distributions from a seed distribution for a fixed surface M2M^{2} of constant Gaussian curvature. On the other hand, one can apply Bäcklund transformations to generate sequences of surfaces M2M^{2} on which the motion takes place. The concrete application of those two types of Bäcklund transformations will be discussed elsewhere. It would be interesting to investigate whether the generation of those two types of sequences commutes, that is, whether the order in which the sequences of motions for a given surface and the sequences of surfaces for a given motion are generated is of importance. In this connection, it is observed that the commutativity of Bäcklund transformations is an important topic in soliton theory [12].

References

References

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