# Stability of a Poiseuille-type flow for a MHD model of an incompressible
polymeric fluid

A. M. Blokhin  Sobolev Institute of Mathematics, 630090, 4 Acad. Koptuyg
Avenue, Novosibirsk, 630090, Russia
Mechanics and Mathematics Department, Novosibirsk State University, 630090, 1
Pirogova Str., Novosibirsk, Russia
<EMAIL_ADDRESS>D. L. Tkachev  Sobolev Institute of Mathematics, 630090,
4 Acad. Koptuyg Avenue, Novosibirsk, 630090, Russia
Mechanics and Mathematics Department, Novosibirsk State University, 630090, 1
Pirogova Str., Novosibirsk, Russia
<EMAIL_ADDRESS>
###### Abstract

We study a generalization of the Pokrovski–Vinogradov model for flows of
solutions and melts of an incompressible viscoelastic polymeric medium to
nonisothermal flows in an infinite plane channel under the influence of
magnetic field. For the linearized problem (when the basic solution is an
analogue of the classical Poiseuille flow for a viscous fluid described by the
Navier-Stokes equations) we find a formal asymptotic representation for the
eigenvalues under the growth of their modulus. We obtain a necessary condition
for the asymptotic stability of a Poiseuille-type shear flow.

###### keywords:

Incompressible viscoelastic polymeric fluid; rheological relation;
magnetohydrodynamic flow.

https://doi.org/10.1016/j/euromechflu.2019.12.006 2019\. This manuscript
version is made available under the CC-BY-NC-ND 4.0 license
http://creativecommons.org/licenses/by-nc-nd/4.0/

Mathematics Subject Classification 2010: 35S99, 76J99, 76T20

## Introduction

In this work we study a generalization of the structurally phenomenological
Pokrovski–Vinogradov model describing flows of melts and solutions of
incompressible viscoelastic polymeric media to the nonisothermal case under
the influence of magnetic field. In the Pokrovski–Vinogradov model, the
polymeric medium is considered as a suspension of polymer macromolecules which
move in an anisotropic fluid produced, for example, by a solvent and other
macromolecules. The influence of environment on a real macromolecule is
modeled by the action on a linear chain of Brownian particles each of which
represents a large enough part of the macromolecule. Brownian particles often
called “beads” are connected to each other by elastic forces called “springs”.
In the case of slow motions the macromolecule is modeled as a chain of two
particles called “dumbbell”.

The physical representation of linear polymer flows described above results in
the formulation of the Pokrovski–Vinogradov rheological model [2, 21, 22]:

$\rho(\frac{\partial}{\partial t}v_{i}+v_{k}\frac{\partial}{\partial
x_{k}}v_{i})=\frac{\partial}{\partial x_{k}}\sigma_{ik},\quad\frac{\partial
v_{i}}{\partial x_{i}}=0,$ (1)
$\sigma_{ik}=-p\delta_{ik}+3\frac{\eta_{0}}{\tau_{0}}a_{ik},$ (2)
$\frac{d}{dt}a_{ik}-v_{ij}a_{jk}-v_{kj}a_{ji}+\frac{1+(k-\beta)I}{\tau_{0}}a_{ik}=\frac{2}{3}\gamma_{ik}-\frac{3\beta}{\tau_{0}}a_{ij}a_{jk},$
(3)

where $\rho$ is the polymer density, $v_{i}$ is the $i$-th velocity component,
$\sigma_{ik}$ is the stress tensor, $p$ is the hydrostatic pressure,
$\eta_{0}$, $\tau_{0}$ are the initial values of the shear viscosity and the
relaxation time respectively for the viscoelastic component, $v_{ij}$ is the
tensor of the velocity gradient, $a_{ik}$ is the symmetrical tensor of
additional stresses of second rank, $I=a_{11}+a_{22}+a_{33}$ is the first
invariant of the tensor of additional stresses,
$\gamma_{ik}=\frac{v_{ik}+v_{ki}}{2}$ is the symmetrized tensor of the
velocity gradient, $k$ and $\beta$ are the phenomenological parameters taking
into account the shape and the size of the coiled molecule in the dynamics
equations of the polymer macromolecule.

Structurally, the model consists of the incompressibility and motion equations
(1) as well as the rheological relations (2), (3) connecting kinematic
characteristics of the flow and its inner thermodynamic parameters.

Some generalizations of model (1) – (3) provide good results in numerical
simulations for viscosymetric flows [16]. For example, such a generalization
is a model for which in equation (2) we add a term taking into account the so-
called shear viscosity and the parameter $\beta$ is additionally dependent on
the first invariant of the anisotropy tensor. Therefore, we may believe that
modifications of the basic Polrovski–Vinogradov model could be useful for
modeling the polymer motion in complex deformation conditions, e.g., for
stationary and non-stationary flows in circular channels, flows in channels
with a fast change of sectional area and flows with a free boundary. An
important feature of such flows is their two- and three dimensional character.

In this work, we consider one of such generalizations that takes into account
the influence of the heat and the magnetic field on the polymeric fluid motion
(see Sect. 1 for more details). Our main interest is an analogue of the
Poiseuille flow which is the well-known shear flow in an infinite channel. It
turns out that in our case this flow has a number of features. For example,
computations show that for some values of parameters the velocity profile is
stretched in the direction opposite to the forces of pressure (see Sect. 1).

Our main results are given in Sect. 2. Firstly, we get an asymptotic
representation of the spectrum of the problem linearized about the the chosen
basic solution which is the Poiseuille-type flow. Secondly, as the result we
get a condition whose fulfilment guarantees that the basic solution is
asymptotically stable by Lyapunov in the chosen class of perturbations
periodic with respect the variable changing along the infinite plane channel.
The last section is devoted to the proof of the theorems formulated in Sect.
2.

Overall, our work continues the study of Lyapunov’s stability of shear flows
for both the original Pokrovski–Vinogradov model and its various
generalizations described in [6, 7, 8, 9, 10, 11].

It should be noted that for the case of viscous fluid there is the known
Krylov’s result [17] about the linear Lyapunov’s instability of the Poiseuille
flow for large enough Reynolds numbers confirming Heisenberg’s hypothesis [15]
(a refinement of this result was obtained in [14]).

##  1 Nonlinear model of the polymeric fluid flow in a plane channel under the
presence of an external magnetic field

Using the results from [1, 19, 23, 25, 26] and [4], let us formulate a
mathematical model describing magnetohydrodynamic flows of an incompressible
polymeric fluid for which, as in [24], in the equation for the inner energy we
introduce some dissipative terms. In a dimensionless form this model reads (we
keep the notations from [5]):

$div\vec{\boldsymbol{u}}=u_{x}+v_{y}=0,$ (4) $div\vec{H}=L_{x}+M_{y}=0,$ (5)
$\frac{d\vec{\boldsymbol{u}}}{dt}+\nabla
P=div(Z\Pi)+\sigma_{m}(\vec{H},\nabla)\vec{H}+Gr(Z-1)\begin{pmatrix}0\\\
1\end{pmatrix},$ (6) $\frac{da_{11}}{dt}-2A_{1}u_{x}-2a_{12}u_{y}+L_{11}=0,$
(7) $\frac{da_{22}}{dt}-2A_{2}v_{y}-2a_{12}v_{x}+L_{22}=0,$ (8)
$\frac{da_{12}}{dt}-A_{1}v_{x}-A_{2}u_{y}+\frac{\widetilde{K}_{I}a_{12}}{\bar{\tau}_{0}(Z)}=0,$
(9)
$\frac{dZ}{dt}=\frac{1}{Pr}\Delta_{x,y}Z+\frac{A_{r}}{Pr}ZD_{\Gamma}+\frac{A_{m}}{Pr}\sigma_{m}D_{m},$
(10)
$\frac{d\vec{H}}{dt}-(\vec{H},\nabla)\vec{\boldsymbol{u}}-b_{m}\Delta_{x,y}\vec{H}=0.$
(11)

where $t$ is the time, $u$, $v$ and $L$, $1+M$ are the components of the
velocity vector $\vec{\boldsymbol{u}}$ and the magnetic field $\vec{H}$
respectively in the Cartesian coordinate system $x,y$;

$P=\rho+\sigma_{m}\frac{L^{2}+(1+M)^{2}}{2},$

$\rho$ is the pressure; $a_{11}$, $a_{22}$, $a_{12}$ are the components of the
symmetrical anisotropy tensor of second rank;

$\Pi=\frac{1}{Re}(a_{ij}),\quad i,j=1,2;\quad
L_{ii}=\frac{K_{I}a_{ii}+\beta(a_{ii}^{2}+a_{12}^{2})}{\bar{\tau}_{0}(Z)},\quad
i=1,2;$ $K_{I}=W^{-1}+\frac{\bar{k}}{3}I,\quad\bar{k}=k-\beta;$

$I=a_{11}+a_{22}$ is the first invariant of the anisotropy tensor; $k$,
$\beta$ ($0<\beta<1$) are the phenomenological parameters of the rheological
model (see [21]); $A_{i}=W^{-1}+a_{ii}$, $i=1,2$; $Z=\frac{T}{T_{0}}$; $T$ is
the temperature, $T_{0}$ is an average temperature (room temperature; we will
further assume that $T_{0}=300$ K);
$\widetilde{K}_{I}=K_{I}+\beta I$; $\bar{\tau}_{0}(Z)=\frac{1}{ZJ(Z)}$,
$J(Z)=\exp\\{\bar{E}_{A}\frac{Z-1}{Z}\\}$,
$\bar{E}_{A}=\frac{E_{A}}{T_{0}}$, $E_{A}$ is the activation energy;
$Re=\frac{\rho u_{H}l}{\eta_{0}^{*}}$ is the Reynolds number;
$W=\frac{\tau_{0}^{*}u_{H}}{l}$ is the Weissenberg number;
$Gr=\frac{Ra}{Pr}$ is the Grasshoff number;
$Pr=\frac{lu_{H}\rho
c_{v}}{\varepsilon}=\frac{c_{v}\eta_{0}^{*}Re}{\varepsilon}$ is the Prandtl
number;
$Ra=\frac{lbgT_{0}Pr}{u_{H}^{2}}$ is the Rayleigh number;
$A_{r}=\frac{\alpha u_{H}^{2}Pr}{ReT_{0}c_{v}}=\frac{\alpha
u_{H}^{2}\eta_{0}^{*}}{T_{0}\varepsilon}$,
$A_{m}=\frac{\alpha_{m}u_{H}^{2}Pr}{T_{0}c_{v}}$;
$D_{\Gamma}=a_{11}u_{x}+(v_{x}+u_{y})a_{12}+a_{22}v_{y}$;
$D_{m}=L^{2}u_{x}+L(1+M)(v_{x}+u_{y})+(1+M)^{2}v_{y}$;
$\rho(=const)$ is the medium density;
$\varepsilon$ is the coefficient of thermal conductivity of the polymeric
fluid;
$b$ is the coefficient thermal expansion of the polymeric fluid;
$g$ is the gravity constant;
$\eta_{0}^{*}$, $\tau_{0}^{*}$ are the initial values for the shear viscosity
and the relaxation time for the room temperature $T_{0}$ (see [21, 22]);
$l$ is the characteristic length, $u_{H}$ is the characteristic velocity;
$\sigma_{m}=\frac{\mu\mu_{0}H_{0}^{2}}{\rho u_{H}^{2}}$ is the magnetic
pressure coefficient;
$b_{m}=\frac{1}{Re_{m}}$, $Re_{m}=\sigma_{m}\mu\mu_{0}u_{H}l$ is the magnetic
Reynolds number;
$\mu_{0}$ is the magnetic penetration in vacuum;
$\mu$ is the magnetic penetration of the polymeric fluid;
$\sigma$ is the electrical conductivity of the medium;
$\alpha$ is the thermal equivalent of work (see [9]);
$\alpha_{m}$ is the magnetothermal equivalent of work;
$c_{v}$ is the heat capacity;
$\frac{d}{dt}=\frac{\partial}{\partial
t}+(\vec{\boldsymbol{u}},\nabla)=\frac{\partial}{\partial
t}+u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}$,
$\Delta_{x,y}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}$ is the Laplace operator.
The variables $t$, $x$, $y$, $u$, $v$, $p$, $a_{11}$, $a_{22}$, $a_{12}$, $L$,
$M$ in system (4)–(11) correspond to the following values: $\frac{l}{u_{H}}$,
$l$, $u_{H}$, $\rho u_{H}^{2}$, $\frac{W}{3}$, $H_{0}$, where $H_{0}$ is the
characteristic magnitude of the magnetic field (see fig. 1).

$\scriptstyle C^{-}$$\scriptstyle x$$\scriptstyle
C^{+}$$S_{1}^{-}$$\scriptstyle h$$S$$S_{1}^{+}$$\scriptstyle-1/2$$\scriptstyle
0$$\scriptstyle 1/2$$\scriptstyle y$$\scriptstyle
H_{1}^{+}=\,(0,1)$$\scriptstyle H_{1}^{-}=\,(0,1)$$\scriptstyle
H_{1}^{-}=\,(0,1)$$\scriptstyle H_{1}^{+}=\,(0,1)$$\scriptstyle
M_{1}^{-}=\,0$$\scriptstyle L_{1}^{-}=\,0$$\scriptstyle
a_{11},a_{12},a_{22},Z$$\scriptstyle u,v,L,1+M$$\scriptstyle
M_{1}^{+}=\,0$$\scriptstyle L_{1}^{+}=\,0$

Figure 1: Plane channel

######  Remark 1.1.

The magnetohydrodynamic equations (4) – (11) are derived with the use of the
Maxwell equations (see [23, 26]). The magnetic induction vector $\vec{B}$ is
represented as

$\vec{B}=\mu\mu_{0}\vec{H}=(1+\chi)\mu_{0}\vec{H},$ (12)

where $\chi$ is the magnetic susceptibility, and (see [1, 20])
$\chi=\frac{\chi_{0}}{Z}$, $\chi_{0}$ is the magnetic susceptibility for the
room temperature $T_{0}$(= 300 K). We will further assume that for the
polymeric fluid $\mu=1$ ($\chi_{0}=0$).

######  Remark 1.2.

Our main problem is the problem of finding solutions to the mathematical model
(4)–(11) describing magnetohydrodynamic flows of an incompressible polymeric
fluid in a plane channel with the depth $1(l)$ and bounded by the horizontal
walls which are the electrodes $C^{+}$ and $C^{-}$ along which we have
electric currents with the current strength $J^{+}$ and $J^{-}$ respectively
(see Fig. 1).

External with respect to the chanel $S$ areas $S_{1}^{+}$, $S_{1}^{-}$ are
under the influence of the magnetic fields with components $L_{1}^{+}=0$,
$M_{1}^{+}$, and
$M_{1}^{+}|_{y=\frac{1}{2}+0}=-1+\frac{1+M(\frac{1}{2})}{1+\chi_{0}^{+}}$,
$L_{1}^{-}=0$, $M_{1}^{-}$ and
$M_{1}^{-}|_{y=-\frac{1}{2}+0}=-1+\frac{1+M(-\frac{1}{2})}{1+\frac{\chi_{0}^{-}}{1+\bar{\theta}}}$.
Values of temperature $Z$ on the sides of the chanel will be defined below
with boundary conditions (13). Acquired correlations between boundary values
of $M_{1}^{+}+1$, $1+M(\frac{1}{2})$, and $M_{1}^{-}+1$, $1+M(-\frac{1}{2})$
correspondingly arise due to the equality (12) and continuity of the normal
component for the magnetic induction vector on the sides of the chanel.

The domains $S_{1}^{+}$ and $S_{1}^{-}$ external to the channel are magnets
with the magnetic susceptibilities $\chi_{1}^{+}$ and $\chi_{1}^{-}$. On the
walls of the channel the following boundary conditions hold:

$\left\\{\begin{array}[]{l}y=\pm\frac{1}{2}:\quad\vec{u}=0\quad(\mbox{no-slip
condition}),\\\ y=\frac{1}{2}:\qquad Z=1\quad(T=T_{0}),\\\
y=-\frac{1}{2}:\quad
Z=1+\bar{\theta}\quad(\bar{\theta}=\frac{\theta}{T_{0}},\,\theta=T-T_{0}).\end{array}\right.$
(13)

We have the temperature $T=T_{0}$ in the domain $S_{1}^{+}$ where as on the
electrode $C^{+}$ we have:

$y=-\frac{1}{2}:Z=1+\bar{\theta},\bar{\theta}=\frac{\theta}{T_{0}},\theta=T-T_{0},$

i.e., for $\bar{\theta}>0$ there is heating from below ($T$ is the temperature
in the domain $S_{1}^{-}$ and on the electrode $C^{-}$), and for
$\bar{\theta}<0$ there is heating from above.

######  Remark 1.3.

We will consider the electrodes $C^{+}$ and $C^{-}$ as the boundaries between
two uniform isotropic magnetics. Therefore, on the boundaries $C^{+}$ and
$C^{-}$ the following known conditions hold (see [1, 18]):

$\left\\{\begin{array}[]{l}y=\frac{1}{2}(C^{+}):\quad L=-J^{+},\quad
M_{y}=0,\\\ y=-\frac{1}{2}(C^{-}):\quad L=-J^{-},\quad
M_{y}=0.\end{array}\right.$ (14)

We get the boundary condition $M_{y}=0$ at $y=\pm\frac{1}{2}$ by assuming that
relation (5) holds for $y=\pm\frac{1}{2}$ and by taking into account the
conditions $L=-J^{+}$ $(y=\frac{1}{2})$ and $L=-J^{-}$ $(y=-\frac{1}{2})$ (see
(14)) that gives us $M_{y}=0$ ($y=\pm\frac{1}{2}$).

######  Remark 1.4.

Let us show that

$\left\\{\begin{array}[]{l}d=L_{x}+M_{y}=0\quad\mbox{for }y=\pm\frac{1}{2},\\\
d=0\quad\mbox{for }t=0,|y|<\frac{1}{2},x\in R^{1};\\\ d\to 0\quad\mbox{for
}|x|\to\infty,t>0,|y|<\frac{1}{2},x\in R^{1},\end{array}\right.$

i.e., relation (5) follows from equations (4), (11).

To prove this we apply the operator div to equation (11). Taking into account
(4), we get

$d_{t}+(\vec{u},\nabla)d-b_{m}\Delta_{x,y}d=0.$

Consequently,

$(d^{2})_{t}+div(d^{2}\vec{u}-2b_{m}d\cdot\nabla d)+2b_{m}|\nabla d|^{2}=0.$

Integrating this relation with respect to $x$ from $-\infty$ to $+\infty$ and
with respect to $y$ from $-\frac{1}{2}$ to $\frac{1}{2}$ gives

$\frac{d}{dt}\left\\{\int_{-\frac{1}{2}}^{\frac{1}{2}}\int_{-\infty}^{+\infty}d^{2}(t,x,y)dxdy\right\\}+2b_{m}\int_{-\frac{1}{2}}^{\frac{1}{2}}\int_{-\infty}^{+\infty}|\nabla
d(t,x,y)|^{2}dxdy=0.$

This implies

$\int_{-\frac{1}{2}}^{\frac{1}{2}}\int_{-\infty}^{+\infty}d^{2}(t,x,y)dxdy\leq
0,$

i.e., $d=0$ for $t>0$, $|y|<\frac{1}{2}$, $x\in R^{1}$.

If we consider $d$ as a function from a wider class of functions bounded on
each infinite set $\\{(t,x,y)|0\leq t\leq T,-\infty<x<\infty,-\frac{1}{2}\leq
y\leq\frac{1}{2}$} (the parameter $T$ is being varied), then the fact that $d$
vanishes follows from the maximum principle for the heat equation

$d_{t}-b_{m}\Delta_{x,y}d=0.$

This equation can be obtained by rewriting relation (11) in the form

$\frac{d\vec{H}}{dt}-rot(\vec{u}\times\vec{H})-b_{m}\Delta_{x,y}\vec{H}=0$

and using the operator div.

Stationary solutions of the mathematical model (4) – (11) were studied in [5].
Particular solutions (analogous to Poiseuille and Couette solutions for the
Navier–Stoks equations system) were constructed there in the following form:

$\left\\{\begin{array}[]{l}\vec{U}(t,x,y)=\hat{\vec{U}}(y),\\\
p(t,x,y)=\hat{P}(y)+\hat{p}_{0}-\hat{A}x,\end{array}\right.$ (15)

where
$\vec{U}=(u,v,a_{11},a_{12},a_{22},Z,L,M)^{T}$,
$\hat{\vec{U}}(y)=(\hat{u}(y),\hat{v}(y),\hat{a}_{11}(y),\hat{a}_{12}(y),\hat{a}_{22}(y),\hat{Z}(y),\hat{L}(y),\hat{M}(y))^{T}$,
$\hat{P}(y)$ ($\hat{P}(0)=0$) is some function that we will define further,
$\hat{p}_{0}$ is the pressure on the channel axis for $y=0$, $x=0$ and
$\hat{A}(>0)$ is the dimensionless constant drop of pressure on the segment
$h$.

From (4) – (11), (13), (14) we have the following relations for getting
functions $\hat{u}(y)$, $\hat{a}_{11}(y)$, $\hat{a}_{12}(y)$,
$\hat{a}_{22}(y)$, $\hat{Z}(y)$, $\hat{L}(y)$, $\hat{M}(y)$, $\hat{P}(y)$:

$\displaystyle\frac{d}{dy}(\hat{Z}\hat{a}_{12}+(1+\hat{\lambda}\sigma_{m}Re\hat{L})=(\hat{Z}\hat{a}_{12}+(1+\hat{\lambda}\sigma_{m}Re\hat{L})^{{}^{\prime}}=-\hat{D},$
(16)
$\displaystyle(\hat{P}-\frac{\hat{Z}\hat{a}_{22}}{Re}+\sigma_{m}\frac{\hat{L}^{2}}{2})^{{}^{\prime}}=Gr(\hat{Z}-1),\quad\hat{P}(0)=0,$
$\displaystyle\hat{u}^{{}^{\prime}}=\frac{\tilde{K}_{\hat{I}}J(\hat{Z})\hat{Z}\hat{a}_{12}}{\hat{A}_{2}},\quad\hat{u}(\pm\frac{1}{2})=0,$
$\displaystyle K_{\hat{I}}\hat{a}_{22}+\beta(\hat{g}+\hat{a}_{22}^{2})=0,$
$\displaystyle
K_{\hat{I}}\hat{a}_{11}+\beta(\hat{g}+\hat{a}_{11}^{2})-2\hat{g}\frac{\tilde{K}_{\hat{I}}}{\hat{A}_{2}}=0,$
$\displaystyle\hat{Z}^{{}^{\prime\prime}}+(a_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}(1+\hat{\lambda})\hat{L})\hat{u}^{{}^{\prime}}=0,\,\hat{Z}(\frac{1}{2})=1,\,\hat{Z}(-\frac{1}{2})=1+\bar{\theta},$
$\displaystyle
b_{m}\hat{L}^{{}^{\prime\prime}}+(1+\hat{\lambda})\hat{u}^{{}^{\prime}}=0,\quad\hat{L}(\pm\frac{1}{2})=-J^{\pm},$
$\displaystyle\hat{M}=\hat{\lambda}=\chi_{0}^{+}=\frac{\chi_{0}^{-}}{1+\bar{\theta}}=const.$

Here $\hat{D}=Re\cdot\hat{A}$, $K_{\hat{I}}=W^{-1}+\frac{\bar{k}}{3}\hat{I}$,
$\hat{I}=\hat{a}_{11}+\hat{a}_{22}$,
$\tilde{K}_{\hat{I}}=K_{\hat{I}}+\beta\hat{I}$, $\hat{g}=\hat{a}_{12}^{2}$,
$\hat{A}_{2}=W^{-1}+\hat{a}_{22}$,
$\chi_{0}^{+}$ and $\chi_{0}^{-}$ are the magnetic susceptibilities for
$T=T_{0}$ in the domains $S_{1}^{+}$ and $S_{1}^{-}$ respectively. A detailed
analysis of relations (16) was performed in [5].

Some solutions or more precisely their components $\hat{u}$, $\hat{Z}$ and
$\hat{L}$ are represented in Fig. 2, 3, 4, 5. Moreover, in the first (main)
case (if we use the terminology from [5]) $\hat{A}=1$, $\hat{\lambda}=1$,
$\sigma_{m}=1$, $Re=1$, $W=1$, $\beta=0.5$, $A_{r}=1$, $A_{m}=1$,
$\bar{\theta}=1$, $b_{m}=1$, $\bar{E}_{A}=1$, $J^{+}=2$, $J^{-}=1$, and in the
second, third and forth cases we change one of the parameters $\hat{A}$,
$J^{+}$ and $\theta$ respectively by leaving other parameters unchanged.

Figure 2: Main case.

Figure 3: Solution for $\hat{A}=3$.

Figure 4: Solution for $J^{+}=-1$.

Figure 5: Solution for $\bar{\theta}=-0.95$.

Let us list most important features of the behavior of stationary polymeric
fluid flows. In [4] the influence of the parameter
$\bar{E}_{A}=\frac{E_{A}}{T_{0}}$ (connected with the activation energy
$E_{A}$) on the form of the velocity profile was found. Unlike [4], in our
case the velocity profile loses symmetry which is a feature of the parabolic
velocity profile of the Poiseuille flow for viscous fluid [19]. This means
that the solutions of (4) – (11), (13), (14) have a wider range of interesting
properties.

In the main case (see Fig. 2) the velocity profile is elongated in the
direction opposite to that of pressure forces (due to the influence of
magnetic field!). In Fig. 3 the pressure drop (its absolute value) is
enlarged. This implies that the velocity profile is turned to the right. In
Fig. 4 the absolute value of pressure is again small. But since the current’s
direction on the top electrode is now opposite to the previous one, the
velocity profile is again turned to the right. At last, strong cooling of the
bottom channel’s boundary implies that the fluid velocity in the bottom part
of the channel becomes close to zero (see Fig. 5).

We will further study Lyapunov’s stability with respect to small perturbations
of the stationary flow described above. Denoting small perturbations of all
values by the same symbols, after linearization we obtain the following linear
problem:

$\left\\{\begin{aligned}
u_{t}&+\hat{u}u_{x}-\hat{Z}(\alpha_{11})_{x}+\hat{Z}(\alpha_{22})_{x}-\hat{Z}(\alpha_{12})_{y}+\hat{u}^{{}^{\prime}}v-\hat{Z}^{{}^{\prime}}\alpha_{12}+\Gamma_{1}=0,\\\
v_{t}&+\hat{u}v_{x}-\hat{Z}(\alpha_{12})_{x}+\Gamma_{2}=0,\\\
(\alpha_{11})_{t}&+\hat{u}(\alpha_{11})_{x}-2\hat{\alpha}_{1}u_{x}-2\hat{\alpha}_{12}u_{y}+\hat{\alpha}_{11}^{{}^{\prime}}v+R_{33}\alpha_{11}+\\\
&+R_{34}\alpha_{12}+R_{35}\alpha_{22}+r_{11}Z=0,\\\
(\alpha_{12})_{t}&+\hat{u}(\alpha_{12})_{x}-\hat{\alpha}_{1}v_{x}-\hat{\alpha}_{2}u_{y}+\hat{\alpha}_{12}^{{}^{\prime}}v+R_{43}\alpha_{11}+\\\
&+R_{54}\alpha_{12}+R_{45}\alpha_{22}+r_{12}Z=0,\\\
(\alpha_{22})_{t}&+\hat{u}(\alpha_{22})_{x}-2\hat{\alpha}_{12}v_{x}-2\hat{\alpha}_{2}v_{y}+\hat{\alpha}_{22}^{{}^{\prime}}v+R_{53}\alpha_{11}+\\\
&+R_{54}\alpha_{12}+R_{55}\alpha_{22}=0,\\\
Z_{t}&+\hat{u}Z_{x}+\hat{Z}^{\prime}v=\frac{1}{Pr}\Delta_{x,y}Z+\frac{Ar}{Pr}\hat{u}^{\prime}\hat{a}_{12}Z+\frac{Ar}{Pr}\hat{Z}(\hat{a}_{11}u_{x}+\\\
&+\hat{a}_{12}v_{x}+\hat{a}_{12}u_{y}+\hat{a}_{22}v_{y}+\hat{u}^{\prime}a_{12})+\frac{Am}{Pr}\sigma_{m}(\hat{L}^{2}u_{x}+\\\
&+\hat{L}(1+\hat{\lambda})v_{x}+\hat{L}(1+\hat{\lambda})u_{y}+\hat{u}^{\prime}((1+\hat{\lambda})L+\hat{L}M)+(1+\hat{\lambda})^{2}v_{y}),\\\
L_{t}&+\hat{u}L_{x}+v\hat{L}^{\prime}-\hat{L}u_{x}-(1+\hat{\lambda})u_{y}-\hat{u}^{\prime}M-b_{m}\Delta_{x,y}L=0,\\\
M_{t}&+\hat{u}M_{x}-\hat{L}v_{x}-(1+\hat{\lambda})v_{y}-b_{m}\Delta_{x,y}M=0,\\\
u_{x}&+v_{y}=0,\quad t>0,\quad|y|<\frac{1}{2},\quad x\in
R^{1};\end{aligned}\right.$ (17) $u=v=Z=L=M_{y}=0\quad\mbox{
}y=\pm\frac{1}{2},\,t>0,x\in R^{1};$ (18)

where

$\displaystyle\alpha_{ij}$
$\displaystyle=\frac{a_{ij}}{Re},\hat{\alpha}_{ij}=\frac{\hat{a}_{ij}}{Re},i,j=1,2,$
$\displaystyle\hat{\alpha}_{i}$
$\displaystyle=\hat{\alpha}_{ii}+\kappa^{2},\kappa^{2}=\frac{1}{WRe}i=1,2,$
$\displaystyle\Gamma_{1}$
$\displaystyle=\Omega_{x}-\hat{\alpha}_{11}Z_{x}-\hat{\alpha}_{12}Z_{y}-\hat{\alpha}_{12}^{\prime}Z+\sigma_{m}(1+\hat{\lambda})\omega_{m}-\sigma_{m}\hat{L}M,$
$\displaystyle\Gamma_{2}$
$\displaystyle=\Omega_{y}-\hat{\alpha}_{12}Z_{x}-\hat{\alpha}_{2}Z_{y}-(\hat{\alpha}_{12}^{{}^{\prime}}+Gr)Z-\sigma_{m}\hat{L}\omega_{m}+\sigma_{m}\hat{L}^{\prime}L,$
$\displaystyle\Omega$
$\displaystyle=P-\hat{Z}\alpha_{22},\omega_{m}=M_{x}-L_{y}$ $\displaystyle
R_{33}$
$\displaystyle=\hat{\bar{\chi}}_{0}(\hat{K}_{\hat{I}}+\hat{a}_{11}(\frac{\bar{k}}{3}+2\beta)),R_{34}=-2\hat{u}^{\prime}+2\beta\hat{a}_{12}\chi_{0}^{*},$
$\displaystyle R_{35}$
$\displaystyle=\frac{\bar{k}}{3}\hat{a}_{11}\hat{\bar{\chi}}_{0},\hat{\bar{\chi}}_{0}=\frac{1}{\bar{\tau}_{0}(\hat{Z})},$
$\displaystyle r_{11}$
$\displaystyle=(\hat{\bar{\chi}}_{0})^{\prime}\frac{2\hat{\alpha}_{12}^{2}\tilde{K}_{\hat{I}}}{\hat{\alpha}_{2}},\hat{\bar{\chi}}_{0}^{`}=\hat{\bar{\chi}}_{0}\frac{\bar{E}_{A}+\hat{Z}}{\hat{Z}^{2}},\hat{\tilde{K}}_{\hat{I}}=\hat{K}_{\hat{I}}+\beta(\hat{a}_{11}+\hat{a}_{22}),$
$\displaystyle R_{43}$
$\displaystyle=\hat{a}_{12}\hat{\bar{\chi}}_{0}(\frac{\bar{k}}{3}+\beta),R_{45}=-\hat{u}^{\prime}+\hat{a}_{12}\hat{\bar{\chi}}_{0}(\frac{\bar{k}}{3}+\beta),R_{44}=\hat{\bar{\chi}}_{0}^{{}^{\prime}}\hat{\tilde{K}}_{\hat{I}},$
$\displaystyle r_{12}$
$\displaystyle=(\hat{\bar{\chi}}_{0})^{\prime}\hat{\alpha}_{12}\hat{\tilde{K}}_{\hat{I}},$
$R_{53}=\hat{\bar{\chi}}_{0}\hat{a}_{22}\frac{\bar{k}}{3},R_{54}=2\beta\hat{a}_{12}\hat{\bar{\chi}}_{0},R_{55}=\hat{\bar{\chi}}_{0}(\hat{K}_{\hat{I}}+\hat{a}_{22}(\frac{\bar{k}}{3}+2\beta)).$
(19)

######  Remark 1.5.

The relation

$L_{x}+M_{y}=0$

is not included into system (17) because due to the last three equations in
(17) it holds for $t>0$ if it was true for $t=0$. That is, relation (5) is, in
fact, a constraint on the initial data for $l$ and $M$.

######  Remark 1.6.

Unlike [5], the components $M_{1}^{+}$ and $M_{1}^{-}$ are not zero in the
domains $S_{1}^{+}$ and $S_{1}^{-}$ (see Fig. 1):

$\displaystyle M_{1}^{+}\big{|}_{y=\frac{1}{2}+0}$
$\displaystyle=-1+\frac{1+M(\frac{1}{2})}{1+\chi_{0}^{+}},$ (20)
$\displaystyle M_{1}^{-}\big{|}_{y=-\frac{1}{2}-0}$
$\displaystyle=-1+\frac{1+M(-\frac{1}{2})}{1+\frac{\bar{\chi}_{0}^{-}}{(1+\bar{\theta})}},$

where $M(\frac{1}{2})$ and $M(-\frac{1}{2})$ are the values of $M$ on the top
and bottom electrodes respectively.

######  Remark 1.7.

Let us assume that the domains $S_{1}^{\pm}$ are filled with nonconducting
mediums. Then, in view of the Maxwell equation [18], the small perturbations
of $M_{1}^{\pm}$ satisfy the Laplace equation and additional conditions at
infinity:

$\displaystyle\Delta_{x,y}M_{1}^{+}$ $\displaystyle=0\,\mbox{in
}S_{1}^{+},\,M_{1}^{+}\to 0\,\mbox{for }y\to\infty,$ (21)
$\displaystyle\Delta_{x,y}M_{1}^{-}$ $\displaystyle=0\,\mbox{in
}S_{1}^{-},\,M_{1}^{-}\to 0\,\mbox{for }y\to-\infty.$

If the components $M_{1}^{+}$ are periodic functions with respect to $x$,
i.e.,

$M_{1}^{\pm}(x,y)=\tilde{M}_{1}^{\pm}(y)e^{i\omega x},\quad\omega\in R,$ (22)

then from relations (21) we get

$\displaystyle\tilde{M}_{1}^{+}(y)$
$\displaystyle=M_{1}^{+}\big{|}_{y=\frac{1}{2}+0}e^{-|\omega|(y-\frac{1}{2})},$
$\displaystyle\tilde{M}_{1}^{-}(y)$
$\displaystyle=M_{1}^{-}\big{|}_{y=-\frac{1}{2}-0}e^{|\omega|(y+\frac{1}{2})}.$

Thereby, due to formulas (20) the components $L_{1}^{\pm}$ and $1+M_{1}^{\pm}$
of the tension vector $\vec{H}$ of the magnetic field are defined in the
domains $S_{1}^{\pm}$, and $L_{1}^{\pm}=0$.

##  2 Periodic perturbations. Linearized problem. Formulation of main results

We will be looking for solutions of system (17) in the special form

$\vec{U}(t,x,y)=\vec{\tilde{U}}(y)\exp\\{\lambda t+i\omega x\\},$ (23)

where $\lambda=\eta+i\xi$, $\xi,\omega\in R^{1}$,
$\vec{U}=(u,v,\alpha_{11},\alpha_{12},\alpha_{22},\Omega,Z,L,M)^{T}$. We will
below drop tildes. Then, it follows from (17) that

$\displaystyle u^{{}^{\prime}}$
$\displaystyle=\frac{\hat{\alpha}_{12}^{{}^{\prime}}-i\omega\hat{\alpha}_{1}}{\hat{\alpha}_{2}}v+\frac{\lambda+i\omega\hat{u}+R_{44}}{\hat{\alpha}_{2}}\alpha_{12}+\frac{R_{43}}{\hat{\alpha}_{2}}\alpha_{11}+\frac{R_{45}}{\hat{\alpha}_{2}}\alpha_{22}+\frac{r_{12}}{\hat{\alpha}_{2}}Z,$
$\displaystyle v^{{}^{\prime}}$ $\displaystyle=-i\omega u,$
$\displaystyle(\lambda$
$\displaystyle+i\omega\hat{u}-\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}}+R_{33})\alpha_{11}-2\hat{\alpha}_{1}i\omega
u-\left(\frac{2\hat{\alpha}_{12}(\hat{\alpha}_{12}^{{}^{\prime}}-i\omega\hat{\alpha}_{1})}{\hat{\alpha}_{2}}-\hat{\alpha}_{11}^{{}^{\prime}}\right)v-$
$\displaystyle-\left(\frac{2\hat{\alpha}_{12}(\lambda+i\omega\hat{u}+R_{44})}{\hat{\alpha}_{2}}-R_{34}\right)\alpha_{12}-\left(\frac{2\hat{\alpha}_{12}R_{45}}{\hat{\alpha}_{2}}-R_{35}\right)\alpha_{22}-$
$\displaystyle-\left(\frac{2\hat{\alpha}_{12}r_{12}}{\hat{\alpha}_{2}}-r_{11}\right)Z=0,$
$\displaystyle\alpha_{12}^{{}^{\prime}}$
$\displaystyle=\frac{\lambda+i\omega\hat{u}}{\hat{Z}}u+\frac{\hat{u}^{{}^{\prime}}}{\hat{Z}}v-\frac{\hat{Z}^{{}^{\prime}}}{\hat{Z}}\alpha_{12}+\frac{i\omega}{\hat{Z}}\Omega-i\omega\alpha_{11}+i\omega\alpha_{22}-\frac{i\omega\hat{\alpha}_{11}+\hat{\alpha}_{12}^{{}^{\prime}}}{\hat{Z}}Z-$
$\displaystyle-\frac{\hat{\alpha}_{12}}{\hat{Z}}Z^{{}^{\prime}}+\frac{\sigma_{m}(1+\hat{\lambda})}{\hat{Z}}(i\omega
M-L^{{}^{\prime}})-\frac{\sigma_{m}\hat{L}^{{}^{\prime}}}{\hat{Z}}M,$
$\displaystyle(\lambda+i\omega\hat{u}+R_{55})\alpha_{22}-(2\hat{\alpha}_{12}i\omega-\hat{\alpha}_{22}^{{}^{\prime}})v+2\hat{\alpha}_{2}i\omega
u+R_{53}\alpha_{11}+R_{54}\alpha_{12}=0,$ $\displaystyle\Omega^{{}^{\prime}}$
$\displaystyle=-(\lambda+i\omega\hat{u})v+i\omega\hat{Z}\alpha_{12}+(\hat{\alpha}_{12}i\omega+\hat{\alpha}_{22}^{{}^{\prime}}+Gr)Z+\hat{\alpha}_{22}Z^{{}^{\prime}}+$
$\displaystyle+\sigma_{m}\hat{L}(i\omega
M-L^{{}^{\prime}})-\sigma_{m}\hat{L}^{{}^{\prime}}L,$
$\displaystyle\frac{1}{Pr}Z^{{}^{\prime\prime}}$
$\displaystyle=(\frac{q_{2}}{Pr}-\frac{A_{r}}{Pr}(\hat{u}^{{}^{\prime}}\hat{a}_{12}+\hat{Z}\frac{\hat{a}_{12}}{\hat{\alpha}_{2}}r_{12})-\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\frac{r_{12}}{\hat{\alpha}_{2}})Z-$
$\displaystyle-u(\frac{A_{r}}{Pr}\hat{Z}(\hat{a}_{11}i\omega-\hat{a}_{22})i\omega+\frac{A_{m}}{Pr}\sigma_{m}(\hat{L}^{2}-(1+\hat{\lambda})^{2})i\omega)+$
$\displaystyle+v\bigg{(}\hat{Z}^{{}^{\prime}}-\frac{A_{r}}{Pr}\hat{Z}(\hat{a}_{22}i\omega+\hat{a}_{12}\frac{\hat{\alpha}_{12}-i\omega\hat{\alpha}_{1}}{\hat{\alpha}_{2}})-\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\times$
$\displaystyle\times(i\omega+\frac{\hat{\alpha}_{12}^{{}^{\prime}}-i\omega\hat{\alpha}_{1}}{\hat{\alpha}_{2}})\bigg{)}-\frac{A_{m}}{Pr}\sigma_{m}(1+\hat{\lambda})\hat{u}^{{}^{\prime}}L-\frac{A_{m}}{Pr}\sigma_{m}\hat{u}^{{}^{\prime}}\hat{L}M-$
$\displaystyle-\alpha_{12}\bigg{(}\frac{A_{r}}{Pr}\hat{Z}\hat{a}_{12}+\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\frac{\lambda+i\omega\hat{u}+R_{44}}{\hat{\alpha}_{2}}\bigg{)}-$
$\displaystyle-\alpha_{11}\big{(}\frac{A_{r}}{Pr}\hat{Z}\hat{a}_{12}+\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\big{)}\frac{R_{43}}{\hat{\alpha}_{2}}-\alpha_{22}\big{(}\frac{A_{r}}{Pr}\hat{Z}\hat{a}_{12}+\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\big{)}\frac{R_{45}}{\hat{\alpha}_{2}},$
$\displaystyle L^{{}^{\prime\prime}}$
$\displaystyle=q_{3}L-\frac{i\omega\hat{L}}{b_{m}}u+v\bigg{(}\frac{\hat{L}^{{}^{\prime}}}{b_{m}}-\frac{1+\lambda}{b_{m}}\frac{(\hat{\alpha}_{12}^{{}^{\prime}}-i\omega\hat{\alpha}_{1})}{\hat{\alpha}_{2}}\bigg{)}-\frac{1+\hat{\lambda}}{b_{m}}\frac{\lambda+i\omega\hat{u}+R_{44}}{\hat{\alpha}_{2}}\alpha_{12}-$
$\displaystyle-\frac{1+\hat{\lambda}}{b_{m}}\frac{R_{43}}{\hat{\alpha}_{2}}\alpha_{11}-\frac{1+\hat{\lambda}}{b_{m}}\frac{r_{12}}{\hat{\alpha}_{2}}Z-\frac{\hat{u}^{{}^{\prime}}}{b_{m}}M-\frac{1+\hat{\lambda}}{b_{m}}\frac{R_{45}}{\hat{\alpha}_{2}}\alpha_{22},$
(24) $\displaystyle M^{{}^{\prime\prime}}$
$\displaystyle=q_{3}M+\frac{1+\hat{\lambda}}{b_{m}}i\omega
u-\frac{i\omega\hat{L}}{b_{m}}v,$

where $q_{2}=Pr(\lambda+i\omega\hat{u})+\omega^{2}$,
$q_{3}=\frac{\lambda+i\omega\hat{u}+b_{m}\omega^{2}}{b_{m}}$.

The following statements are true.

######  Theorem 2.1.

If problem (17), (18) has a solution in form (23) (the parameter $\omega$ is
constant!), then we have the following asymptotic representation for
$\lambda$:

$\lambda_{k}=\left[\int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi\right]^{-1}\bigg{(}\int_{-\frac{1}{2}}^{\frac{1}{2}}-\frac{1}{2}\bigg{[}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\bigg{(}\frac{i\omega\hat{u}+R_{44}}{\hat{\alpha}_{2}}+\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}^{2}}\bigg{)}+\\\
+i\omega\hat{u}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}+\frac{\hat{\alpha}_{12}}{\hat{Z}}\times\\\
\times(A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda}))\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}+\frac{\sigma_{m}(1+\hat{\lambda})^{2}}{b_{m}\sqrt{\hat{Z}\hat{\alpha}_{2}}}\bigg{]}d\xi+k\pi
i\bigg{)}+O(\frac{1}{k}),\quad k\to\infty,$ (25)

where we use $O$ as a big O notation.

From representation (25) we obtain a necessary condition for the asymptotic
stability of the Poiseuille-type flow described in Sect.

######  Theorem 2.2.

For the asymptotic stability of the Poiseuille-type flow it is necessary that
the following inequality holds true:

$\int_{-\frac{1}{2}}^{\frac{1}{2}}\bigg{[}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\hat{\bar{\chi}}_{0}\left(\frac{1}{\hat{\alpha}_{2}}\big{(}W^{-1}+\frac{k+2\beta}{3}(\hat{a}_{11}+\hat{a}_{22})\big{)}+\hat{\bar{\chi}}_{0}\frac{2\hat{\alpha}_{12}}{\hat{\alpha}_{2}^{2}}\hat{a}_{12}\frac{k+2\beta}{3}\right)+\\\
+\frac{\hat{\alpha}_{12}}{\hat{Z}}\big{(}A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda})\big{)}\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}+\frac{\sigma_{m}(1+\hat{\lambda})^{2}}{b_{m}\sqrt{\hat{Z}\hat{\alpha}_{2}}}\bigg{]}d\xi>0.$
(26)

##  3 Proof of theorems 1 and 2

Due to the third and fifth equations of system (24) we can get representations
for the components $\alpha_{11}$, $\alpha_{22}$:

$\displaystyle\alpha_{11}$
$\displaystyle=\frac{2\hat{\alpha}_{12}}{\hat{\alpha}_{2}}\alpha_{12}+\alpha_{11}^{*},$
(27) $\displaystyle\alpha_{22}$ $\displaystyle=\alpha_{22}^{*},$

where the functions $\alpha_{11}^{*}$, $\alpha_{22}^{*}$ are expressed through
$u,v,Z,\alpha_{12}$ with coefficients proportional to $\frac{1}{\lambda}$. We
will further show that such values do not influence on the first term in the
asymptotic representation for the eigenvalues $\lambda$ as
$|\lambda|\to\infty$ and, hence, they can be omitted.

Using relations (27) and denoting

$\tilde{Y}=(\tilde{y}_{1},\tilde{y}_{2},\tilde{y}_{3},\tilde{y}_{4},\tilde{y}_{5},\tilde{y}_{6},\tilde{y}_{7},\tilde{y}_{8},\tilde{y}_{9},\tilde{y}_{10})^{T}=(u,v,\alpha_{12},\Omega,Z,Z^{{}^{\prime}},L,L^{{}^{\prime}},M,M^{{}^{\prime}})^{T},$

system (24) can be rewritten in the following form:

$\displaystyle\tilde{y}_{1}^{{}^{\prime}}$
$\displaystyle=\frac{\hat{\alpha}_{12}^{{}^{\prime}}-i\omega\hat{\alpha}_{1}}{\hat{\alpha}_{2}}\tilde{y}_{2}+\frac{\lambda+i\omega\hat{u}+R_{44}+\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}}}{\hat{\alpha}_{2}}\tilde{y}_{3}+\frac{r_{12}}{\hat{\alpha}_{2}}\tilde{y}_{5},$
(28) $\displaystyle\tilde{y}_{2}^{{}^{\prime}}$
$\displaystyle=-i\omega\tilde{y}_{1},$
$\displaystyle\tilde{y}_{3}^{{}^{\prime}}$
$\displaystyle=\frac{\lambda+i\omega\hat{u}}{\hat{Z}}\tilde{y}_{1}+\frac{\hat{u}^{{}^{\prime}}}{\hat{Z}}\tilde{y}_{2}-(\frac{\hat{Z}^{{}^{\prime}}}{\hat{Z}}+\frac{i\omega
2\hat{\alpha}_{12}}{\hat{\alpha}_{2}})\tilde{y}_{3}+\frac{i\omega}{\hat{Z}}\tilde{y}_{4}-\frac{i\omega\hat{\alpha}_{11}+\hat{\alpha}_{12}^{{}^{\prime}}}{\hat{Z}}\tilde{y}_{5}-$
$\displaystyle-\frac{\hat{\alpha}_{12}}{\hat{Z}}\tilde{y}_{6}+\left(\frac{\sigma_{m}(1+\hat{\lambda})}{\hat{Z}}i\omega-\frac{\sigma_{m}\hat{L}^{{}^{\prime}}}{\hat{Z}}\right)\tilde{y}_{9}-\frac{\sigma_{m}(1+\hat{\lambda})}{\hat{Z}}\tilde{y}_{8},$
$\displaystyle\tilde{y}_{4}^{{}^{\prime}}$
$\displaystyle=-(\lambda+i\omega\hat{u})\tilde{y}_{2}+i\omega\hat{Z}\tilde{y}_{3}+(\hat{\alpha}_{12}i\omega+\hat{\alpha}_{22}^{{}^{\prime}}+Gr)\tilde{y}_{5}-\sigma_{m}\hat{L}^{{}^{\prime}}\tilde{y}_{7}-\sigma_{m}\hat{L}\tilde{y}_{8}+$
$\displaystyle+\sigma_{m}Li\omega\tilde{y}_{9}+\hat{\alpha}_{22}\tilde{y}_{6},$
$\displaystyle\tilde{y}_{5}^{{}^{\prime}}$ $\displaystyle=\tilde{y}_{6},$
$\displaystyle\frac{1}{Pr}\tilde{y}_{6}^{{}^{\prime}}$
$\displaystyle=(\lambda+i\omega\hat{u}+\frac{\omega^{2}}{Pr}-\frac{A_{r}}{Pr}(\hat{u}^{{}^{\prime}}\hat{a}_{12}+\hat{Z}\frac{\hat{a}_{12}}{\hat{\alpha}_{2}}r_{12})-\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\frac{r_{12}}{\hat{\alpha}_{2}})\tilde{y}_{5}-$
$\displaystyle-\tilde{y}_{1}(\frac{A_{r}}{Pr}\hat{Z}(\hat{a}_{11}-\hat{a}_{22})i\omega+\frac{A_{m}}{Pr}\sigma_{m}(\hat{L}^{2}-(1+\hat{\lambda})^{2})i\omega)+$
$\displaystyle+\tilde{y}_{2}\bigg{(}\hat{Z}^{{}^{\prime}}-\frac{A_{r}}{Pr}\hat{Z}\hat{\alpha}_{12}(i\omega+\frac{\hat{\alpha}_{12}^{{}^{\prime}}-i\omega\hat{\alpha}_{1}}{\hat{\alpha}_{2}})-\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\times$
$\displaystyle\times(i\omega+\frac{\hat{\alpha}_{12}^{{}^{\prime}}-i\omega\hat{\alpha}_{1}}{\hat{\alpha}_{2}})\bigg{)}-\frac{A_{m}}{Pr}\sigma_{m}(1+\hat{\lambda})\hat{u}^{{}^{\prime}}\tilde{y}_{7}-\frac{A_{m}}{Pr}\sigma_{m}\hat{u}^{{}^{\prime}}\hat{L}\tilde{y}_{9}-$
$\displaystyle-\bigg{(}\frac{A_{r}}{Pr}\hat{Z}\hat{a}_{12}+\frac{A_{m}}{Pr}\sigma_{m}\hat{L}(1+\hat{\lambda})\frac{\lambda+i\omega\hat{u}+R_{44}+\frac{2\hat{\alpha}_{12}R_{53}}{\hat{\alpha}_{2}}}{\hat{\alpha}_{2}}\bigg{)}\tilde{y}_{3},$
$\displaystyle\tilde{y}_{7}^{{}^{\prime}}$ $\displaystyle=\tilde{y}_{8}$
$\displaystyle\tilde{y}_{8}^{{}^{\prime}}$
$\displaystyle=(\frac{\lambda+i\omega\hat{u}}{b_{m}}+\omega^{2})\tilde{y}_{7}-\frac{i\omega\hat{L}}{b_{m}}\tilde{y}_{1}+\bigg{(}\frac{\hat{L}^{{}^{\prime}}}{b_{m}}-\frac{1+\lambda}{b_{m}}(\hat{\alpha}_{12}-i\omega\hat{\alpha}_{1})\bigg{)}\tilde{y}_{2}-$
$\displaystyle-\frac{1+\hat{\lambda}}{b_{m}}\frac{\lambda+i\omega\hat{u}+R_{44}+\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}}}{\hat{\alpha}_{2}}\tilde{y}_{3}-\frac{1+\hat{\lambda}}{b_{m}}{r_{12}}{\hat{\alpha}_{2}}\tilde{y}_{5}-\frac{\hat{u}^{{}^{\prime}}}{b_{m}}\tilde{y}_{9},$
$\displaystyle\tilde{y}_{9}^{{}^{\prime}}$ $\displaystyle=\tilde{y}_{10},$
$\displaystyle\tilde{y}_{10}^{{}^{\prime}}$
$\displaystyle=\left(\frac{\lambda+i\omega\hat{u}}{b_{m}}+\omega^{2}\right)\tilde{y}_{9}+\frac{1+\hat{\lambda}}{b_{m}}i\omega\tilde{y}_{1}-\frac{i\omega\hat{L}}{b_{m}}\tilde{y}_{2}.$

Let us rewrite system (28) in the more compact form

$\tilde{Y}^{{}^{\prime}}=(\lambda D+P)\tilde{Y},$ (29)

where the matrix $D$ is rather sparse:

$\displaystyle d_{13}$
$\displaystyle=\frac{1}{\hat{\alpha}_{2}},\,d_{31}=\frac{1}{\hat{Z}},\,d_{42}=-1,\,d_{63}=-\frac{1}{\hat{\alpha}_{2}}(A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda})),$
$\displaystyle d_{65}$
$\displaystyle=Pr,\,d_{83}=-\frac{1+\hat{\lambda}}{b_{m}\hat{\alpha}_{2}},\,d_{87}=\frac{1}{b_{m}},\,d_{10,9}=\frac{1}{b_{m}},$

all other elements equal zero.
We can make the change

$\tilde{Y}=TY,$ (30)

where the elements of the matrix $T$ read:

$\displaystyle t_{11}$
$\displaystyle=-t_{12}=-\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}},\,t_{71}=-t_{72}=(A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda}))\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}},$
$\displaystyle t_{81}$
$\displaystyle=-t_{82}=\frac{1+\hat{\lambda}}{b_{m}}\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}},\,t_{58}=\frac{1}{Pr},\,t_{7,10}=b_{m},$
$\displaystyle t_{31}$
$\displaystyle=t_{32}=t_{45}=t_{67}=t_{89}=t_{93}=t_{10,4}=-t_{26}=1,$

and other elements are zero.
This enables one to write down the matrix $D$ in the upper Jordan form:

$T^{-1}DT=W=diag\bigg{\\{}-\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}},\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}},0,0,block\begin{pmatrix}0&1\\\
0&0\end{pmatrix},block\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\\\
block\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\bigg{\\}}.$ (31)

Then, system (29) is transformed in the following way:

$Y^{{}^{\prime}}=(\lambda W+C)Y,$ (32)

where $C=T^{-1}PT-T^{-1}T^{{}^{\prime}}$.

Lets write only two elements of the matrix $C$ which will be used below:

$\displaystyle c_{11}$
$\displaystyle=-\frac{1}{2}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\left(\frac{i\omega\hat{u}+R_{44}+\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}}}{\hat{\alpha}_{2}}\right)+\frac{1}{2}\bigg{(}-i\omega\hat{u}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}-\frac{\hat{Z}^{{}^{\prime}}}{\hat{Z}}-\frac{i\omega\hat{\alpha}_{12}}{\hat{\alpha}_{2}}-\frac{\hat{\alpha}_{12}}{\hat{Z}}\times$
(33)
$\displaystyle\times(A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda}))\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}-\frac{\sigma_{m}(1+\hat{\lambda})^{2}}{b_{m}\sqrt{\hat{Z}\hat{\alpha}_{2}}}\bigg{)}-\frac{1}{2}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\left(\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}\right)^{{}^{\prime}},$
$\displaystyle c_{22}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\left(\frac{i\omega\hat{u}+R_{44}+\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}}}{\hat{\alpha}_{2}}\right)+\frac{1}{2}\bigg{(}-i\omega\hat{u}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}-\frac{\hat{Z}^{{}^{\prime}}}{\hat{Z}}-\frac{i\omega\hat{\alpha}_{12}}{\hat{\alpha}_{2}}-\frac{\hat{\alpha}_{12}}{\hat{Z}}\times$
$\displaystyle\times(A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda}))\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}-\frac{\sigma_{m}(1+\hat{\lambda})^{2}}{b_{m}\sqrt{\hat{Z}\hat{\alpha}_{2}}}\bigg{)}-\frac{1}{2}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\left(\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}\right)^{{}^{\prime}}.$

Let us get an asymptotic representation for the fundamental matrix of system
(32). For this purpose we split the matrix $C$ into blocks corresponding to
representation (31) of the matrix $W$: the first diagonal block corresponds to
the nonzero diagonal elements $-\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}$ and
$\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}$ whereas the second diagonal block
corresponds, on the contrary, to zero elements. Then, system (32) can be
written in the more convenient form

$Y^{{}^{\prime}}=\begin{pmatrix}Y_{I}\\\
Y_{II}\end{pmatrix}^{{}^{\prime}}=(\lambda W+C)\begin{pmatrix}Y_{I}\\\
Y_{II}\end{pmatrix},$ (34)

where

$C=\begin{pmatrix}C_{I}^{I}&C_{II}^{I}\\\
C_{I}^{II}&C_{II}^{II}\end{pmatrix}.$ (35)

Using splitting (35) of the matrix $C$, we get the following system for the
vector $Y_{II}$:

$(Y_{II})^{{}^{\prime}}=C_{II}^{II}Y_{II}+C_{I}^{II}Y_{I}.$ (36)

Assuming that the vector $Y_{I}$ is known, we can write down a system of
fundamental solutions associated with the homogeneous system:

$Y_{II}=\sum_{i=3}^{10}c_{i}Y_{II}^{i},$

where $c_{i}$ are arbitrary complex constants, $i=3,\dots,10$,
$Y^{i}_{II}\big{|}_{y=-\frac{1}{2}}=\begin{pmatrix}0\\\ \vdots\\\ 0\\\ 1\\\
0\\\ \vdots\\\ 0\end{pmatrix}$ is the $i$th component, and the general
solution of system (36) is

$Y_{II}=\sum_{i=3}^{10}c_{i}Y_{II}^{i}+\int_{-\frac{1}{2}}^{y}Y(y)Y^{-1}(s)C_{I}^{II}Y_{I}ds,$
(37)

where $Y(y)$ is the fundamental matrix composed by the vectors $Y_{II}^{i}$.

Due to representation (37) the system for other component $Y_{I}$ can be
written as

$Y_{I}^{{}^{\prime}}=\lambda\begin{pmatrix}-\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}&0\\\
0&\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}\end{pmatrix}Y_{I}+C_{I}^{I}Y_{I}+C_{II}^{I}\left(\sum_{i=3}^{10}c_{i}Y_{II}^{i}\right)+C_{II}^{I}\times\\\
\times\int_{-\frac{1}{2}}^{y}Y(y)Y^{-1}(s)C_{I}^{II}Y_{I}(s)ds.$ (38)

Considering $C_{II}^{I}(\sum\limits_{i=3}^{10}c_{i}Y_{II}^{i})$ as a free
term, we can find the fundamental matrix for system (38) in the following
form:

$\hat{Y}=\left(P_{0}(y)+\frac{1}{\lambda}P_{1}(y)+\frac{1}{\lambda^{2}}P_{2}(y)+\dots\right)\left(\delta_{ij}e^{\lambda\Gamma_{j}(y)}\right)+\frac{M_{0}}{\lambda^{2}}+\frac{M_{1}}{\lambda^{3}}+\dots,\,i,j=1,2,$
(39)

where $\delta_{ij}$ is the Kronecker symbol,
$\Gamma_{1}(y)=e^{-\int\limits_{-\frac{1}{2}}^{y}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}$,
$\Gamma_{2}(y)=e^{\int\limits_{-\frac{1}{2}}^{y}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}$.

######  Remark 3.1.

The first term in (39) is the representation obtained by G. Birkhoff [3] for
the differential equation

$Y_{I}^{{}^{\prime}}=\lambda\begin{pmatrix}-\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}&0\\\
0&\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}\end{pmatrix}Y_{I}+C_{I}^{I}Y_{I}.$

Substituting the matrix $\hat{Y}$ into equation (38), we get the following
correlation (the term $C_{II}^{I}(\sum\limits_{i=3}^{10}c_{i}Y_{II}^{i})$ is
omitted):

$\displaystyle\left(P_{0}^{{}^{\prime}}(y)+\frac{1}{\lambda}P_{1}^{{}^{\prime}}(y)+\frac{1}{\lambda^{2}}P_{2}^{{}^{\prime}}(y)+\dots\right)\left(\delta_{ij}e^{\lambda\Gamma_{j}(y)}\right)_{i,j=1,2}+$
(40)
$\displaystyle+\lambda\left(P_{0}(y)+\frac{1}{\lambda}P_{1}(y)+\frac{1}{\lambda^{2}}P_{2}(y)+\dots\right)\Lambda\left(\delta_{ij}e^{\lambda\Gamma_{j}(y)}\right)_{i,j=1,2}+$
$\displaystyle+\frac{M_{0}^{{}^{\prime}}(y)}{\lambda^{2}}+\frac{M_{1}^{{}^{\prime}}(y)}{\lambda^{3}}+\dots=$
$\displaystyle=\lambda\Lambda\left(P_{0}(y)+\frac{1}{\lambda}P_{1}(y)+\frac{1}{\lambda^{2}}P_{2}(y)+\dots\right)\left(\delta_{ij}e^{\lambda\Gamma_{j}(y)}\right)_{i,j=1,2}+$
$\displaystyle+\lambda\Lambda\left(\frac{M_{0}(y)}{\lambda^{2}}+\frac{M_{1}(y)}{\lambda^{3}}+\dots\right)+$
$\displaystyle+C_{I}^{I}\left(P_{0}(y)+\frac{1}{\lambda}P_{1}(y)+\frac{1}{\lambda^{2}}P_{2}(y)+\dots\right)\left(\delta_{ij}e^{\lambda\Gamma_{j}(y)}\right)_{i,j=1,2}+$
$\displaystyle+C_{I}^{I}\left(\frac{M_{0}(y)}{\lambda^{2}}+\frac{M_{1}(y)}{\lambda^{3}}+\dots\right)+$
$\displaystyle+C_{II}^{I}\int_{-\frac{1}{2}}^{y}Y(y)Y^{-1}(s)C_{I}^{II}\bigg{[}\left(P_{0}(s)+\frac{1}{\lambda}P_{1}(s)+\frac{1}{\lambda^{2}}P_{2}(s)+\dots\right)\times$
$\displaystyle\times\left(\delta_{ij}e^{\lambda\Gamma_{j}(s)}\right)_{i,j=1,2}+\frac{M_{0}(s)}{\lambda^{2}}+\frac{M_{1}(s)}{\lambda^{3}}+\dots\bigg{]}ds,\quad\Lambda=diag\\{-\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}},\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}\\}.$

Comparing the coefficients by the same powers of $\lambda$ which may contain
the matrix $(\delta_{ij}e^{\lambda\Gamma_{j}(s)})_{i,j=1,2}$ or be independent
on it and integrating by parts a needed number of times, we, in particular,
get

$P_{0}\Lambda=\Lambda P_{0},$ (41) $P_{0}^{{}^{\prime}}+P_{1}\Lambda=\Lambda
P_{1}+C_{I}^{I}P_{0},$ (42) $P_{1}^{{}^{\prime}}+P_{2}\Lambda=\Lambda
P_{2}+C_{I}^{I}P_{1}+C_{II}^{I}C_{I}^{II}P_{0}\Lambda^{-1},$ (43) $\Lambda
M_{0}=C_{II}^{I}YC_{I}^{II}(-\frac{1}{2}).$ (44)

Then, it follows from equality (41) that $P_{0}(y)$ is a diagonal matrix,

$P_{0}(y)=\begin{pmatrix}p_{1}(y)&0\\\ 0&p_{2}(y)\end{pmatrix},$

and equality (42) for the case of diagonal elements gives the Cauchy problems

$p_{i}^{{}^{\prime}}=c_{ii}^{I}p_{i},\quad p_{i}(-\frac{1}{2})=1,\quad i=1,2,$
(45)

where $c_{ii}^{I}$ are diagonal elements of the matrix $C_{I}^{I}$ (see
formulas (33)).

Solving problems (45) gives us the functions $p_{i}(y)$:
$p_{i}(y)=e^{\int\limits_{-\frac{1}{2}}^{y}c_{ii}^{I}(\xi)d\xi}$, $i=1,2$ (the
top index $I$ will be omitted below).

Then, equality (42) enables one to determine the non-diagonal elements of the
matrix $P_{1}(y)$ which, in turn, enables one to find the diagonal elements
from equality (43). By finite induction we can find all the matrices
$P_{i}(y)$, $i=2,\dots$. Equalities analogous to (44) enables one to determine
the matrices $M_{i}$, $i=0,1,\dots$

Due to the method of variation of constants the presence of the free term
$C_{II}^{I}(\sum\limits_{i=3}^{10}c_{i}Y_{II}^{i})$ results in representation
(39) for the additional terms containing as multipliers the powers of
$\frac{1}{\lambda}$: $\frac{1}{\lambda^{k}}$, $k=1,2,\dots$. It becomes then
clear that such terms do not influence on the main term in the asymptotic
representation of the spectrum. This means that we can use only the main term
in representation (39):

$Y=P_{0}(y)(\delta_{ij}e^{\lambda\Gamma_{j}(y)})_{i,j=1,2}.$ (46)

Recalling the fundamental matrix of equation (34) and taking into account
(46), we get the main term in the asymptotic representation of the fundamental
matrix $W_{Y}$ of system (34):

$W_{Y}=\begin{pmatrix}e^{-\lambda\int\limits_{-\frac{1}{2}}^{y}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}p_{1}(y)&0&0&\dots&0\\\
0&e^{\lambda\int\limits_{-\frac{1}{2}}^{y}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}p_{2}(y)&0&\dots&0\\\
0&0&y_{3}^{3}&\dots&y_{10}^{3}\\\ \dots&\dots&\dots&\dots&\dots\\\
0&0&y_{3}^{10}&\dots&y_{10}^{10}\\\ \end{pmatrix},$ (47)

where $y_{i}^{j}$, $i,j=3,\dots,10$ are components of the fundamental system
for equation (36) composed by the columns of the matrix $Y(y)$.

######  Remark 3.2.

In this work we do not give a justification of the fact that representation
(39) is really asymptotic series as well as we do not justify the described
representation of the fundamental matrix $W_{Y}$ (more precisely, its “full
variant”). This is a subject for future research. We only note that this fact
was established by G. Birkhoff [3] for equation (40) (when there are no terms
with coefficients $M_{0}$, $M_{1}$, $\dots$ in the right-hand side of equality
(39), i.e., the part of the series corresponding to the integral term in
equation (38)), and in each half-plane $Re\lambda>0$ and $Re\lambda<0$ the
asymptotic series are different from each other [12].

By the way, considering the integral term as a free term and using the method
of variation of constants is another way of finding the matrices $M_{0}$,
$M_{1}$, $\dots$ (the idea of getting fundamental matrices by the method of
variation of constants is described in [13]).

Recalling the boundary conditions for $y=\pm\frac{1}{2}$, we can note that
relations (18) after change (30) of the variable $\tilde{Y}$ are transformed
in the following way:

$y_{1}=y_{2},\,y_{6}=0,\,y_{8}=0,\,y_{10}=0,\,y_{4}=0,\quad\mbox{
}y=\pm\frac{1}{2},\,t>0.$ (48)

Or, taking into account representation (47), they can be written as the
equality

$det\begin{pmatrix}L\\\ LW_{Y}(\frac{1}{2})\end{pmatrix}=0,$ (49)

where

$L=\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&-1&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&1&0&0&0&0&0&0&0\end{pmatrix}.$

After elementary transforms of the determinant and the use of the Laplace
theorem about the representation of the determinant as multiplications of its
minors we see that (49) is equivalent to the equality

$\displaystyle
e^{\lambda\int\limits_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}p_{2}(\frac{1}{2})-e^{-\lambda\int\limits_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}p_{1}(\frac{1}{2})=0,\,\mbox{
}$ (50) $\displaystyle
e^{\lambda\int\limits_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}e^{\int\limits_{-\frac{1}{2}}^{\frac{1}{2}}c_{22}(\xi)d\xi}-e^{-\lambda\int\limits_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi}e^{\int\limits_{-\frac{1}{2}}^{\frac{1}{2}}c_{11}(\xi)d\xi}=0.$

Recalling formulas (33), we get the spectrum representation

$\lambda_{k}=\left[\int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}d\xi\right]^{-1}\bigg{(}\int_{-\frac{1}{2}}^{\frac{1}{2}}-\frac{1}{2}\bigg{[}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\bigg{(}\frac{i\omega\hat{u}+R_{44}}{\hat{\alpha}_{2}}+\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}^{2}}\bigg{)}+\\\
+i\omega\hat{u}\frac{1}{\sqrt{\hat{Z}\hat{\alpha}_{2}}}+\frac{\hat{\alpha}_{12}}{\hat{Z}}\times\\\
\times(A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda}))\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}+\frac{\sigma_{m}(1+\hat{\lambda})^{2}}{b_{m}\sqrt{\hat{Z}\hat{\alpha}_{2}}}\bigg{]}d\xi+k\pi
i\bigg{)}+O(\frac{1}{k}),\quad k\to\infty,$ (51)

where the symbol $O$ denotes a big O.
The proof of Theorem 1 is thus complete.

######  Remark 3.3.

Using representation (39), we can get an asymptotic representation of
$\lambda_{k}$ with an arbitrary order of accuracy defined by the powers
$\frac{1}{k}$ (see also [3]).

Now, as a consequence of representation (51), we get the following result. If
the Poiseuille-type flow described in Sect. 1 is asymptotically stable, then
the following inequality necessarily hold:

$Re\lambda_{k}=\int_{-\frac{1}{2}}^{\frac{1}{2}}\bigg{[}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}(\frac{R_{44}}{\hat{\alpha}_{2}}+\frac{2\hat{\alpha}_{12}R_{43}}{\hat{\alpha}_{2}^{2}})+\\\
+\frac{\hat{\alpha}_{12}}{\hat{Z}}\big{(}A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda})\big{)}\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}+\frac{\sigma_{m}(1+\hat{\lambda})^{2}}{b_{m}\sqrt{\hat{Z}\hat{\alpha}_{2}}}\bigg{]}d\xi>0,$

or taking into account formula (19),

$Re\lambda_{k}=\int_{-\frac{1}{2}}^{\frac{1}{2}}\bigg{[}\sqrt{\frac{\hat{\alpha}_{2}}{\hat{Z}}}\hat{\bar{\chi}}_{0}\left(\frac{1}{\hat{\alpha}_{2}}\big{(}W^{-1}+\frac{k+2\beta}{3}(\hat{a}_{11}+\hat{a}_{22})\big{)}+\hat{\bar{\chi}}_{0}\frac{2\hat{\alpha}_{12}}{\hat{\alpha}_{2}^{2}}\hat{a}_{12}\frac{k+2\beta}{3}\right)+\\\
+\frac{\hat{\alpha}_{12}}{\hat{Z}}\big{(}A_{r}\hat{Z}\hat{a}_{12}+A_{m}\sigma_{m}\hat{L}(1+\hat{\lambda})\big{)}\sqrt{\frac{\hat{Z}}{\hat{\alpha}_{2}}}+\frac{\sigma_{m}(1+\hat{\lambda})^{2}}{b_{m}\sqrt{\hat{Z}\hat{\alpha}_{2}}}\bigg{]}d\xi>0.$
(52)

The proof of Theorem 2 is complete.

The authors are grateful to A.V. Yegitov for his help in the preparation of
the manuscript of the paper.
This work is supported by the Russian Foundation for Basic Research (the grant
numbers 17-01-00791 and 19-01-00261 ).

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#  LingMess: Linguistically Informed Multi Expert Scorers
for Coreference Resolution

Shon Otmazgin1 Arie Cattan1 Yoav Goldberg1,2
1Computer Science Department, Bar Ilan University
2Allen Institute for Artificial Intelligence
<EMAIL_ADDRESS>

###### Abstract

Current state-of-the-art coreference systems are based on a single pairwise
scoring component, which assigns to each pair of mention spans a score
reflecting their tendency to corefer to each other. We observe that different
kinds of mention pairs require different information sources to assess their
score. We present LingMess, a linguistically motivated categorization of
mention-pairs into 6 types of coreference decisions and learn a dedicated
trainable scoring function for each category. This significantly improves the
accuracy of the pairwise scorer as well as of the overall coreference
performance on the English Ontonotes coreference corpus and 5 additional
datasets.111The codebase to train and run LingMess is available in
https://github.com/shon-otmazgin/lingmess-coref. Also, our recent F-coref
Python package (Otmazgin et al., 2022) includes a simple and efficient
implementation of LingMess in https://github.com/shon-otmazgin/fastcoref.

##  1 Introduction

Coreference resolution is the task of clustering textual mentions that refer
to the same discourse entity. This fundamental task requires many decisions.
In this work, we argue that different _kinds_ of decisions involve different
challenges. To illustrate that, consider the following text:

_“ Lionel Messi has won a record seven Ballon d’Or awards. He signed for Paris
Saint-Germain in August 2021. “I would like to thank my family”, said the
Argentinian footballer. Messi holds the records for most goals in La Liga”_

To correctly identify that the pronoun “He” refers to “Lionel Messi”, models
need to model the discourse, while linking “my” to “I” may rely more heavily
on morphological agreement. Likewise, linking “the Argentinian footballer” to
“Lionel Messi” requires world knowledge, while linking “Messi” to “Lionel
Messi” may be achieved by simple lexical heuristics.

Indeed, pre-neural coreference resolution works often considered the types of
a mention-pair, either by incorporating this information as model features, or
by tailoring specific rules or specific models for each mention pair (see
related work section).

However, neural-network based coreference models are all based on a single
pairwise scorer that is shared for all mention pairs, regardless of the
different challenges that needs to be addressed by each pair type (Lee et al.,
2017, 2018; Joshi et al., 2019; Kantor and Globerson, 2019; Joshi et al.,
2020; Xu and Choi, 2020; Xia et al., 2020; Toshniwal et al., 2020;
Thirukovalluru et al., 2021; Kirstain et al., 2021; Dobrovolskii, 2021).

Figure 1: Architecture of our multi expert model. Given two spans _“Lionel
Messi”_ and _“He”_ , we sum four scores: individual mention scores (black),
$f_{m}(\emph{``Lionel Messi''})$, $f_{m}(\emph{``He''})$, and pairwise scores,
shared antecedent score (white) $f_{a}(\emph{``Lionel Messi''},\emph{``He''})$
and the relevant “expert” score (blue) $f_{a}^{\textsc{Pron-
Ent}}(\emph{``Lionel Messi''},\emph{``He''})$.

Category |  Co-referring example |  Non Co-referring example
---|---|---
Pron-Pron-C |  _A couple of my law clerks were going to … and I was afraid I was going to…_ |  _The Lord God said to my Lord: “Sit by me at my right side , and I will put your enemies …_ ”
Pron-Pron-NC |  _“ I made a similar line and I produced it cheaper” , he says._ |  _She is my Goddess …_
Ent-Pron |  _Spain, Argentina, Thailand and Indonesia were doing too little to prevent … across their borders._ |  _Tonight, to kick off the effort, CNN will premiere its first prime - time newscast in years._
Match |  _… says Paul Amos, CNN executive vice president for programming. Accordingly, CNN is …_ |  _Hertz and Avis can not benefit Budget’s programs, ” said Bob Wilson, Budget’s vice president …_
Contains |  _He reportedly showed DeLay a videotape that made him weep. Tom DeLay then …_ |  _Give SEC authority to halt securities trading, (also opposed by new SEC chairman) …_
Other |  _They also saw the two men who were standing with him. When Moses and Elijah were leaving …_ |  _The company is already working on its own programming … the newspaper said._

Table 1: Example of each category, taken from Ontonotes development set. We
define the categories of mention pairs as follows. Pron-Pron-C: compatible
pronouns based on their attributes such as gender, number and animacy (see
Appendix C for more details), Pron-Pron-NC: incompatible pronouns, Ent-Pron: a
pronoun and another span, Match: non-pronoun spans with the same content
words, Contains: one contains the content words of the other, Other: all other
pairs. Content words exclude stop words, see Appendix C for the full list of
stop words.

In this work, we suggest that modeling different mention pairs by different
sub-models (in our case, different learned scoring functions) depending on
their types is beneficial also for neural models. We identify a set of
decisions: (a) linking compatible pronouns (Pron-Pron-C); (b) linking
incompatible pronouns (Pron-Pron-NC); (c) linking pronouns to entities (Ent-
Pron); (d) linking entities which share the exact lexical form (Match); (e)
linking entities where the lexical form of one contains the lexical form of
the other (Contains); (f) other cases (Other). Each of these classes is easy
to identify deterministically, each contains both positive and negative
instances, and each could benefit from a somewhat different decision process.
Table 1 demonstrates the classes.222More fine-grained distinctions are of
course also possible, but we leave exploration of them to future work.

We present Linguistically Informed Multi Expert Scorers (LingMess), a
coreference model which categorizes each pairwise decision into one of these
classes, and learns, in addition to a shared scoring function, also a separate
scoring function for each pair type. At inference time, for each pair of
mentions being scored, we deterministically identify the pair’s type, and use
the corresponding scoring function.333For computational efficiency on the GPU,
we find it beneficial to compute all the scoring functions and mask away the
not needed values.

Specifically, we extend the recent _s2e_ ’s model (Kirstain et al., 2021) by
adding per-category scoring, but the method is general and may work with other
coreference models as well. As illustrated in Figure 1, the final coreference
score between two spans is composed—in addition to the individual mention
scores—of two pairwise scores: a shared antecedent-compatibility score and an
“expert” antecedent compatibility score which depends on the linguistic-type
of the pair.

We show that this significantly improves the coreference performance on
Ontonotes (Pradhan et al., 2012) and 5 additional datasets. We also inspect
the performance of the model for each category separately, showing that some
classes improve more than others. This analysis provides a finer-grained
understanding of the models and points out directions for future research.

##  2 Background: the s2e Model

The _s2e_ model (Kirstain et al., 2021) achieves the current best coreference
scores among all practical neural models.444We define “practical models” as
those that require a constant number transformer-based document encodings per
passage, as opposed to a constant number of document encodings per mention.
The CorefQA model (Wu et al., 2020) achieves a substantially higher score, but
requires to run a separate BERT inference for each mention, making it highly
impractical.

Given a sequence of tokens $x_{1},\ldots,x_{n}$, each mention pair $(c,q)$ is
scored using a scoring function $F_{\textsc{s}}(c,q)$555s state for shared
described below, where $c$ is a “candidate span” and $q$ is a “query span”
which appears before $c$ in the sentence. The span encodings are based on
contextualized word embeddings obtained by a Longformer encoder, see Kirstain
et al. (2021) for details. These pairwise scores are then used to form
coreference chains (see “inference” below).

The scoring function $F_{\textsc{s}}$ is further decomposed:

$\displaystyle
F_{\textsc{s}}(c,q)=\begin{cases}f_{m}(c)+f_{m}(q)+f_{a}(c,q)&c\neq\varepsilon\\\
0&c=\varepsilon\end{cases}$

where $\varepsilon$ is the null antecedent, and $f_{m}$ and $f_{a}$ are
parameterized functions, scoring each individual span ($f_{m}$) and the
pairwise interaction ($f_{a}$).

For each possible mention $q$, the learning objective optimizes the sum of
probabilities over the true antecedent $\hat{c}$ of $q$:

$\displaystyle
L_{\textsc{s}}(q)=\log\sum_{\hat{c}\in\mathcal{C}(q)\cap\textsc{gold}(q)}P_{\textsc{s}}(\hat{c}\mid
q)$

where $\mathcal{C}(q)$ is the set of all candidate antecedents666All spans
before $q$ that passed some pruning threshold. with a null antecedent
$\varepsilon$. $\textsc{gold}(q)$ is the set of the true antecedents of $q$.
$P_{\textsc{s}}(c\mid q)$ is computed as a softmax over $F_{\textsc{s}}(c,q)$
scores for $c$ values in $\mathcal{C}(q)$:

$\displaystyle P_{\textsc{s}}(\hat{c}\mid
q)=\frac{\exp{F_{\textsc{s}}(\hat{c},q)}}{\sum\limits_{c^{\prime}\in\mathcal{C}(q)}\exp{F_{\textsc{s}}(c^{\prime},q)}}$

##  3 LingMess

Clustering coreferring entities typically involves many different phenomena,
which we argue should be addressed in a different manner. Therefore, our core
contribution is proposing to allocate a dedicated scorer $f^{t}_{a}(c,q)$ for
each phenomenon type $t$, in addition to the shared pairwise scorer
$f_{a}(c,q)$. The overall architecture of our model is shown in Figure 1.

Concretely, we extend the _s2e_ model with six additional antecedent scorers
$f_{a}^{t}$ where $t\in\\{\textsc{Pron-Pron-C},\textsc{Pron-Pron-
NC},\textsc{Ent-Pron},$ $\textsc{Match},\textsc{Contains},\textsc{Other}\\}$,
the six categories we list in Table 1.

The pairwise scoring function now becomes:

$\displaystyle F(c,q)=$
$\displaystyle\begin{cases}f_{m}(c)+f_{m}(q)+f(c,q)&c\neq\varepsilon\\\
0&c=\varepsilon\\\ \end{cases}$ $\displaystyle f(c,q)=$ $\displaystyle
f_{a}(c,q)+f^{T(c,q)}_{a}(c,q)$

where $T(c,q)$ is a deterministic, rule-based function to determine the
category $t$ of the pair $(c,q)$. The pairwise scoring function $f(c,q)$
scoring $c$ as the antecedent of $q$, is now composed of a shared scorer
$f_{a}(c,q)$ and an “expert” scorer $f^{t}_{a}(c,q)$ which differs based on
the type of the pair $c,q$. Each of the seven pairwise scoring functions
($f_{a}$ and the six $f_{a}^{t}$) is parameterized separately using its own
set of matrices. The transformer-based encoder and the mention scorer $f_{m}$
are shared between all the antecedent scorers. See Appendix A.2 for full model
architecture.

#### Training

For each span $q$, our model optimizes the objective function
$L_{\textsc{Coref}}$ over the sum of probabilities of all true antecedents of
$q$:

$\displaystyle
L_{\textsc{Coref}}(q)=\log\sum_{\hat{c}\in\mathcal{C}(q)\cap\textsc{gold}(q)}P(\hat{c}\mid
q)$

Here, $P(\hat{c}\mid q)$ is a softmax over $F(\hat{c},q)$ scores, that is our
new score function described in Figure 1.

$\displaystyle P(\hat{c}\mid
q)=\frac{\exp{F(\hat{c},q)}}{\sum\limits_{c^{\prime}\in\mathcal{C}(q)}\exp{F(c^{\prime},q)}}$

This scorer is also the one used in inference. However, this objective does
not explicitly push each category (“expert”) to specialize. For example, for
the Pron-Pron-C cases, it would be useful to explicitly train the model to
distinguish between the possible antecedents of that type only (without
regarding other antecedents), as well as to explicitly distinguish between a
pronoun antecedent and a null antecedent. To this end, we extend the training
objective by also training each “expert” separately:

$\displaystyle
L_{t}(q)=\log\sum_{\hat{c}\in\mathcal{C}_{t}(q)\cap\textsc{gold}(q)}P_{t}(\hat{c}\mid
q)$

|  | MUC |  | B3 |  | CEAF$\phi_{4}$ |  | LEA |  |
---|---|---|---|---|---|---|---|---|---|---
|  | R | P | F1 |  | R | P | F1 |  | R | P | F1 |  | R | P | F1 |  | Avg. F1
_s2e_ |  | 85.2 | 86.6 | 85.9 |  | 77.9 | 80.3 | 79.1 |  | 75.4 | 76.8 | 76.1 |  | 75.8 | 78.3 | 77.0 |  | 80.3
LingMess |  | 85.1 | 88.1 | 86.6 |  | 78.3 | 82.7 | 80.5 |  | 76.1 | 78.5 | 77.3 |  | 76.3 | 80.9 | 78.5 |  | 81.4

Table 2: Performance on the test set of the English OntoNotes 5.0 dataset. The
averaged F1 of MUC, B3, CEAF$\phi$ is the main evaluation metric. Our model
outperforms the _s2e_ model (Kirstain et al., 2021) by 1.1 CoNLL F1. The
performance gain is statistically significant according to a non-parametric
permutation test ($p<0.05$).

| _s2e_ | LingMess
---|---|---
WikiCoref (Ghaddar and Langlais, 2016) | 59.7 | 62.6
GAP (Webster et al., 2018) | 88.3 | 89.6
WinoGender (Rudinger et al., 2018) | 70.5 | 77.3
WinoBias (Zhao et al., 2018) | 84.3 | 85.1
BUG (Levy et al., 2021) | 72.2 | 74.6

Table 3: Performance on the test set of various coreference datasets. The
reported metrics are CoNLL F1 for WikiCoref, F1 for GAP and Accuracy for
WinoGender, WinoBias and BUG.

| _s2e_ |  | LingMess
---|---|---|---
| P | R | F1 |  | P | R | F1
Pron-Pron-C | 88.8 | 71.3 | 79.1 |  | 88.0 | 85.1 | 86.5
Pron-Pron-NC | 84.2 | 55.8 | 67.1 |  | 88.3 | 68.7 | 77.3
Ent-Pron | 78.8 | 68.7 | 73.4 |  | 80.4 | 69.8 | 74.7
Match | 85.6 | 90.2 | 87.8 |  | 85.3 | 93.7 | 89.3
Contains | 72.4 | 80.9 | 76.4 |  | 77.4 | 78.9 | 78.1
Other | 60.1 | 70.2 | 64.7 |  | 71.7 | 64.2 | 67.7

Table 4: Pairwise performance by category, on the test set of the English
OntoNotes 5.0 dataset. LingMess surpasses the _s2e_ model (Kirstain et al.,
2021) for most categories by a substantial margin.

$\displaystyle P_{t}(\hat{c}\mid
q)=\frac{\exp{F_{t}(\hat{c},q)}}{\sum\limits_{c^{\prime}\in\mathcal{C}_{t}(q)}\exp{F_{t}(c^{\prime},q)}}$

$\displaystyle
F_{t}(c,q)=\begin{cases}f_{m}(c)+f_{m}(q)+f^{t}_{a}(c,q)&c\neq\varepsilon\\\
0&c=\varepsilon\end{cases}$

Note that for $L_{t}(q)$ we replace $\mathcal{C}(q)$ with
$\mathcal{C}_{t}(q)$, considering only the potential antecedents that are
compatible with the span $q$ for the given type. For example, for
$L_{\textsc{Match}}$ and a span $q$, we will only consider candidates $c$
which appear before $q$ with the exact same content words as $q$. Our final
objective for each mention span $q$ is thus:

$\displaystyle L(q)$
$\displaystyle=L_{\textsc{Coref}}(q)+L_{\textsc{Experts}}(q)$ $\displaystyle
L_{\textsc{Experts}}(q)$ $\displaystyle=\sum_{t}L_{t}(q)+L_{\textsc{s}}(q)$

#### Inference

At inference time, we compute the score of each mention based on the shared
scorer as well as the per-type scorer matching the mention type. We then form
the coreference chains by linking each mention $q$ to its most likely
antecedent $c$ according to $F(c,q)$. We do not use higher-order inference as
it has been shown to have a marginal impact (Xu and Choi, 2020).

##  4 Experiments

Our baseline is the s2e model trained on the English OntoNotes 5.0 dataset by
its authors (Kirstain et al., 2021). We train LingMess also on OntoNotes, and
evaluate both models on OntoNotes, WikiCoref, GAP, WinoGender, WinoBias and
BUG. Implementation details are described in Appendix B.

#### Performance

Table 2 presents the performance of LingMess on the test set of Ontonotes.
LingMess achieves 81.4 F1 on Ontonotes, while the _s2e_ baseline achieves
80.3. Our performance gain is statistically significant according to a non-
parametric permutation test ($p$ < 0.05). Additionally, Table 3 shows that
LingMess outperforms the _s2e_ model on WikiCoref (+2.9) GAP (+1.3),
WinoGender (+6.8), WinoBias (+0.8) and BUG (+2.4), indicating a better out-of-
domain generalization.

#### Importance of per-category scoring.

To assess that the improvement of LingMess is due to the decomposition into
our set of categories and not to the added parameters, we do two experiments.
First, we train a random baseline, which randomly assigns a category for each
pair777For each pair of mentions $(c,q)$, we take the modulo of the sum of the
ASCII code of the last character of the last token of $c$ and $q$. and obtain
similar results as the baseline. Second, we train our model by optimizing only
the overall loss $L_{\textsc{Coref}}$ and not $L_{\textsc{Experts}}$. This
achieves lower results than the baseline, due to low mention recall.

In addition to the standard coreference evaluation, we report pairwise
performance for each category. Given a mention-pair $(c,q)$, if $F(c,q)$ is
higher than 0, we treat it as a positive prediction, otherwise negative. We
then measure precision, recall and F1 based on gold clusters labels. Table 4
shows the pairwise performance of the _s2e_ model and LingMess. LingMess
outperforms _s2e_ by a significant margin for all categories (e.g +7.4 F1 for
Pron-Pron-C, +10.2 F1 for Pron-Pron-NC, etc.).888These gains in this pairwise
metric are higher than the CoNLL metrics reported in Table 2, because the
CoNLL metrics are based on the final clusters, after aggregation of individual
pairwise decisions. The performance varies across the different categories,
suggesting aspects of the coreference problem where future work can attempt to
improve.

#### The importance of the shared scorer.

To investigate the role of the shared scorer, we trained the LingMess model
with only the per-type pairwise scorers, excluding the shared pairwise scorer
$F_{\textsc{s}}(c,q)$ and its accompanying loss term $L_{\textsc{s}}(q)$. This
resulted in a significant decrease in performance (-0.9), specifically in the
recall of the mention detection component. However, adding the shared scorer
was able to mitigate this degradation by balancing the different “experts”
pairwise scorers.

##  5 Related Work

Many pre-neural works consider the various linguistic phenomena involved in
coreference resolution as a different challenge. The early coreference system
by Zhou and Su (2004) divided the antecedents candidates into distinct
coreference categories (e.g., Name Alias, Apposition, Definite Noun, and a few
more) and defined tailored rules for each category. Later, Lee et al. (2013)
proposed the multi-sieves deterministic model, where each sieve adds
coreference links between mention pairs from a specific linguistic category
(e.g string match, compatible pronoun, etc.). Haghighi and Klein (2009)
performed an error analysis of their coreference model according to different
types of antecedent decisions, such as Proper Noun-Pronoun, Pronoun-Pronoun,
etc. Based on this analysis, they focus on fixing the pronoun antecedent
choices by adding syntactic features. More recently, Lu and Ng (2020) analyze
empirically the performance of neural coreference resolvers on various fine-
grained resolution categories of mentions (e.g gendered pronoun vs. 1st and
2nd pronoun). They find that while models perform well on name and nominal
mention pairs with some shared content words, they still struggle with
resolving pronouns, particularly relative pronouns.

Early supervised statistical models train a feature-based classifier that
incorporates the type of antecedent decision (e.g. pronoun-entity, string
match) as features at the mention-pair level (Soon et al., 2001; Bengtson and
Roth, 2008; Clark and Manning, 2015, 2016). Subsequently, Denis and Baldridge
(2008) demonstrate that training separate classifiers that specialize in
particular types of mentions (e.g third person pronouns, speech pronouns,
proper names, definite descriptions, and all other) provides significant
performance improvements. Lassalle and Denis (2013) took that observation a
step further and proposed a more advanced method for model specialization by
learning to separate types of mention into optimal classes and their proper
feature space.

In our work, we make progress in coreference systems specialization direction,
and show that the incorporation of linguistic information is helpful also in
the context of end-to-end neural models.

##  6 Conclusion

We present LingMess, a coreference model that significantly improves accuracy
by splitting the scoring function into different categories, and routing each
scoring decision to its own category based on a deterministic, linguistically
informed heuristic. This indicates that while end-to-end training is very
effective, linguistic knowledge and symbolic computation can still be used to
improve results.

## Limitations

In this paper, we consider a set of 6 linguistic categories of mention pairs,
as listed in Table 1. These categories might not be optimal for the task,
while a different set of finer-grained categories may result to a higher
performance gain. Another aspect that can be considered as a limitation is the
computation of the categories for every possible pairs. Although the model
considers only the top-scoring spans, this additional computation layer
increases training and inference time over the baseline (see Appendix B.3 for
the exact time). Our linguistic heuristics could be improved by, e.g., running
a parser and considering head words. However, we chose not to do so in this
work as this will further increase runtime.

## Acknowledgements

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme,
grant agreement No. 802774 (iEXTRACT). Arie Cattan is partially supported by
the PBC fellowship for outstanding PhD candidates in data science.

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##  Appendix A Model Architecture

Given a sequence of tokens $x_{1},...,x_{n}$ from an input document, a
transformer-based (BERT-like) encoder first forms contextualized
representation vectors, $\mathbf{x_{1}},...,\mathbf{x_{n}}$ for each token in
the sequence.

###  A.1 The _s2e_ Model

#### Mention scorer

Given a span $q=(\mathbf{x_{i}},\mathbf{x_{j}})$, represented by its start and
end tokens, the score for $q$ being a mention is defined as follow:

$\displaystyle f_{m}(q)$
$\displaystyle=\mathbf{m}_{s}(\mathbf{x_{i}})\cdot\mathbf{v}_{s}+\mathbf{m}_{s}(\mathbf{x_{j}})\cdot\mathbf{v}_{e}$
$\displaystyle+\mathbf{m}_{s}(\mathbf{x_{i}})\cdot\mathbf{B}_{m}\cdot\mathbf{m}_{s}(\mathbf{x_{j}})$

where $\mathbf{m}_{s}(\mathbf{x})$ and $\mathbf{m}_{e}(\mathbf{x})$ are two
non-linear functions to obtain start and end representations for each token
$x$, and $f_{m}(q)$ is a biaffine product over these representations.

#### Antecedent scorer

Given two spans, $c=(\mathbf{x_{i}},\mathbf{x_{j}})$ and
$q=(\mathbf{x_{k}},\mathbf{x_{l}})$, represented by their start and end
tokens, the score for $c$ being an antecedent of $q$ is computed as follow:

$\displaystyle f_{a}(c,q)$
$\displaystyle=\mathbf{a}_{s}(\mathbf{\mathbf{x_{i}}})\cdot\mathbf{B}_{ss}\cdot\mathbf{a}_{s}(\mathbf{\mathbf{x_{k}}})$
$\displaystyle+\mathbf{a}_{e}(\mathbf{\mathbf{x_{j}}})\cdot\mathbf{B}_{es}\cdot\mathbf{a}_{s}(\mathbf{\mathbf{x_{k}}})$
$\displaystyle+\mathbf{a}_{s}(\mathbf{\mathbf{x_{i}}})\cdot\mathbf{B}_{se}\cdot\mathbf{a}_{e}(\mathbf{\mathbf{x_{l}}})$
$\displaystyle+\mathbf{a}_{e}(\mathbf{\mathbf{x_{j}}})\cdot\mathbf{B}_{ee}\cdot\mathbf{a}_{e}(\mathbf{\mathbf{x_{l}}})$

Similar to the mention scorer, $\mathbf{a}_{s}(\mathbf{x})$ and
$\mathbf{a}_{e}(\mathbf{x})$ are two non-linear functions to obtain start/end
representations for each token, and $f_{a}(c,q)$ is a sum of four bilinear
functions over the start and end representations of $c$ and $q$.

###  A.2 LingMess

#### Mention scorer

Our mention scorer is the same as _s2e_ mention scorer implementation.

#### Antecedent scorer

As mentioned in the paper (§3), in addition to the shared antecedent scorer
$f_{a}(c,q)$, LingMess includes a dedicated antecedent scorer $f^{t}_{a}(c,q)$
for each category $t\in\\{\textsc{Pron-Pron-C},\textsc{Pron-Pron-
NC},\textsc{Ent-Pron},$ $\textsc{Match},\textsc{Contains},\textsc{Other}\\}$.
The overall score for $c=(\mathbf{x_{i}},\mathbf{x_{j}})$ being an antecedent
of $q=(\mathbf{x_{k}},\mathbf{x_{l}})$ becomes the sum of the shared scorer
and the relevant category “expert” scorer:

$\displaystyle f(c,q)=f_{a}(c,q)+f^{T(c,q)}_{a}(c,q)$

where $T(c,q)$ is a deterministic function to determine the category $t$ of
the pair $(c,q)$.

For each category $t$, we define two specific non-linear functions to obtain
start and end representations ($\mathbf{a}^{t}_{s}(\mathbf{x})$ and
$\mathbf{a}^{t}_{e}(\mathbf{x})$) as well as an “expert” antecedent scoring
function $f^{t}_{a}(c,q)$:

$\displaystyle f^{t}_{a}(c,q)$
$\displaystyle=\mathbf{a}^{t}_{s}(\mathbf{\mathbf{x_{i}}})\cdot\mathbf{B}^{t}_{ss}\cdot\mathbf{a}^{t}_{s}(\mathbf{\mathbf{x_{k}}})$
$\displaystyle+\mathbf{a}^{t}_{e}(\mathbf{\mathbf{x_{j}}})\cdot\mathbf{B}^{t}_{es}\cdot\mathbf{a}^{t}_{s}(\mathbf{\mathbf{x_{k}}})$
$\displaystyle+\mathbf{a}^{t}_{s}(\mathbf{\mathbf{x_{i}}})\cdot\mathbf{B}^{t}_{se}\cdot\mathbf{a}^{t}_{e}(\mathbf{\mathbf{x_{l}}})$
$\displaystyle+\mathbf{a}^{t}_{e}(\mathbf{\mathbf{x_{j}}})\cdot\mathbf{B}^{t}_{ee}\cdot\mathbf{a}^{t}_{e}(\mathbf{\mathbf{x_{l}}})$

Overall, our model introduces 6 learnable matrices for each category
($\mathbf{W}^{t}_{\mathbf{a}_{s}}$, $\mathbf{W}^{t}_{\mathbf{a}_{e}}$,
$\mathbf{B}^{t}_{ss}$, $\mathbf{B}^{t}_{es}$, $\mathbf{B}^{t}_{se}$,
$\mathbf{B}^{t}_{ee}$). The transformer-based encoder and the mention scorer
are shared between all the different pairwise scorers.

##  Appendix B Implementation Details

###  B.1 Hyperparameteres

We extend the _s2e_ ’s implementation based on PyTorch (Paszke et al., 2019)
and Transformers (Wolf et al., 2020). We used the same hyperparameters (e.g
learning rate, warmup, etc.) as the _s2e_ model except the hidden size of all
matrices $W$ and $B$. As our method introduces a dedicated antecedent scoring
function $f^{t}_{a}$ function for each category $t$, we reduce the size of
these matrices from 3072 to 2048 to fit training into memory in our hardware.
Similar to the baseline our head method is on top of Longformer-Large Beltagy
et al. (2020), resulting in a total of 590M learnable parameters (the _s2e_
model contains 494M learnable parameters). We used dynamic batching both for
training and inference, specifically 5K tokens in a single batch during
training and 10K tokens at inference.

###  B.2 Evaluation

As mentioned in the paper (§4), we conduct our experiments on the English
portion of the OntoNotes corpus (Pradhan et al., 2012). This dataset contains
2802 documents for training, 343 for development, and 348 for test.

We evaluate our model according to the standard coreference metric: MUC
(Vilain et al., 1995), B3 (Bagga and Baldwin, 1998), CEAF$\phi_{4}$ (Luo,
2005), and LEA (Moosavi and Strube, 2016) using the official CoNLL coreference
scorer.999https://github.com/conll/reference-coreference-scorers. LingMess
achieves 81.6 CoNLL F1 on the development set of Ontonotes. Also, Table 5
presents the pairwise performance on the development set for each category. We
compute statistical significance with a non-parametric permutation test using
Ulmer et al. (2022)’s implementation. Table 6 shows that LingMess consistenly
outperforms the _s2e_ model on GAP.

| Kirstain et al. (2021) |  | LingMess
---|---|---|---
| P | R | F1 |  | P | R | F1
Pron-Pron-C | 91.7 | 77.5 | 84.0 |  | 91.7 | 90.2 | 91.0
Pron-Pron-NC | 88.9 | 66.2 | 75.9 |  | 90.2 | 81.3 | 85.5
Ent-Pron | 82.0 | 74.1 | 77.9 |  | 81.4 | 74.7 | 77.9
Match | 88.3 | 87.5 | 87.9 |  | 88.4 | 92.0 | 90.2
Contains | 69.1 | 77.2 | 72.9 |  | 76.1 | 73.5 | 74.8
Other | 56.8 | 67.5 | 61.7 |  | 70.8 | 64.4 | 67.5

Table 5: Pairwise performance by category, on the dev set of the English OntoNotes 5.0 dataset. | Masc | Fem | Bias | Overall
---|---|---|---|---
Kirstain et al. (2021) | 90.6 | 85.8 | 0.95 | 88.3
LingMess | 91.3 | 87.8 | 0.96 | 89.6
Table 6: Performance on the test set of the GAP coreference dataset. The
reported metrics are F1 scores.

###  B.3 Runtime and Memory

| Runtime | Memory
---|---|---
Kirstain et al. (2021) | 28 | 5.4
LingMess | 43 | 5.9

Table 7: Inference time(Seconds) and memory(GiB) on 343 docs of OntoNotes
development set. Using Dynamic batching, 10K tokens in a single batch.
Hardware, NVIDIA Tesla V100 SXM2

Our model was trained for 129 epochs on a single 32GB NVIDIA Tesla V100 SXM2
GPU. The training took 23 hours. At shown in Table 7, the run-time at
inference time in LingMess is longer than in the _s2e_ model because of the
category selection for every possible pair of mentions. The memory consumption
remains quite similar to the baseline.

##  Appendix C Determining pair types

Our method routes each pair of spans to their corresponding category scorer.
This decision is based on the linguistic properties of the spans. Given a
mention-pair $(c,q)$, we defined a rule based function $T(c,q)$ that
determines the category of that pair. If $c$ and $q$ are both pronouns, if
they are compatible according to gender, number and animacy (see Table 8 for
the full list), the metnion pair will be routed to Pron-Pron-C, otherwise
Pron-Pron-NC. If $c$ is pronoun and $q$ is a non-pronoun span (or vise-versa)
we route the mention-pair to Pron-Ent. We route the remaining pairs to their
corresponding categories (Match, Contains or Other) by considering only
content words, excluding the following stop words: _{ ’s, a, all, an, and, at,
for, from, in, into, more, of, on, or, some, the, these, those}_. Accordingly,
the mentions “ _the U.S. and Japan_ ” and “ _Japan and the U.S._ ” are
considered Match, “ _This lake of fire_ ” and “ _the lake of fire_ ” are
considered Contains and “ _Bill Clinton_ ” and “ _The President_ ” are
considered Other.

ID | Pronouns
---|---
1 | _I, me, my, mine, myself_
2 | _you, your, yours, yourself, yourselves_
3 | _he, him, his, himself_
4 | _she, her, hers, herself_
5 | _it, its, itself_
6 | _we, us, our, ours, ourselves_
7 | _they, them, their, themselves_
8 | _that, this_

Table 8: List of groups of compatible pronouns, pronouns with the same ID are
considered as compatible.