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Instituto de Fisica 
Universidade de Sao Paulo 
Caixa Postal 66 318 
05315 Sao Paulo, SP, Brazil 
e-mails: guidi@if.usp.br, marchett@if.usp.br
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Upward propagating flames; multi - coalescent pole solutions; 
stability analysis; nonlinear partial differential equations;  
metastable solutions; invariant set dimensionality. 
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\begin{document}
\title{Rakib--Sivashinsky and Michelson--Sivashinsky Equations for
Upward Propagating Flames: A Comparison Analysis } 
\author{ \textbf{Leonardo F. Guidi}\thanks{ 
Supported by FAPESP under grant 98/10745-1. E-mail: {\it
guidi@if.usp.br}} 
\ and \ \textbf{Domingos H. U. Marchetti}\thanks{
Partialy supported by CNPq and FAPESP. E-mail:{\it \
marchett@if.usp.br}} \\ 
%EndAName
Instituto de F\'{\i}sica\\
Universidade de S\~{a}o Paulo\\
Caixa Postal 66 318\\
05315 S\~{a}o Paulo, SP, Brazil}
\date{\today }
\maketitle
%\pacs{}

\begin{abstract}
We establish a comparison between Rakib--Sivashinsky and
Michelson--Sivashinsky quasilinear parabolic differential equations governing
the weak thermal limit of upward flame front propagating in a channel. For
the former equation, we give a complete description of all steady solutions
and present their local and global stability analysis. For the latter,
multi--coalescent unstable steady solutions are introduced and shown to be
exponentially more numerous than the previous known coalescente solutions.
This fact is argued to be responsible for the disagreement of the observed
dynamics in numerical experiments with the exact (linear) stability analysis
and also gives the ingredients to describe the quasi--stable behavior of
parabolic steadily propagating flame with centered tip.
\end{abstract}


\newpage

Rakib--Sivashinsky quasilinear parabolic differential equation governing the
weak thermal limit of an upward flame interface propagating in a channel is
given by 
\begin{equation}
\phi _{t}=\varepsilon \phi _{xx}-\frac{1}{2}\phi _{x}^{2}+\phi -\stackrel{\_
}{\phi }\,,  \label{RSeq}
\end{equation}
where $y=\phi (t,x)$, $0<x<\pi $, defines an instantaneous flame profile in
dimensionless variables, $\stackrel{\_}{\phi }=
{\displaystyle{1 \over \pi }} \displaystyle\int 
_{0}^{\pi }\phi (t,x)\,dx$ denotes the space average, $\varepsilon >0$ is a
physical parameter (Markstein length) and Neumann (adiabatic) boundary
condition at channel walls is imposed: $\phi _{x}(t,0)=\phi _{x}(t,\pi )=0$.
According to Darrieus--Landau's hydrodynamic flame theory\cite{RS} there are
two competing sources of instabilities in this model given by the last two
terms of (\ref{RSeq}). A plane flame front separating the cold and hot gas
is subject to Rayleigh--Taylor instability, due to the thermal expansion,
and to the buoyancy effect caused by external acceleration. These two
ingredients lead the front to become convex toward the cold gas. As in \cite
{RS}, the acceleration vector points toward positive $y$--values and the
upward propagating direction is negative. For the purpose of comparison such
convention will be maintained.

Paraboloid profiles with the tip located around the center of the channel
are observed, both experimentally and in computational simulations, to
remain ``stable'' for long time (see e.g. \cite{MiS} and references
therein). There 
are, however, experiments whose paraboloid tip eventually slides to the
channel wall. A mathematical description of the former configuration as a
``quasi--equilibrium transient state'' was provided by Berestycki, Kamin and
Sivashinsky \cite{BKS} in which work stationary solutions of (\ref{RSeq}),
their respective stability properties and the nonlinear dynamics were
described for sufficiently small $\varepsilon $. Metastable dynamics has
been studied in details in \cite{SW}. The proofs of the other theorems
stated in \cite{BKS} remain, to the best of our knowledge, unpublished.

Sivashinsky'{}s previous equation of weak thermal expansion\cite{S,MS} 
\begin{equation}
\phi _{t}=\varepsilon \phi _{xx}-\frac{1}{2}\phi _{x}^{2}+I(\phi )\,,
\label{MSeq}
\end{equation}
hasn'{}t considered buoyancy effect and Darrieus--Landau instability
has been taken 
into account replacing $\phi -\stackrel{\_}{\phi }$ in (\ref{RSeq}) by a
linear singular integral operator\cite{nota0} given by a multiplication by $
\left| k\right| $ in the Fourier representation 
\[
I\left( \phi \right) (t,x)=\frac{1}{\pi }\sum_{k=1}^{\infty }k\,\int_{-\pi
}^{\pi }\cos \left[ k\left( x-y\right) \right] \,\phi (t,y)\,dy\,. 
\]
In contradistinction, the only trace of $I(\phi )$ term in equation (\ref
{RSeq}) comes from the removal of $k=0$ mode 
\[
\phi -\stackrel{\_}{\phi }=\frac{1}{\pi }\sum_{k=1}^{\infty }\int_{-\pi
}^{\pi }\cos \left[ k\left( x-y\right) \right] \,\phi (t,y)\,dy\,\,. 
\]
Michelson--Sivashinsky equation (\ref{MSeq}) has been studied by many
authors after Thual--Frish--H\'{e}non's application\cite{TFH} of pole
decomposition. For a more recent survey of this method, see Vaymblat and
Matalon\cite{VM1,VM2} and references therein.

Despite of the fact that equations (\ref{RSeq}) and (\ref{MSeq}) differ in
many respects, \ it is our purpose to expose the similarities and
distinctions of their solutions. We shall see, by the introduction of so
called bi--coalescent steady states, that equation (\ref{MSeq}) may also
admit quasi--equilibrium transient ``parabolic'' profile with centered tip.
Our comparison relies on two recent analysis. Firstly, Vaymblat and Matalon 
\cite{VM1} have determined all steady solutions whose poles coalesce into a
line parallel to the imaginary axis and solved explicitly the eigenvalue
problem of (\ref{MSeq}) linearized about those states. They conclude, in a
second paper\cite{VM2}, the existence of only one linearly stable
steady coalescent pole solution.

For equation (\ref{RSeq}) with a different parametrization, we have given
\cite{GM} a complete description of all equilibrium solutions and
provided their local and global stability in an appropriated Sobolev space.
Since our results are major extensions of those stated in \cite{BKS} and the
mathematical presentation in \cite{GM} may cause certain difficulties in
translating to the present application, we shall here restate them with a
brief explanations of their proofs. A detailed presentation including the
metastable states analysis will appear elsewhere \cite{MG}.

We shall first restrict ourselves to the Rakib--Sivashinsky equation. The
details presented after each statement (detached by brackets) are
essentials to establish the subsequent comparison but may be skipped
in a first reading. Multi--coalescent steady states of
Michelson--Sivashinsky equation will be considered next. A conclusion
will be presented at the end. 

To discuss our results let us consider the equation 
\begin{equation}
u_{t}=\varepsilon u_{xx}-u\,u_{x}+u\,\,,  \label{u}
\end{equation}
for the derivative $u=\phi _{x}$, with Dirichlet boundary conditions $
u(t,0)=u(t,\pi )=0$. There is a one--to--one correspondence between 
\begin{equation}
\theta (t,x):=\int_{0}^{x}u(t,y)\,dy\,,  \label{phiu}
\end{equation}
with $u$ a solution of (\ref{u}), and\ a solution $\phi $ of (\ref{RSeq})
given as follows. Notice that $\theta $ satisfy (\ref{RSeq}) with $\stackrel{
\_}{\phi }$ replaced by $\varepsilon \phi _{xx}(t,0)$. By definition, $
\theta (t,0)=0$ for all $t\geq 0$ and their steady solutions do not
propagate (they are equilibrium). We have $\theta (t,x)=\phi (t,x)-\phi
(t,0)\,$, $\theta _{x}=\phi _{x}$ and $\theta _{xx}=\phi _{xx}$. For the
opposite relation, we have 
\[
\phi +\int_{0}^{t}\stackrel{\_}{\phi }(s)\,e^{t-s}\,ds=\theta +\varepsilon
\int_{0}^{t}\theta _{xx}(s,0)\,e^{t-s}\,ds\,, 
\]
because both sides satisfy equation (\ref{RSeq}) without $\stackrel{\_}{\phi 
}$ term, which yields 
\begin{equation}
\phi (t,x)=\theta (t,x)-\int_{0}^{t}\left( \stackrel{\_}{\theta }(\tau
)\,-\varepsilon \,\theta _{xx}(\tau ,0)\right) \,\,ds\,.
\label{phitheta} 
\end{equation}
This and (\ref{phiu}) will be used to discuss the steadily propagating
solutions of (\ref{RSeq}).

If $A$ denotes the operator given by r.h.s. of (\ref{u}) linearized about
the trivial solution $u_{0}=0$, let ${\cal B}^{1/2}$ be the Banach space
equipped with the graph norm $\left\| f\right\| _{1/2}=
\displaystyle\int 
_{0}^{\pi }\left| A^{1/2}f(x)\right| ^{2}\,dx$ (equivalent to the Sobolev
space $H_{0}^{1}(0,\pi)$). In \cite{GM}, Theorems 3.2 and 5.10, we have proven:

\noindent 
{\it The initial value problem (\ref{u}) with }$u(0,\cdot )=u_{0}\in
{\cal B}^{1/2}$
{\it \ has a unique solution for all }$t>0${\it \ and the trajectories }$
\left\{ u(t,\cdot )\right\} _{t\geq 0}${\it \ lie in a compact set in }$
{\cal B}^{1/2}${\it .}

By (\ref{phiu}) and (\ref{phitheta}), this provides the existence and
uniqueness of the initial value problem (\ref{RSeq}) for all $t>0$.
Compactness property will be useful for the global stability analysis.

The steadily propagating solutions of (\ref{RSeq}) can be obtained from the
equilibrium solution of (\ref{u}). The quantity inside the parenthesis
in (\ref{phitheta}) remains positive under the dynamics of (\ref{u})
and the 
front propagates upward (toward negative $y$--values). As a consequence, the
flame front propagates steadily with velocity 
\begin{equation}
V=-\,\stackrel{\_}{\vartheta }\,+\varepsilon \,\vartheta _{xx}(0)=-\, 
%\stackrel{\_\_}
\frac{1}{2} \; \overline{\vartheta ^{2}},  \label{vv} 
\end{equation}
provided $\vartheta (x)=\int_{0}^{x}v(y)\,dy$, where $v$ is a non trivial
solution of 
\begin{equation}
\varepsilon v_{xx}-v\,v_{x}+v=0\,,  \label{v}
\end{equation}
with $v(0)=v(\pi )=0$. The next result is Theorem 4.1 of \cite{GM}:

\noindent 
{\it For }$\varepsilon >1${\it , }$v_{0}=0${\it \ is the unique
solution of (\ref{v}). For }$\varepsilon <1${\it \ such that }$ 
{\displaystyle{1 \over k+1}}
\leq \varepsilon <
{\displaystyle{1 \over k}}
${\it \ holds for some }$k\in N${\it , there exist }$2k${\it \ nontrivial
solutions }$u_{j}^{\pm }${\it , }$j=1,\ldots ,k${\it , with }$j-1${\it \
zeros in }$\left( 0,\pi \right) ${\it \ and such that }$%
u_{j}^{-}(x)=-u_{j}^{+}(\pi -x)${\it \ for }$j${\it \ odd and }$%
u_{j}^{-}(x)=u_{j}^{+}(\pi /j+x)${\it \ mod }$\pi ${\it \ for }$j${\it \
even. Each pair }$u_{j}^{\pm }${\it \ bifurcate from the trivial solution }$%
v_{0}=0${\it \ at }$\varepsilon _{j}=1/j^{2}${\it \ with }$%
\lim\limits_{\varepsilon \uparrow \varepsilon _{j}}u_{j}^{\pm }=0${\it . In
the phase space }${\Bbb R}^{2}${\it , }$\left( u_{j}^{\pm \prime
},u_{j}^{\pm }\right) ${\it \ describes closed orbits around }$\left(
0,0\right) ${\it \ whose distance from the origin increases monotonically as 
}$\varepsilon ${\it \ decreases.}

\noindent [Equation (\ref{v}) can be written as a dynamical
system  
\begin{equation}
\left\{ 
\begin{array}{lll}
w \acute{}
& = & \varepsilon ^{-1}p\left( w-1\right)  \\ 
p \acute{}  = & w\,,
\end{array}
\right.   \label{ds}
\end{equation}
where $p=v$ and $w=v{\acute{}}$. Since (\ref{ds}) remains unaltered by
changing $p\rightarrow -p$ and $ 
x\rightarrow -x$, the orbits are symmetric with respect to the $w$--axis and
Dirichlet boundary conditions $p(0)=p(\pi )=0$ hold for any periodic orbit
with period $2\pi $ such that $p=0$ at $x=0$. In \cite{GM} we have shown
that {\bf (i)} ${\Bbb R}^{2}$ is foliated by non--overlapping orbits 
\[
\gamma _{w_{0}}=\left\{ (w(x),p(x)):x\in {\Bbb R}\;{\rm and}
\;(w(0),p(0))=\left( w_{0},0\right) \right\} 
\]
labeled by the coordinate $w_{0}$ of the positive $w$--axis; {\bf (ii)} as $
\varepsilon $ varies, the orbit varies continuously from one to another; 
{\bf (iii) }the orbits are closed if $w_{0}<1$ and open if $w_{0}\geq 1$; 
{\bf (iv)} the origin $\left( 0,0\right) $ is enclosed by any closed orbits.
%For this, note that equation (\ref{ds}) yields 
%\[
%\frac{dp}{dw}=\varepsilon \frac{w}{w-1}\,,
%\]
%which can be integrated out 
%\begin{equation}
%\frac{1}{2\varepsilon }p^{2}=w-w_{0}+\ln \left( \frac{1-w}{1-w_{0}}\right)
%\,.  \label{traj}
%\end{equation}
The trajectories $\gamma _{w_{0}}$, $w_{0}\geq 0$, are portrayed in
Figure \ref{phsp}. 


\begin{figure}[!ht] 
\begin{center}
\epsfig{file=figure1.eps, height=7.5cm, width=12.5cm}
\end{center}
\caption{Trajectories of the dynamical system (\ref{ds}).} 
\label{phsp} 
\end{figure}

The period $T=T(\varepsilon ,w_{0})$ of a closed orbit varies continuously
in $\varepsilon $ and $w_{0}$ and is given by 
\[
T=\int_{\gamma _{w_{0}}}dx=2\int \frac{dp}{w}\,, 
\]
where the second integral is calculated in the semi--orbit. Hence, $T=2\pi $
defines implicitly a continuous function $w_{0}=w_{0}(\varepsilon )$, whose
corresponding semi--orbit $\gamma _{w_{0}}=\left\{ \left( w(x),p(x)\right)
:0\leq x\leq \pi \right\} $ determines the solution $v(x)=p(x)$ of
(\ref{v}). The question to be addressed now is how many $2\pi
$--periodic orbits 
there are for each $\varepsilon $. If $w_{0}$ is small, $\gamma _{w_{0}}$
can be approximated by an ellipsoid: $\varepsilon ^{-1}p^{2}+w^{2}=w_{0}^{2}$
; 
\begin{equation}
T_{{\rm ellipse}}=4\int_{0}^{\sqrt{\varepsilon }}\frac{dp}{\sqrt{
1-\varepsilon ^{-1}p^{2}}}=2\pi \sqrt{\varepsilon }  \label{Te}
\end{equation}
uniformly in $w_{0}$ and we have 
\begin{equation}
\lim\limits_{w_{0}\rightarrow 0}T(\varepsilon ,w_{0})=2\pi \sqrt{\varepsilon 
}\,.  \label{Tw0}
\end{equation}
In \cite{GM}, we have shown 
\begin{equation}
\frac{\partial T}{\partial w_{0}}>0\,  \label{delT}
\end{equation}
holds form any $\varepsilon >0$. Consequently, $T=T(\varepsilon ,w_{0})$ is
a (strictly) monotonic increasing function of $w_{0}$ and $\varepsilon $,
since $T(\varepsilon ,w_{0})=\sqrt{\varepsilon }T(1,w_{0})$. As $
T$ increases when $\varepsilon $ and $w_{0}$ increase, equation (\ref{Tw0})
implies that there is no nontrivial solutions ($v\neq 0$) for $\varepsilon
\geq 1$, since the period of each nontrivial orbit exceeds $2\pi $. Now, let 
\[
G_{1}(\varepsilon ,w_{0})=T(\varepsilon ,w_{0})-2\pi \,, 
\]
be defined for $0<\varepsilon \leq 1$ and $0\leq w_{0}<1$. Under the
condition (\ref{delT}), there exist a {\bf unique} function $g_{1}:\left[ 0,1
\right] \longrightarrow {\Bbb R}_{+}$ with $g_{1}(1)=0$ such that $%
G_{1}(\varepsilon ,g(\varepsilon ))=0$ for all $0<\varepsilon \leq 1$. In
view of (\ref{delT}), $g(\varepsilon )$ is a monotonically (strictly)
decreasing function and $\lim_{\varepsilon \rightarrow 0}g(\varepsilon )=1$.

Now, $v\equiv 0$ is the unique equilibrium solution of (\ref{u}) for any $
\varepsilon \geq 1$. For $\varepsilon <1$ such that $1/(k+1)^{2}\leq
\varepsilon <1/k^{2}$ holds for some integer $k\geq 1$, one can apply the
implicit function theorem to equation 
\[
G_{j}(\varepsilon ,w_{0}):=T(\varepsilon ,w_{0})-\frac{2\pi }{j}=0\,,
\]
with $j=1,\ldots ,k$ and conclude, exactly as in the case for $G_{1}$, the
existence of a unique monotone decreasing function $g_{j}:\left[ 0,1/j^{2}
\right] \longrightarrow {\Bbb R}_{+}$ with $g_{j}(1/j^{2})=0$ and $
\lim\limits_{\varepsilon \rightarrow 0}g_{j}(\varepsilon )=1$, such that $
G_{j}(\varepsilon ,g_{j}(\varepsilon ))=0$. The branch $g_{j}(\varepsilon )$
defines a nontrivial equilibrium solution $v_{j}$ of (\ref{u}) given by the $
p$--component of the orbit $\gamma _{g_{j}(\varepsilon )}$ with $\left(
g_{j}(\varepsilon ),0\right) $ at $x=0$ and $p(x)=0$ at each semi--period $
x=m\pi /j$, $m=1,\ldots ,j$. Hence, if $\varepsilon $ is such that $
1/(k+1)^{2}\leq \varepsilon <1/k^{2}$, there exist $k$ equilibrium (not
identically $0$) solutions of (\ref{u}), $v_{1}^{+},\,\ldots ,$ $v_{k}^{+}$,
with $v_{j}^{+}$ having $j-1$ zeros in the interval $(0,\pi )$. Notice that $
g_{j}(\varepsilon )=v_{j}^{\prime }(0)>0$, $j=1,\ldots ,k$, and $v_{1}(x)>0$
for all $0<x<\pi $. These will be useful for the stability analysis. There
is an additional set of $k$ equilibrium solution $v_{1}^{-},\,\ldots
,v_{k}^{-}$, with $v_{j}^{-}(x)=p(\pi /j+x)$.]

When $\varepsilon $ is small enough the orbit spends most of time at right
semi--plane, $p(x)\simeq x$ mod $2\pi /j$ and $\vartheta _{j}^{\pm
}(x):=\int_{0}^{x}v_{j}^{\pm }(y)\,dy\,$\ has\ wrinkles of parabolic shape
separated by ``cusps''. From (\ref{vv}) the steadily propagating solutions
of (\ref{RSeq}) has the profile of $\vartheta _{j}$ propagating with the
velocity $V$: 
\begin{equation}
\varphi _{j}(x)=\vartheta
_{j}(x)-\,
%\stackrel{\_\_}
\frac{t}{2} \; \overline{\vartheta 
_{j}^{2}}\,.  \label{phij} 
\end{equation}

The plane flame front solution $\varphi _{0}={\rm Const}$ is an
asymptotically stable solution if $\varepsilon \geq 1$ for both
equations (\ref{RSeq}) and (\ref{MSeq}). Its local stability can be
read directly from 
the spectrum $\sigma (A)$ of the operator $A$, given by (\ref{RSeq})
and (\ref{MSeq}) linearized about $\varphi _{0}$, which consist of
simple eigenvalues 
\begin{equation}
\lambda _{n}=1-\varepsilon n^{2}\qquad {\rm and}\qquad \eta
_{n}=n-\varepsilon n^{2},\qquad n=1,2\ldots \,,\,  \label{eigenv}
\end{equation}
respectively, with corresponding eigenfunction $\sqrt{1/\pi }\cos nx$. So, $
\lambda _{n}<0$ and $\eta _{n}<0$ if $\varepsilon >1$. At $\varepsilon =1$,
the trivial solution $\varphi _{0}$ bifurcates \cite{nota} into a steadily
propagating front in which, when extended periodically to the real line as
an even function, there is one ``parabolic'' tip centered at one wall and a
single ``cusp'' at another. The next result of \cite{GM}, Theorems $5.1$ and 
$5.14$, states that such configuration is the only one globally
asymptotically stable solution of (\ref{RSeq}) for all $\varepsilon <1$.
Parabolic front with centered tip or wrinkled flame fronts may be described
as quasi--equilibrium states discussed in refs. \cite{MS,BKS,SW}. By (\ref
{phitheta}), it is enough to examine the stability of the nontrivial
equilibrium solutions of (\ref{u}).

\noindent 
{\it If }$u_{0}\in {\cal B}^{1/2}${\it \ and }$\varepsilon >1${\it , then the
initial value problem (\ref{u}) with }$u(0,\cdot )=u_{0}${\it \ satisfies }$
\lim\limits_{t\rightarrow \infty }\left\| u(t,\cdot )\right\| _{1/2}=0$ and
the trivial solution is globally stable{\it . For }$\varepsilon <1${\it \ so
that }$1/(k+1)^{2}\leq \varepsilon <1/k^{2}${\it \ holds, there exist }$\rho
>0${\it \ such that, if }$\left\| u_{0}-v\right\| _{1/2}\leq \rho ${\it ,
then }$\lim\limits_{t\rightarrow \infty }\left\| u(t,\cdot )-v\right\|
_{1/2}=0${\it \ for }$v=v_{1}^{\pm }${\it \ and, for any sequence }$\left\{
u_{n}\right\} _{n\geq 1}${\it \ with }$\lim\limits_{n\rightarrow \infty
}\left\| u_{n}-v\right\| _{1/2}=0${\it , we have }$\sup\limits_{t>0}\left\|
u(t,\cdot )-v\right\| _{1/2}\geq \delta >0${\it \ for all }$n${\it \ and }$v$
{\it \ equal to }$0${\it \ or any equilibrium }$v_{j}^{\pm }${\it , }$
j=2,\ldots ,k${\it . Moreover, there is an open dense set }$U\subset
{\cal B}^{1/2}$ 
{\it \ containing the origin and }$\left\{ v_{j}^{\pm }\right\} _{j=1}^{k}$
{\it such that, if }$u_{0}\in U${\it \ and }$\varepsilon <1${\it , then }$
\lim\limits_{t\rightarrow \infty }\left\| u(t,\cdot )-v_{1}^{\pm }\right\|
_{1/2}=0${\it .}

\noindent [Let us first examine the linear stability. 
If $u(t,x)=v(x)+\zeta (t,x)$ then equation (\ref{u}) can be
written as 
\[
\zeta _{t}=L\zeta +\zeta \zeta _{x}\,,
\]
where\cite{nota3} 
\[
L\xi =L\left[ v\right] \xi =\varepsilon \xi ^{\prime \prime }-v\xi ^{\prime
}+(1-v^{\prime })\xi 
\]
is the linearization of the right hand side of (\ref{u}) about $v$. Acting
on the space of functions $\xi $ satisfying $\xi (0)=\xi (\pi )=0$, $L$ is
symmetric, $\left( \eta ,L\xi \right) _{\rho }=\left( L\eta ,\xi \right)
_{\rho }$, with respect to the inner product 
\[
\left( \eta ,\xi \right) _{p}=
\displaystyle\int 
_{0}^{\pi }\eta (x)\,\xi (x)\,\rho (x)\,dx
\]
with weight $\rho (x)=\exp \left\{ -\varepsilon
^{-1}\int_{0}^{x}v(y)\,dy\right\} $. \ As a consequence, we can apply the
comparison theorem to establish the following criterium (see \cite{GM} for
details).

\noindent Stability Criterium. {\it If} $\varphi $ {\it is the solution of} 
\[
L\left[ v\right] \varphi =0\,, 
\]
{\it on} $0<x<\pi $, {\it satisfying} $\varphi (0)=0$ {\it and} $\varphi
^{\prime }(0)=1$, {\it the largest eigenvalue} $\lambda $ {\it of }$L\left[ v
\right] $ {\it is negative if} $\varphi >0$ {\it on} $\left( 0,\pi \right) $ 
{\it and positive if there exist} $x^{\prime }$ {\it such that }$0<x^{\prime
}<\pi $ {\it and} $\varphi (x^{\prime })=0$.

For an equilibrium solution $v$ of (\ref{u}), let 
\[
\chi =c\left( 2v-v^{\prime \prime }\right) \,,  
\]
where $c>0$ is chosen so that $\chi ^{\prime }(0)=1$. It follows from
equilibrium equation $\varepsilon v^{\prime \prime }=-\left( 1-v^{\prime
}\right) v$ that $\chi (0)=0$ and $\chi >0$ whenever $v>0$ (recall $v(0)=0$
and $\left( 1-v^{\prime }\right) >0$ for all closed orbits). Moreover, if $
v>0$, an explicit calculation gives (see \cite{GM}) 
\[
L\left[ v\right] \chi =-2\left( v^{\prime }\right) ^{2}v<0\,\,,  
\]
and this implies, by applying the comparison theorem once more, 
\[
\varphi (x)\geq \chi (x) 
\]
for all $0\leq x<\pi $. It thus follows from the stability criterium that $
v_{1}$ is a stable equilibrium solution for all $\varepsilon <1$ (recall $
v_{1}(x)>0$ for all $0<x<\pi $, implying $\chi (x)>0$ in the same domain).
To show $v_{2},\ldots ,v_{k}$ are unstable, we observe
\begin{equation}\label{Lv}
L\left[ v_{j}\right] v_{j}^{\prime } =\left( \varepsilon v_{j}^{\prime
\prime }+(1-v_{j}^{\prime 
})v_{j}\right) ^{\prime }=0\,, 
\end{equation}
by the equilibrium equation. It thus follows that $v_{j}$ with $j\geq 2$ is
unstable in view of the equilibrium criterium and the fact that $v_{j}$ has
at least one zero in $\left( 0,\pi \right) $ and the same holds for $\varphi 
$ by the Wronskian positivity $W(\varphi ,v_{j}^{\prime };x)>0$. It is
important to note that $v_{j}^{\prime }$ fails to be an eigenfunction of $L
\left[ v_{j}\right] $ with $0$ eigenvalue because it does not satisfies the
boundary conditions required. Approximate eigenfunction with exponentially
small (in $\varepsilon $) eigeinvalue can, however, be constructed using
boundary layer techniques (see e. g. \cite{SW}). Equation (\ref{Lv}) holds
also for any stationary solution $v$ of Michelson--Sivashinsky equation.

We turn to the global stability. A Liapunov function for equation (\ref{u})
can be constructed via the generalized Euler--Lagrange method due to
Zelenyak, Lavrentiev and Vishnevskii \cite{ZLV}. Let 
\[
U(u)=\int_{0}^{\pi }\Phi (u,u_{x})\,dx 
\]
where $\Phi (p,w)=-p^{2}/(2\varepsilon )+\left( 1-w\right) \ln \left(
1-w\right) +w$ is an appropriated ``Lagrangian'' . The total derivative of $
U $ with respect to $t$ is obtained by the calculus of variation 
\begin{eqnarray*}
\stackrel{\cdot }{U}\left( u\right) &=&-\int_{0}^{\pi }\left(  
\frac{\partial}{\partial x} \frac{\partial \Phi }{\partial
u_{x}}-\frac{\partial \Phi v
}{\partial u} \right) u_{t}\,dx \\   
&=&-\int_{0}^{\pi }\rho (u_{x})\left( \varepsilon  
u_{xx}-u\,u_{x}+u\right) u_{t}\,dx  
\end{eqnarray*}
where $\rho (w)=\varepsilon ^{-1}/(1-w)$ is a positive weight. Note $
\stackrel{\cdot }{U}$ is negative in view of (\ref{u}) and $U$ is a Liapunov
functional. Due to the fact that the trajectories $\left\{ u(t,\cdot
)\right\} _{t\geq 0}$ lie in a compact set, LaSalle's invariance principle
can be applied to show that all solutions of (\ref{u}) in ${\cal B}^{1/2}$
converge to an equilibrium solution as $t\rightarrow \infty $. From
equations (\ref{phitheta}), (\ref{phiu}) and the two results stated before
one concludes that $\varphi _{0}=0$ is globally asymptotically stable
solution of (\ref{RSeq}) for $\varepsilon \geq 1$. In addition, if $
\varepsilon <1$, there is an open dense set ${\cal U}\subset {\cal B}^{1/2}$
of initial condition such that all solutions $\phi (t,x)$, with initial
condition on ${\cal U}$, is asymptotically of the form $\vartheta _{1}+Vt$
where $\vartheta _{1}=\int_{0}^{x}v_{1}(y)\,dy$, the velocity $V=\stackrel{\_\_
}{-\vartheta _{1}^{2}}/2$ and $v_{1}$ is the stable equilibrium
solution of (\ref{u}).]

\medskip

\begin{center}
\noindent MULTI--COALESCENT\ POLE\ SOLUTIONS
\end{center}

\medskip

The same scenario seems to hold for equation (\ref{MSeq}), at least if one
restricts to the space of coalescent pole solutions. According to \cite
{TFH,VM1}, if $\varepsilon $ is such that $%
{\displaystyle{1 \over 2n+1}}
\leq \varepsilon <
{\displaystyle{1 \over 2n-1}}
$, $n=1,2,\ldots $, there exist$\ n$ steadily propagating fronts, $\varphi
_{j}(t,x)=\vartheta _{j}(x)+Vt$, with $\vartheta _{j}(x)=
\displaystyle\int 
_{0}^{x}v_{j}(y)\,dy$ and 
\[
v_{j}(x)=\varepsilon \sum_{k=1}^{2j}\cot \left( \frac{x-z_{k}}{2}\right) \,
\]
for $j=1,\ldots ,n$, each corresponding to the coalescent $j$--pole solution
of (\ref{MSeq}), but only the one with largest number of poles, $\varphi _{n}
$, is asymptotically stable. When $\varepsilon $ crosses $1/(2n+1)$ from
above, $\varphi _{n}$ becomes unstable and a new solution $\varphi _{n+1}$
bifurcates from the former solution.

$\varphi _{j}$ was called coalescent $j$--pole solution because of the
dynamic of poles $\left\{ z_{k}(t)\right\} _{k=1}^{2j}$, induced by (\ref
{MSeq}), tends to align them parallel to the imaginary axis. In \cite
{TFH,VM1}, (\ref{MSeq}) was considered with periodic boundary conditions.
With Newmann (adiabatic) boundary conditions there are two sets of steadily
propagating fronts $\left\{ \varphi _{j}^{\pm }\right\} _{j=1}^{n}$
distinguished by the location of their poles: $\varphi _{j}^{+}$ and $
\varphi _{j}^{-}$ has poles aligned at $\Re e\,z=0$ and $\Re e\,z=\pi $,
respectively.

Note that there is a relation between the number of steadily propagating
fronts and of positive eigenvalues in the spectrum $\sigma (A)=\left\{ \eta
_{j}\right\} _{j\geq 1}$ of the linear operator $Av=\varepsilon v^{\prime
\prime }+I(v)$, since the latter increases by one unit at $\varepsilon
_{n}=1/n$ and the former at $\widetilde{\varepsilon }_{n}=1/\left(
2n-1\right) $. For equation (\ref{RSeq}), the number of steady solutions and
the dimension of the unstable manifold ${\cal M}_{n}$ of the trivial
solution $\varphi _{0}={\rm const}$, are equal with the instability value
now located at $1/n^{2}$. There, in contradistinction, $\varphi _{1}^{\pm }$
are the only asymptotically stable solution for all $\varepsilon <1$.

Although (\ref{phij}) holds for $\varphi
_{j}^{\pm }$ with $v_{j}^{\pm }$ a coalescent pole solution of 
\begin{equation}
\varepsilon v_{xx}-v\,v_{x}+I(v)=0\,,  \label{vI}
\end{equation}
$v_{j}^{\pm }$ does not vanish in $\left( 0,\pi \right) $. One may define a
coalescent $j$--pole solution $v_{j}^{(k)\pm }$ with $k-1$ zeros in $\left(
0,\pi \right) $, $k=1,2,\ldots $, by setting $v_{j}^{(k)\pm }(x)=
{\displaystyle{1 \over k}}
v_{j}^{\pm }(kx)$ mod $\pi $. Note $v_{j}^{(k)\pm }$ solves (\ref{vI})
provided $v_{j}^{\pm }$ solves the same equation with $\varepsilon $
replaced by $k\varepsilon $. As a consequence, a sets of steadily
propagating fronts $\left\{ \varphi _{j}^{(k)\pm }\right\} _{j=1}^{n}$
exists if $\varepsilon $ is such that $
{\displaystyle{1 \over 2n+1}}
\leq k\varepsilon <
{\displaystyle{1 \over 2n-1}}
$ holds for some $n=1,2,\ldots $ (see Figure \ref{bif}). In total,
there are $2 \mathop{\displaystyle\sum} _{m=1}^{n}\left[ {2n / (2m-1)}
\right] = o(n^{2})$ coalescent steady solutions for $\varepsilon \gtrsim 
{\displaystyle{1 \over 2n+1}}
$, where $\left[ z\right] $ means the integer part of real number $z$. The
solutions $\varphi _{j}^{(k)\pm }$ with $k\geq 2$, are not stable and may
represent the cellular profile observed experimentally provided an associate
quasi--equilibrium solution described in \cite{MS,BKS,SW} can be defined.

In the following, for each $\varepsilon $ such that $
{\displaystyle{1 \over n+1}}
\leq \varepsilon <
{\displaystyle{1 \over n}}
$ holds, a new family ${\cal F}_{n}$ of steadily propagating flame front
solutions, denominated multi--coalescent $n$--pole solutions, will be
introduced. Our preliminary investigation indicates that there are at
least an exponential number $c^{n-1}$ of solutions in ${\cal F}_{n}$ and all,
but $2$ of them, seems to be 
unstable by numerical computation. As a consequence, the invariant set $
{\cal K}_{n}^{MS}=\bigcup_{\varphi }{\cal M}_{n}(\varphi )$, defined as the
union of the unstable manifold of all equilibrium solutions, for
equation (\ref{MSeq}) differs enormously from the invariant set ${\cal
K}_{n}^{RS}=
{\cal M}_{n}(\varphi _{0})$ for equation (\ref{RSeq}). Here, $n$ indicates
the number of bifurcations with respect to the trivial solution $\varphi _{0}
$. In particular, ${\cal K}_{n}^{MS}$ may have dimension exponentially more
numerous than the dimension of ${\cal K}_{n}^{RS}$ (for comparison,
see Figures \ref{bifur} and \ref{bif}). We believe that this crucial
distinction is responsible for the disagreement between the numerical
study by Gutman--Sivashinky \cite{GS} and the exact calculation by
Vaymblat--Matalon \cite{VM1}. 

The bi--coalescent $n$--pole solutions $\varphi _{n_{0},n_{\pi }}$ are
indexed by $(n_{0},n_{\pi })$ with $n_{0}+n_{\pi }=n$ indicating the number
of pairs of complex conjugate poles\cite{nota4} in each line $\Re e\,z=0$
and $\Re e\,z=\pi $. Note that the system of equations governing the
dynamics of the poles $z_{j}=x_{j}+iy_{j}$, $j=1,\ldots ,2n$, in the complex
plane, given by 
\[
\stackrel{\cdot }{z_{j}}=-\varepsilon \sum_{l\neq j}\cot \left( \frac{
z_{j}-z_{l}}{2}\right) -i\,\frac{y_{j}}{\left| y_{j}\right| }\,,
\]
preserves the location of real part $x_{j}$ since, in this case, $\Re
e\,\cot \left( 
{\displaystyle{z_{j}-z_{l} \over 2}}
\right) =0$. The poles of the bi--coalescent solution $\varphi
_{n_{0},n_{\pi }}$ thus satisfies 
\begin{equation}
\stackrel{\cdot }{x_{j}}=0\qquad {\rm and}\qquad \stackrel{\cdot }{y_{j}}
=F_{j}  \label{xy}
\end{equation}
where 
\begin{equation}
F_{j}=\varepsilon \sum_{l\neq j}\left( \coth \left( \frac{y_{j}-y_{l}}{2}
\right) \right) ^{\eta _{jl}}-\frac{y_{j}}{\left| y_{j}\right| }\,.
\label{Fj}
\end{equation}
with $\eta _{jl}:=\cos \left( x_{j}-x_{l}\right) $ taking $\pm 1$ values
according the poles $z_{j}$ and $z_{l}$ are in the same or different line.
Because of the real parte $x_{j}$ of the pole gives the ``cusp'' position of
a propagating flame profile, the bi--coalescent solution has its tip
centered somewhere in the interior of channel.

Thual, Frisch and H\'{e}non \cite{TFH} have proven that, provided $n$ is
such that $\varepsilon \left( 2n-1\right) <1$, there exist one and only one
coalescent steady solution and any solution of (\ref{xy}) with $\eta _{jl}=1$
for all $j$ and $\,l$, tends toward this steady state as $t\rightarrow
\infty $. The proof of these properties is based on the existence of a
Liapunov function with negative curvature in every direction. Here, there
exists a Liapunov function for bi--coalescent solutions 
\[
U=\varepsilon \sum_{j,l:j\neq l}\ln \left| \frac{e^{(y_{j}-y_{l})/2}+\eta
_{jl}\,e^{-(y_{j}-y_{l})/2}}{2}\right| -\sum_{j}\left| y_{j}\right| \,\,,
\]
satisfying $\stackrel{\cdot }{U}=
\mathop{\displaystyle\sum}
_{j=1}^{2n}F_{j}^{2}\geq 0$ whose Hessian matrix $
H=[U_{y_{i}y_{j}}]_{i,j=1}^{2n}$ cannot be \ proven to be negative definite
in the case of strictly bi--coalescent solution ($n_{0},\,n_{\pi }\neq 0,n/2$
) since its Ger\v{s}gorin discs may have non--vanishing intersection with
the semi--plane $\Re e\,z\geq 0$.  As $U$ may have several local maxima and
saddle points in this case (see Figure 2), any solution of equations (\ref
{xy}) tends toward to a steady bi--coalescent state as $t\rightarrow \infty $
but uniqueness cannot be guaranteed.

Let the poles $z_{1},\ldots ,z_{2n}$ of a bi--coalescente steady solution $
\varphi _{n_{0},n_{\pi }}$ be indexed as follows: $x_{j}=0$ if $j=1,\ldots
,n_{0}$, $x_{j}=\pi $ if $j=n_{0}+1,\ldots ,n$ and $y_{n+j}=-y_{j}$ for $
j=1,\ldots ,n$. The case with $n_{0}=n_{\pi }=n/2$ plays special role to
describe the stability of coalescent solutions. Note that, if $%
y_{j}=y_{j+n/2}$ for $j=1,\ldots ,n/2$, then 
\begin{eqnarray*}
v_{n/2,n/2}(x) &=&\varepsilon \sum_{j=1}^{n/2}\sum_{\eta \in \left\{
-1,1\right\} }\left\{ \cot \left( \frac{x-i\eta y_{j}}{2}\right) +\cot
\left( \frac{x-\pi -i\eta y_{j}}{2}\right) \right\}  \\
&=&2\varepsilon \sum_{j=1}^{n/2}\sum_{\eta \in \left\{ -1,1\right\} }\cot
\left( x-i\eta y_{j}\right) 
\end{eqnarray*}
corresponds to a coalescent $n/2$--pole solution $v_{n/2}^{2}$ with one zero
in $\left( 0,\pi \right) $. As $\varepsilon $ varies from $1/\left(
2n-1\right) $ to $1/\left( 2n+1\right) $, a point $\left\{ z_{j}\right\}
_{j=1}^{2n}$ satisfying $y_{j}=y_{j+n/2}$ can be shown to change from saddle
point to a global maximum of $U$, turning a local maximum somewhere in
between.
\begin{figure}[!ht] 
\begin{center}
\epsfig{file=figure2.eps, height=7.5cm, width=12.5cm}
\end{center}
\caption{Bifurcation diagram for the distance $\Delta \varphi $
from cusp to tip of steadily state solutions of (\ref{RSeq}). Solid
and dashed lines refer, respectively, to stable 
and unstable solutions.}
\label{bifur} 
\end{figure}

\begin{figure}[!ht] 
\begin{center}
\epsfig{file=figure6.eps, height=7.5cm, width=12.5cm}
\end{center}
\caption{Bifurcation diagram for the distance $\Delta \varphi $
from cusp to tip of steadily state solutions of (\ref{MSeq}). Solid
and dashed lines refer, respectively, to stable 
and unstable coalescente solutions. Doted-dashed lines refer to
unstable multi--coalescente solutions.} 
\label{bif} 
\end{figure}

Two conclusions can immediately be extracted from these observations.
Because (\ref{xy}) tends to align the poles along either the line $\Re e\,z=0
$ or $\Re e\,z=\pi $, the coalescent $n$--pole solution $\varphi _{n}$ is
more stable than $\varphi _{n/2}^{2}$ since $\varphi _{n/2}^{2}$ is
equivalent to a bi--coalescent $n$--pole solution $\varphi _{n/2,n/2}$ which
is unstable under small perturbation which involves the real part $x_{j}$ of
the poles. Moreover, we may construct from a coalescent $n$--pole solution $
\varphi _{n}^{k}$ with $k-1$ zeros in $\left( 0,\pi \right) $ a family
of bi--coalescent solutions $\varphi _{n_{0},n_{\pi 
/k}}^{k}$ with $n_{0}$ poles aligned in $\Re e\,z=0$ and $n-n_{0}$ poles
aligned in $\Re e\,z=\pi /k$ which agree with the coalescent solution at $
\varepsilon \gtrsim 1/\left( 2kn+k\right) $ if $n_{0}=$ $n_{\pi /k}=n/2$ and 
$y_{j}=y_{j+n/2}$. Proceeding in a similar fashion, one can introduce
muti--coalescent $n$--pole solutions $\varphi _{n_{0},n_{2\pi
/k},\ldots n_{2(k-1)\pi /k} }$, with $n_{2j\pi /k} \ge 1$ poles at $\Re
e\,z=2j\pi /k$ and $
\mathop{\displaystyle\sum}
_{j=0}^{k-1}n_{2j\pi /k}=n$ and which vanishes at $k-1$ points in $\left(
0,\pi \right) $. Hence, a trajectory $\left\{ z_{j}(t)\right\} _{t\geq 0}$
of (\ref{xy}) in the phase space ${\Bbb C}^{n}$, with $\left\{
z_{j}(0)\right\} $ close to the poles of a multi--coalescent steady solution
may go along many intermediate steady states before it reaches the final
equilibrium.


\medskip 
%\newpage

\begin{center}
\noindent CONCLUSIONS
\end{center}

\medskip

Whether the cellular structure (flame profile with many wrinkles), observed 
numerically by Gutman--Sivashinky \cite{GS} and experimentally by \cite{G},
could be produced by the instability of the coalescent pole solutions for $
\varepsilon $ small, has been debated in the
literature (see e.g.\cite{KOP,RAS}). The work of Vaymblat--Matalon
\cite{VM1,VM2} has resolved the controversies by proving that there
always exist 
a unique (linearly) stable coalescent pole solution for $\varepsilon <1$. In 
\cite{VM1}, the discrepancy between the numerical and the exact results is
explained as an artefact of truncation and we shall not observe different
profiles if more modes were included. According to Joulin \cite{J}, once
equation (\ref{MSeq}) is incapable to describe observed wrinkled propagating
flames (as an equilibrium solution) it should be replaced by another model.

A different scenario has been presented for Rakib--Sivashinsky equation.
Numerical integrations of (\ref{RSeq}) have agreed with the analytic
prediction since the beginning, although truncated equation has been used.
Besides, parabolic profile with centered tip and cellular profiles can be
successfully explained as metastable states \cite{MiS,BKS,SW}. So, the
question to be addressed is why equation (\ref{MSeq}) is more sensitive than
(\ref{RSeq}) to be treated numerically and whether the cellular profiles can
be described as a quasi--equilibrium solution of (\ref{MSeq}).

Based in the present analysis of equation (\ref{RSeq}) and in the existence
of multi--coalescent steady states of (\ref{MSeq}), we argue in the following
that many questions remain to be investigated before Michelson--Sivashinsky
equation is abandoned.

Using the analysis in \cite{GM}, global existence in a Sobolev space $
H_{0}^{1}$ (for all times $t>0$) and uniqueness can be established for
equation (\ref{MSeq}). A basic question is: {\it Does the solution} $\phi $ 
{\it of (\ref{MSeq}) with initial condition }$\phi _{0}${\it \ in a dense
subset of} $H_{0}^{1}$ {\it converge, as} $t\rightarrow \infty ${\it \ and
for all} $\varepsilon <1$, {\it to one of coalescent steady state described
in \cite{VM1}}?

To answer this question a geometric analysis, as given for equation 
(\ref{RSeq}), must be carried out for equation (\ref{MSeq}). A
family ${\cal F}_{n}$ of steady multi--coalescent poles solutions have been
described. It may be difficult to determine whether ${\cal F}_{n}$ exhausts
all steady solutions of (\ref{MSeq}) but it is already a remarkable
difference between both equations. If all states of ${\cal F}_{n}$, except
the states $\varphi _{n}^{\pm }$ described by \cite{VM1}, are shown to be
unstable, then the unstable manifold ${\cal K}_{n}^{MS}$ containing the
origin would have dimension at least exponentially large with $n$. 

The method of \cite{ZLV} may be useful to construct a Liapunov function. To
establish the existence of a dense subspace of initial conditions from which
the solutions of (\ref{MSeq}) converge to a steady solution requires,
besides a Liapunov function, that the trajectories remain in a compact set.
For this, it is enough that the trajectories remain bounded, which follows
if the maximum principle can be shown to be adapted for equation (\ref{MSeq}).

Finally, the existence of multi--coalescent pole solutions would explain the
discrepancy between the numerical integration by Sivashinky and the linear
stability analysis in \cite{VM1}. For Rakib--Sivashinsky equation, there
exist metastable solutions whose time interval they remain ``stable'' becomes
exponentially long when $\varepsilon $ is small, creating the illusion that
they have reached the equilibrium. If metastable states can be constructed
from the bi--coalescent states is a question to be investigate. It would, in
particular, describe the quasi--stable behavior of parabolic steadily
propagating flame with centered tip. Moreover, the effect of truncation
would become more sensitive than for equation (\ref{RSeq}) in view of the
fact there is exponentially more numerous (meta)states available.



\begin{thebibliography}{BKS}
\bibitem[BKS]{BKS}  H. Berestyck, S. Kamin and G. Sivashinsky,
C. R. Acad. Sci. Paris, {\bf t. 321} 185
(1995).

\bibitem[C]{C}  Carmen Chicone, J. Diff. Eqns. \textbf{69},
310-321 (87).

\bibitem[G]{G}  E. Groff, Combust. Flame \textbf{48} 51-62 (1982)

\bibitem[GM]{GM}  L. R. Guidi and D. H. U. Marchetti, Commun. Math.
Phys. \textbf{219}, 671-702 (2001).

\bibitem[GS]{GS}  Gutman and Sivashinsky, Phys. \textbf{D 43} 129-139 (1990) 

%\bibitem[H]{H}  Daniel Henry, ``Geometric theory of semilinear
%parabolic equations''. Lecture Notes in Mathematics \textbf{840},
%Springer-Verlag, 1981.

\bibitem[J]{J}  G. Joulin, Phys. Rev. \textbf{E 50}
2030-2047 (1994).

\bibitem[KOP]{KOP} O. Kupervasser, Z. Olami and I. Procaccia,
Phys. Rev. Lett. \textbf{76}, 146-149 (1996).

\bibitem[MG]{MG}  D. H. U. Marchetti and L. R. Guidi, in preparation

\bibitem[MiS]{MiS}  A. B. Mikishev and G. I. Sivashinsky,
Phys. Lett. \textbf{A 175}, 409-414 (1993).

\bibitem[MS]{MS}  D. M. Michelson and G. I. Sivashinsky,
Acta Astronautica \textbf{4}, 1207 (1977).

\bibitem[RS]{RS}  Z. Rakib and G. I. Sivashinsky, Combust. Sci and
Tech. \textbf{54}, 69 (1987). 

\bibitem[RAS]{RAS} O. Rahib, N. Aubry  and G. I. Sivashinsky,
Combust. Theor. Modelling \textbf{2}, 19-41 (1998). 

\bibitem[S]{S}  G. I. Sivashinsky, Acta Astronautica \textbf{4}, 1177 (1977)

\bibitem[SW]{SW}  X. Sun and M. J. Ward, Euro.
Journ. Appl. Math. {\bf 10}, 27-53 (1999).

\bibitem[TFH]{TFH}  O. Thual, U. Frisch and M. H\'{e}non,
J. Physique \textbf{46}, 1485 (1985). 

\bibitem[VM1]{VM1}  D. Vaymblat and M. Matalon, SIAM
J. Appl. Math. \textbf{60}, 679 (2000). 

\bibitem[VM2]{VM2}  D. Vaymblat and M. Matalon, SIAM
J. Appl. Math. \textbf{60}, 703 (2000). 

\bibitem[ZLV]{ZLV}  T. I. Zelenyak, M. M. Lavrentiev Jr. and M. P.
Vishnevskii, ``Qualitative theory of parabolic equations''. VSP,
Utrecht, 1997.

\bibitem{nota0}  $I(\phi )={\rm P.V}.
\displaystyle\int 
{\displaystyle{dy \over x-y}}
\,\phi _{x}(t,y)$ is the composition of minus the Hilbert transform and the
derivative of $\phi $ with respect to $x$.

\bibitem{nota}  In \cite{VM1}, the Darrieus--Landau instability term $I(\phi
)$ of (\ref{MSeq}) has a $1/2$ factor in front so, the bifurcation point is
at $\varepsilon =1/\gamma =1/2$. In \cite{BKS}, equation (\ref{RSeq}) is
defined for $x\in \left( 0,1\right) $ and the bifurcation occurs at $
\varepsilon =\pi ^{-2}$.

%\bibitem{nota1}  This is due to the fact $\varepsilon $ can be set to $1$ in
%(\ref{ds}) provided $x$ is replaced by $\sqrt{\varepsilon }x$ and $p$ is
%replaced by $\sqrt{\varepsilon }p$.

%\bibitem{nota2}  It follows from $T(\varepsilon ,w_{0})=\sqrt{\varepsilon }
%T(1,w_{0})$ that, $T(1,g(\varepsilon ))\rightarrow \infty $ as $\varepsilon
%\rightarrow 0$ and this implies, by the continuity of orbits, $g(\varepsilon
%)\nearrow 1$ because the orbit $\gamma _{1}$ has infinite period: $
%T(\varepsilon ,1)=\infty $ for any $0<\varepsilon \leq 1$.

\bibitem{nota3}  Here, $L$ is defined to be minus the $L$ operator in \cite
{GM}.

\bibitem{nota4}  Because we are looking for real valued solutions of (\ref
{MSeq}), poles must occur in pairs of complex conjugate.

%\bibitem{nota5}  The real parte $x_{j}(t)=\Re e\,z_{j}(t)$ of the pole
%position $z_{j}(t)$ gives the cusp position of the flame profile. 
\end{thebibliography}

\end{document}
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.57503 .30141 L
.57699 .30271 L
.57915 .30381 L
.58144 .30469 L
.58342 .30526 L
.58528 .30566 L
.58634 .30583 L
.58731 .30596 L
.58839 .30606 L
.58892 .3061 L
.58951 .30612 L
.59056 .30615 L
.59152 .30614 L
.59202 .30612 L
.59258 .30609 L
.59316 .30605 L
.5937 .306 L
.59469 .30588 L
.5956 .30574 L
.59765 .3053 L
.59946 .30476 L
.60144 .30398 L
Mistroke
.60331 .303 L
.60504 .30183 L
.60915 .29628 L
.60973 .29347 L
.61027 .29347 L
.61084 .29347 L
.61146 .29347 L
.61358 .29347 L
.62162 .29347 L
.6379 .29347 L
.65264 .29347 L
.68349 .29347 L
.71627 .29347 L
.74786 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0544 .29347 L
.08775 .29347 L
.11908 .29347 L
.14921 .29347 L
.18128 .29347 L
.21215 .29347 L
.24496 .29347 L
.27657 .29347 L
.30698 .29347 L
.33933 .29347 L
.37049 .29347 L
.40044 .29347 L
.43233 .29347 L
.46303 .29347 L
.49566 .29347 L
.5271 .29347 L
.54296 .29347 L
.55734 .29347 L
.56119 .29347 L
.56536 .29347 L
.56727 .29347 L
.56831 .29347 L
.56929 .29347 L
.57012 .29347 L
.57102 .29056 L
.57201 .28868 L
.57291 .2875 L
.57503 .28554 L
.57699 .28423 L
.57915 .28314 L
.58144 .28226 L
.58342 .28168 L
.58528 .28128 L
.58634 .28111 L
.58731 .28098 L
.58839 .28088 L
.58892 .28085 L
.58951 .28082 L
.59056 .28079 L
.59152 .28081 L
.59202 .28082 L
.59258 .28085 L
.59316 .2809 L
.5937 .28095 L
.59469 .28106 L
.5956 .28121 L
.59765 .28164 L
.59946 .28218 L
.60144 .28296 L
Mistroke
.60331 .28394 L
.60504 .28512 L
.60915 .29066 L
.60973 .29347 L
.61027 .29347 L
.61084 .29347 L
.61146 .29347 L
.61358 .29347 L
.62162 .29347 L
.6379 .29347 L
.65264 .29347 L
.68349 .29347 L
.71627 .29347 L
.74786 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0544 .29347 L
.08775 .29347 L
.11908 .29347 L
.14921 .29347 L
.18128 .29347 L
.21215 .29347 L
.24496 .29347 L
.27657 .29347 L
.30698 .29347 L
.33933 .29347 L
.37049 .29347 L
.40044 .29347 L
.43233 .29347 L
.46303 .29347 L
.47863 .29347 L
.49566 .29347 L
.50355 .29347 L
.50788 .29347 L
.51197 .29347 L
.51578 .29347 L
.51682 .29347 L
.51793 .29347 L
.51887 .29347 L
.5199 .29602 L
.52076 .30008 L
.52157 .30231 L
.52242 .30414 L
.52335 .30584 L
.5253 .30874 L
.5271 .31096 L
.53125 .3151 L
.53518 .31823 L
.54254 .32287 L
.55051 .3267 L
.5591 .32982 L
.56668 .33188 L
.57102 .3328 L
.57497 .33348 L
.57915 .33403 L
.58145 .33427 L
.58362 .33444 L
.5846 .3345 L
.58563 .33456 L
.58652 .3346 L
.58748 .33464 L
.58851 .33467 L
.58948 .33468 L
.59007 .33469 L
.59062 .33469 L
Mistroke
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.59167 .33468 L
.59219 .33468 L
.59273 .33467 L
.5937 .33464 L
.59474 .33461 L
.59589 .33455 L
.59795 .33441 L
.59986 .33425 L
.60344 .33381 L
.60728 .33318 L
.61174 .33218 L
.61597 .33095 L
.61974 .32958 L
.62389 .32774 L
.63155 .3231 L
.63575 .31958 L
.63968 .31527 L
.64666 .30051 L
.64764 .29347 L
.64857 .29347 L
.64964 .29347 L
.65061 .29347 L
.65428 .29347 L
.66208 .29347 L
.67029 .29347 L
.68494 .29347 L
.71544 .29347 L
.74787 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0544 .29347 L
.08775 .29347 L
.11908 .29347 L
.14921 .29347 L
.18128 .29347 L
.21215 .29347 L
.24496 .29347 L
.27657 .29347 L
.30698 .29347 L
.33933 .29347 L
.37049 .29347 L
.40044 .29347 L
.43233 .29347 L
.46303 .29347 L
.47863 .29347 L
.49566 .29347 L
.50355 .29347 L
.50788 .29347 L
.51197 .29347 L
.51578 .29347 L
.51682 .29347 L
.51793 .29347 L
.51887 .29347 L
.5199 .29092 L
.52076 .28687 L
.52157 .28463 L
.52242 .2828 L
.52335 .28111 L
.5253 .2782 L
.5271 .27598 L
.53125 .27185 L
.53518 .26871 L
.54254 .26407 L
.55051 .26025 L
.5591 .25712 L
.56668 .25506 L
.57102 .25414 L
.57497 .25347 L
.57915 .25291 L
.58145 .25268 L
.58362 .2525 L
.5846 .25244 L
.58563 .25238 L
.58652 .25234 L
.58748 .2523 L
.58851 .25228 L
.58948 .25226 L
.59007 .25225 L
.59062 .25225 L
Mistroke
.59112 .25225 L
.59167 .25226 L
.59219 .25226 L
.59273 .25227 L
.5937 .2523 L
.59474 .25234 L
.59589 .25239 L
.59795 .25253 L
.59986 .2527 L
.60344 .25313 L
.60728 .25377 L
.61174 .25477 L
.61597 .256 L
.61974 .25736 L
.62389 .2592 L
.63155 .26384 L
.63575 .26736 L
.63968 .27167 L
.64666 .28643 L
.64764 .29347 L
.64857 .29347 L
.64964 .29347 L
.65061 .29347 L
.65428 .29347 L
.66208 .29347 L
.67029 .29347 L
.68494 .29347 L
.71544 .29347 L
.74787 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0544 .29347 L
.08775 .29347 L
.11908 .29347 L
.14921 .29347 L
.18128 .29347 L
.21215 .29347 L
.24496 .29347 L
.27657 .29347 L
.30698 .29347 L
.33933 .29347 L
.37049 .29347 L
.40044 .29347 L
.41709 .29347 L
.43233 .29347 L
.43986 .29347 L
.44192 .29347 L
.44412 .29347 L
.44518 .29347 L
.44619 .29347 L
.44709 .29347 L
.44807 .29803 L
.44894 .30242 L
.44987 .30545 L
.4509 .30806 L
.45186 .31012 L
.45586 .3168 L
.46303 .32525 L
.47108 .3324 L
.47984 .33866 L
.49577 .34762 L
.51143 .35443 L
.52806 .36007 L
.54347 .36408 L
.55794 .36686 L
.56573 .36796 L
.56997 .36845 L
.57392 .36883 L
.57759 .36911 L
.58092 .36931 L
.58282 .3694 L
.58462 .36946 L
.58651 .36952 L
.58758 .36954 L
.58857 .36955 L
.58956 .36956 L
.5905 .36956 L
.59135 .36956 L
.59227 .36956 L
.59327 .36955 L
Mistroke
.59383 .36954 L
.59433 .36953 L
.59625 .36948 L
.59802 .36941 L
.59994 .36932 L
.60343 .36909 L
.60757 .36872 L
.61194 .36821 L
.61976 .36692 L
.62729 .36519 L
.63534 .3627 L
.64294 .35963 L
.64987 .35606 L
.65774 .35084 L
.66501 .34448 L
.67299 .33476 L
.67701 .32797 L
.6793 .32309 L
.68145 .31733 L
.68246 .31397 L
.68341 .3101 L
.68394 .30746 L
.6845 .30382 L
.6855 .29347 L
.68645 .29347 L
.68745 .29347 L
.68926 .29347 L
.69318 .29347 L
.69746 .29347 L
.71212 .29347 L
.74216 .29347 L
.77413 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0544 .29347 L
.08775 .29347 L
.11908 .29347 L
.14921 .29347 L
.18128 .29347 L
.21215 .29347 L
.24496 .29347 L
.27657 .29347 L
.30698 .29347 L
.33933 .29347 L
.37049 .29347 L
.40044 .29347 L
.41709 .29347 L
.43233 .29347 L
.43986 .29347 L
.44192 .29347 L
.44412 .29347 L
.44518 .29347 L
.44619 .29347 L
.44709 .29347 L
.44807 .28891 L
.44894 .28452 L
.44987 .28149 L
.4509 .27888 L
.45186 .27682 L
.45586 .27015 L
.46303 .26169 L
.47108 .25454 L
.47984 .24828 L
.49577 .23932 L
.51143 .23252 L
.52806 .22687 L
.54347 .22286 L
.55794 .22008 L
.56573 .21898 L
.56997 .21849 L
.57392 .21812 L
.57759 .21783 L
.58092 .21763 L
.58282 .21755 L
.58462 .21748 L
.58651 .21743 L
.58758 .2174 L
.58857 .21739 L
.58956 .21738 L
.5905 .21738 L
.59135 .21738 L
.59227 .21739 L
.59327 .2174 L
Mistroke
.59383 .21741 L
.59433 .21742 L
.59625 .21746 L
.59802 .21753 L
.59994 .21762 L
.60343 .21785 L
.60757 .21822 L
.61194 .21874 L
.61976 .22002 L
.62729 .22175 L
.63534 .22424 L
.64294 .22731 L
.64987 .23088 L
.65774 .2361 L
.66501 .24247 L
.67299 .25218 L
.67701 .25897 L
.6793 .26385 L
.68145 .26961 L
.68246 .27298 L
.68341 .27685 L
.68394 .27948 L
.6845 .28312 L
.6855 .29347 L
.68645 .29347 L
.68745 .29347 L
.68926 .29347 L
.69318 .29347 L
.69746 .29347 L
.71212 .29347 L
.74216 .29347 L
.77413 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0544 .29347 L
.08775 .29347 L
.11908 .29347 L
.14921 .29347 L
.18128 .29347 L
.21215 .29347 L
.24496 .29347 L
.27657 .29347 L
.30698 .29347 L
.32268 .29347 L
.3265 .29347 L
.32861 .29347 L
.33061 .29347 L
.33173 .29347 L
.33237 .29347 L
.33296 .29667 L
.33354 .30143 L
.33409 .30412 L
.33511 .30786 L
.33716 .31329 L
.33933 .31777 L
.34706 .32933 L
.35523 .3383 L
.36985 .35073 L
.39991 .36939 L
.43192 .38404 L
.46272 .395 L
.49546 .40408 L
.51185 .40772 L
.52701 .41057 L
.54266 .41296 L
.55735 .41467 L
.56446 .4153 L
.57213 .41583 L
.57627 .41605 L
.57853 .41614 L
.58064 .41622 L
.58259 .41628 L
.58436 .41632 L
.58633 .41635 L
.58741 .41637 L
.58844 .41638 L
.58935 .41638 L
.59033 .41638 L
.59126 .41638 L
.5921 .41638 L
.59309 .41638 L
.59365 .41637 L
.59417 .41636 L
Mistroke
.59613 .41633 L
.5972 .41631 L
.5982 .41629 L
.60008 .41623 L
.60428 .41605 L
.60794 .41584 L
.6113 .4156 L
.61889 .41487 L
.62662 .41383 L
.63382 .41257 L
.64173 .41081 L
.65019 .40841 L
.65827 .40555 L
.66704 .40166 L
.67462 .39749 L
.68298 .39176 L
.69107 .3847 L
.69866 .37619 L
.70668 .36412 L
.71118 .35508 L
.7153 .34435 L
.7171 .33847 L
.71904 .33072 L
.71993 .32641 L
.72088 .32099 L
.72168 .31519 L
.72257 .30577 L
.72358 .29347 L
.72415 .29347 L
.72467 .29347 L
.72663 .29347 L
.73032 .29347 L
.73882 .29347 L
.74671 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0544 .29347 L
.08775 .29347 L
.11908 .29347 L
.14921 .29347 L
.18128 .29347 L
.21215 .29347 L
.24496 .29347 L
.27657 .29347 L
.30698 .29347 L
.32268 .29347 L
.3265 .29347 L
.32861 .29347 L
.33061 .29347 L
.33173 .29347 L
.33237 .29347 L
.33296 .29027 L
.33354 .28551 L
.33409 .28282 L
.33511 .27909 L
.33716 .27365 L
.33933 .26917 L
.34706 .25761 L
.35523 .24864 L
.36985 .23621 L
.39991 .21755 L
.43192 .20291 L
.46272 .19194 L
.49546 .18286 L
.51185 .17922 L
.52701 .17638 L
.54266 .17398 L
.55735 .17227 L
.56446 .17164 L
.57213 .17111 L
.57627 .1709 L
.57853 .1708 L
.58064 .17072 L
.58259 .17067 L
.58436 .17062 L
.58633 .17059 L
.58741 .17058 L
.58844 .17057 L
.58935 .17056 L
.59033 .17056 L
.59126 .17056 L
.5921 .17056 L
.59309 .17057 L
.59365 .17057 L
.59417 .17058 L
Mistroke
.59613 .17061 L
.5972 .17063 L
.5982 .17066 L
.60008 .17071 L
.60428 .17089 L
.60794 .1711 L
.6113 .17134 L
.61889 .17207 L
.62662 .17311 L
.63382 .17438 L
.64173 .17614 L
.65019 .17853 L
.65827 .18139 L
.66704 .18528 L
.67462 .18945 L
.68298 .19519 L
.69107 .20224 L
.69866 .21075 L
.70668 .22283 L
.71118 .23186 L
.7153 .2426 L
.7171 .24847 L
.71904 .25622 L
.71993 .26053 L
.72088 .26595 L
.72168 .27175 L
.72257 .28117 L
.72358 .29347 L
.72415 .29347 L
.72467 .29347 L
.72663 .29347 L
.73032 .29347 L
.73882 .29347 L
.74671 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0387 .29347 L
.0544 .29347 L
.06209 .29347 L
.07032 .29347 L
.07275 .29347 L
.07504 .29347 L
.07604 .29347 L
.0771 .29347 L
.07766 .29347 L
.07827 .30177 L
.07882 .30501 L
.07934 .30738 L
.08146 .31441 L
.08339 .3192 L
.08775 .32761 L
.09555 .33888 L
.10278 .34724 L
.11919 .36245 L
.15017 .38421 L
.18309 .40231 L
.2148 .41692 L
.24532 .42918 L
.27779 .44074 L
.30905 .45069 L
.33911 .45933 L
.37111 .46761 L
.40191 .47475 L
.43466 .48145 L
.4662 .48704 L
.49654 .49153 L
.51189 .49345 L
.52883 .49527 L
.54505 .49667 L
.55992 .49764 L
.56738 .49801 L
.57523 .49829 L
.57932 .49839 L
.58127 .49843 L
.58312 .49847 L
.58417 .49848 L
.5853 .49849 L
.58625 .4985 L
.58729 .49851 L
.58783 .49851 L
.58841 .49852 L
.58903 .49852 L
.58961 .49852 L
.59018 .49852 L
.59072 .49852 L
Mistroke
.59174 .49852 L
.59273 .49852 L
.59329 .49852 L
.5938 .49851 L
.59474 .49851 L
.59574 .4985 L
.59746 .49847 L
.59935 .49844 L
.60355 .49835 L
.60745 .49822 L
.61449 .49789 L
.61843 .49765 L
.62202 .49739 L
.62937 .49674 L
.63716 .49585 L
.64571 .49462 L
.65361 .4932 L
.66905 .48949 L
.68353 .48456 L
.69134 .48112 L
.69954 .47675 L
.7142 .46627 L
.72178 .45893 L
.72986 .44887 L
.73749 .43621 L
.74441 .42035 L
.74826 .40855 L
.75234 .39209 L
.75401 .38343 L
.75581 .3722 L
.75685 .36444 L
.75779 .3561 L
.75963 .33315 L
.76064 .30694 L
.76159 .29347 L
.76259 .29347 L
.76368 .29347 L
.76597 .29347 L
.76809 .29347 L
.77611 .29347 L
.77778 .29347 L
Mfstroke
.02381 .29347 m
.0387 .29347 L
.0544 .29347 L
.06209 .29347 L
.07032 .29347 L
.07275 .29347 L
.07504 .29347 L
.07604 .29347 L
.0771 .29347 L
.07766 .29347 L
.07827 .28517 L
.07882 .28193 L
.07934 .27956 L
.08146 .27253 L
.08339 .26774 L
.08775 .25933 L
.09555 .24806 L
.10278 .2397 L
.11919 .22449 L
.15017 .20274 L
.18309 .18464 L
.2148 .17002 L
.24532 .15777 L
.27779 .14621 L
.30905 .13625 L
.33911 .12761 L
.37111 .11933 L
.40191 .11219 L
.43466 .10549 L
.4662 .09991 L
.49654 .09541 L
.51189 .09349 L
.52883 .09168 L
.54505 .09027 L
.55992 .0893 L
.56738 .08894 L
.57523 .08865 L
.57932 .08855 L
.58127 .08851 L
.58312 .08848 L
.58417 .08846 L
.5853 .08845 L
.58625 .08844 L
.58729 .08843 L
.58783 .08843 L
.58841 .08842 L
.58903 .08842 L
.58961 .08842 L
.59018 .08842 L
.59072 .08842 L
Mistroke
.59174 .08842 L
.59273 .08842 L
.59329 .08843 L
.5938 .08843 L
.59474 .08844 L
.59574 .08845 L
.59746 .08847 L
.59935 .0885 L
.60355 .0886 L
.60745 .08873 L
.61449 .08905 L
.61843 .0893 L
.62202 .08955 L
.62937 .0902 L
.63716 .09109 L
.64571 .09233 L
.65361 .09375 L
.66905 .09745 L
.68353 .10239 L
.69134 .10582 L
.69954 .11019 L
.7142 .12067 L
.72178 .12802 L
.72986 .13807 L
.73749 .15073 L
.74441 .16659 L
.74826 .1784 L
.75234 .19485 L
.75401 .20351 L
.75581 .21474 L
.75685 .22251 L
.75779 .23085 L
.75963 .25379 L
.76064 .28001 L
.76159 .29347 L
.76259 .29347 L
.76368 .29347 L
.76597 .29347 L
.76809 .29347 L
.77611 .29347 L
.77778 .29347 L
Mfstroke
.78912 .29347 m
.79266 .29347 L
.7964 .29347 L
.79735 .29347 L
.79787 .29347 L
.79812 .29347 L
.79836 .29347 L
.79858 .29928 L
.79882 .31452 L
.79906 .32262 L
.79932 .32942 L
.80019 .34572 L
.80122 .35955 L
.80233 .37124 L
.80434 .38788 L
.80832 .41192 L
.81211 .42894 L
.81931 .45317 L
.82697 .4724 L
.83435 .48719 L
.84218 .50032 L
.84974 .5112 L
.857 .52044 L
.86473 .52926 L
.87218 .53697 L
.87933 .54379 L
.88695 .55053 L
.89429 .55658 L
.90208 .56261 L
.90959 .56808 L
.91682 .57308 L
.9245 .57813 L
.93191 .58278 L
.93902 .58707 L
.9466 .59145 L
.95389 .59552 L
.96165 .59969 L
.96863 .60332 L
s
.78912 .29347 m
.7964 .29347 L
.80434 .29347 L
.8118 .29347 L
.81897 .29347 L
.82661 .29347 L
.83013 .29347 L
.83396 .29347 L
.83497 .29347 L
.83553 .29347 L
.8358 .29347 L
.83605 .29347 L
.83627 .29347 L
.83649 .30297 L
.83672 .30915 L
.83696 .31385 L
.83749 .32124 L
.83797 .32663 L
.83981 .34165 L
.84177 .35341 L
.84581 .3717 L
.84961 .38512 L
.85687 .40534 L
.8646 .42232 L
.87205 .43595 L
.8792 .44733 L
.88682 .45807 L
.89416 .46737 L
.90195 .47637 L
.90946 .48434 L
.91669 .49146 L
.92438 .49855 L
.93178 .50496 L
.93889 .51079 L
.94647 .51668 L
.95376 .52208 L
.96152 .52756 L
.96863 .53238 L
s
.78912 .29347 m
.7964 .29347 L
.80434 .29347 L
.8118 .29347 L
.81897 .29347 L
.82661 .29347 L
.83396 .29347 L
.84177 .29347 L
.8493 .29347 L
.85654 .29347 L
.86424 .29347 L
.86807 .29347 L
.86994 .29347 L
.87166 .29347 L
.8725 .29347 L
.87294 .29347 L
.87342 .29347 L
.87365 .29347 L
.87387 .29347 L
.87407 .29347 L
.87429 .30156 L
.87449 .30622 L
.87471 .30974 L
.87509 .31463 L
.876 .32309 L
.87686 .32919 L
.87879 .34009 L
.88263 .35608 L
.88612 .36747 L
.89363 .38681 L
.90086 .40164 L
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cleartomark{restore}if

%%EndProcSet
%%BeginProcSet: Math3.pfa
%!PS-AdobeFont-1.0: Math3 001.200
%%CreationDate: 1/15/98 at 3:54 PM
%%VMusage: 1024 32956  
% Mathematica typeface design by Andre Kuzniarek. Copyright \(c\) 1996-98 Wolfram Research, Inc. [http://www.wolfram.com]. All rights reserved. [Font version 1.20]
% ADL: 800 200 0
%%EndComments
FontDirectory/Math3 known{/Math3 findfont dup/UniqueID known{dup
/UniqueID get 5095645 eq exch/FontType get 1 eq and}{pop false}ifelse
{save true}{false}ifelse}{false}ifelse
20 dict begin
/FontInfo 16 dict dup begin
  /version (001.200) readonly def
  /FullName (Math3) readonly def
  /FamilyName (Math3) readonly def
  /Weight (Medium) readonly def
  /ItalicAngle 0 def
  /isFixedPitch false def
  /UnderlinePosition -133 def
  /UnderlineThickness 20 def
  /Notice (Mathematica typeface design by Andre Kuzniarek. Copyright \(c\) 1996-98 Wolfram Research, Inc. [http://www.wolfram.com]. All rights reserved. [Font version 1.20]) readonly def
  /em 1000 def
  /ascent 800 def
  /descent 200 def
end readonly def
/FontName /Math3 def
/Encoding 256 array
dup 0/NUL put
dup 1/Eth put
dup 2/eth put
dup 3/Lslash put
dup 4/lslash put
dup 5/Scaron put
dup 6/scaron put
dup 7/Yacute put
dup 8/yacute put
dup 9/HT put
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dup 14/Zcaron put
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dup 22/onequarter put
dup 23/onesuperior put
dup 24/threequarters put
dup 25/threesuperior put
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dup 27/brokenbar put
dup 28/minus put
dup 29/multiply put
dup 30/RS put
dup 31/US put
dup 32/Space put
dup 33/Integral put
dup 34/ContourIntegral put
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dup 36/CntrClckwContInteg put
dup 37/ClckwContInteg put
dup 38/SquareContInteg put
dup 39/Intersection put
dup 40/Union put
dup 41/UnionPlus put
dup 42/SquareIntersection put
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dup 44/Wedge put
dup 45/SmallWedge put
dup 46/Vee put
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dup 48/LTriangle put
dup 49/LTriangleEqual put
dup 50/LTriangleBar put
dup 51/NotLTriangle put
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dup 53/NotLTriangleBar put
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dup 55/Parallel put
dup 56/TrplBar put
dup 57/ShortBar put
dup 58/ShortParallelBar put
dup 59/BarExt put
dup 60/Tangental put
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dup 63/EqualTilde put
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dup 70/NotBar put
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dup 73/ShortNotBar put
dup 74/ShortNotParallelBar put
dup 75/ParallelExt put
dup 76/NotTilde put
dup 77/NotTildeTilde put
dup 78/NotTildeEqual put
dup 79/NotEqualTilde put
dup 80/HumpEqual put
dup 81/HumpDownHump put
dup 82/NotHumpEqual put
dup 83/NotHumpDownHump put
dup 84/NotCongruent put
dup 85/DotEqual put
dup 86/RoundImplies put
dup 87/Equal put
dup 88/TrplEqual put
dup 89/FrownEqual put
dup 90/SmileEqual put
dup 91/Frown put
dup 92/Smile put
dup 93/FrownSmile put
dup 94/SmileFrown put
dup 95/NotSmileFrown put
dup 96/LessLess put
dup 97/DblLess put
dup 98/LessSlantEqual put
dup 99/LessFullEqual put
dup 100/LessTilde put
dup 101/NotLess put
dup 102/NotLessLess put
dup 103/NotDblLess put
dup 104/NotLessEqual put
dup 105/NotLessSlantEqual put
dup 106/NotLessFullEqual put
dup 107/NotLessTilde put
dup 108/SquareSubset put
dup 109/SquareSubsetEqual put
dup 110/NotSquareSubset put
dup 111/NotSqrSubsetEqual put
dup 112/GrtrGrtr put
dup 113/DblGrtr put
dup 114/GrtrSlantEqual put
dup 115/GrtrFullEqual put
dup 116/GrtrTilde put
dup 117/NotGrtr put
dup 118/NotGrtrGrtr put
dup 119/NotDblGrtr put
dup 120/NotGrtrEqual put
dup 121/NotGrtrSlantEqual put
dup 122/NotGrtrFullEqual put
dup 123/NotGrtrTilde put
dup 124/SquareSprset put
dup 125/SquareSprsetEqual put
dup 126/NotSquareSprset put
dup 127/DEL put
dup 128/Prcds put
dup 129/PrcdsEqual put
dup 130/PrcdsSlantEqual put
dup 131/PrcdsTilde put
dup 132/NotPrcds put
dup 133/NotPrcdsEqual put
dup 134/NotPrcdsSlantEqual put
dup 135/NotPrcdsTilde put
dup 136/LessGrtr put
dup 137/NotLessGrtr put
dup 138/LessEqualGrtr put
dup 139/Because put
dup 140/NotSqrSprsetEqual put
dup 141/NotSubsetEqual put
dup 142/NotSprset put
dup 143/NotSprsetEqual put
dup 144/Succds put
dup 145/SuccdsEqual put
dup 146/SuccdsSlantEqual put
dup 147/SuccdsTilde put
dup 148/NotSuccds put
dup 149/NotSuccdsEqual put
dup 150/NotSuccdsSlantEqual put
dup 151/NotSuccdsTilde put
dup 152/GrtrLess put
dup 153/NotGrtrLess put
dup 154/GrtrEqualLess put
dup 155/Proportion put
dup 156/Member put
dup 157/NotMember put
dup 158/CircleMinus put
dup 159/CircleDot put
dup 160/EqualParallel put
dup 161/MinusPlus put
dup 162/RTack put
dup 163/DblRTack put
dup 164/LTack put
dup 165/DblLTack put
dup 166/UpTee put
dup 167/DownTee put
dup 168/DownTeeLong put
dup 169/Between put
dup 170/VertTilde put
dup 171/HorizHourglass put
dup 172/RightAngle put
dup 173/MeasuredAngle put
dup 174/SphericalAngle put
dup 175/Diameter put
dup 176/Mho put
dup 177/NotExists put
dup 178/CapDigamma put
dup 179/CapStigma put
dup 180/CapKoppa put
dup 181/CapSampi put
dup 182/CurlyEpsilon put
dup 183/CurlyRho put
dup 184/Digamma put
dup 185/Stigma put
dup 186/Koppa put
dup 187/Sampi put
dup 188/Bet put
dup 189/Gimel put
dup 190/Dalet put
dup 191/Aleph put
dup 192/CapYAcute put
dup 193/YAcute put
dup 194/CapEth put
dup 195/Eth put
dup 196/CapThorn put
dup 197/Thorn put
dup 198/CapLSlash put
dup 199/LSlash put
dup 200/DownQuestion put
dup 201/Cent put
dup 202/Nbspace put
dup 203/Sterling put
dup 204/DownExclamation put
dup 205/Section put
dup 206/Paragraph put
dup 207/Yen put
dup 208/Micro put
dup 209/HBar put
dup 210/DotlessI put
dup 211/ItalDotlessI put
dup 212/DotlessJ put
dup 213/ItalDotlessJ put
dup 214/Dagger put
dup 215/DoubleDagger put
dup 216/Degree put
dup 217/Flat put
dup 218/Natural put
dup 219/Sharp put
dup 220/QuestionBox put
dup 221/ErrorIndicator put
dup 222/Angstrom put
dup 223/LamdaSlash put
dup 224/SolidSquare put
dup 225/Square put
dup 226/MedSolidSquare put
dup 227/MedEmptySquare put
dup 228/SmallSolidSquare put
dup 229/SmallEmptySquare put
dup 230/SolidCircle put
dup 231/Circle put
dup 232/Bullet put
dup 233/SmallCircle put
dup 234/Dot put
dup 235/OpenDot put
dup 236/MedSolidDiamond put
dup 237/MedEmptyDiamond put
dup 238/SmallSolidDiamond put
dup 239/SmallEmptyDiamond put
dup 240/SolidRectangle put
dup 241/Rectangle put
dup 242/SolidUpTriangle put
dup 243/UpTriangle put
dup 244/SolidDownTriangle put
dup 245/DownTriangle put
dup 246/SolidDiamond put
dup 247/Star6 put
dup 248/Star put
dup 249/Bullet3 put
dup 250/Bullet4 put
dup 251/Euler put
dup 252/ImagJ put
dup 253/DoubleGamma put
dup 254/DoublePi put
dup 255/DoubleScriptCapD put
 readonly def
/PaintType 0 def
/FontType 1 def
/StrokeWidth 0 def
/FontMatrix[0.001 0 0 0.001 0 0]readonly def
/UniqueID 5095645 def
/FontBBox{-109 -231 966 869}readonly def
currentdict end
currentfile eexec
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cleartomark{restore}if

%%EndProcSet
%%BeginProcSet: Math3.pfa
%!PS-AdobeFont-1.0: Math3 001.200
%%CreationDate: 1/15/98 at 3:54 PM
%%VMusage: 1024 32956  
% Mathematica typeface design by Andre Kuzniarek. Copyright \(c\) 1996-98 Wolfram Research, Inc. [http://www.wolfram.com]. All rights reserved. [Font version 1.20]
% ADL: 800 200 0
%%EndComments
FontDirectory/Math3 known{/Math3 findfont dup/UniqueID known{dup
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  /isFixedPitch false def
  /UnderlinePosition -133 def
  /UnderlineThickness 20 def
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  /ascent 800 def
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dup 30/RS put
dup 31/US put
dup 32/Space put
dup 33/Integral put
dup 34/ContourIntegral put
dup 35/DblContInteg put
dup 36/CntrClckwContInteg put
dup 37/ClckwContInteg put
dup 38/SquareContInteg put
dup 39/Intersection put
dup 40/Union put
dup 41/UnionPlus put
dup 42/SquareIntersection put
dup 43/SquareUnion put
dup 44/Wedge put
dup 45/SmallWedge put
dup 46/Vee put
dup 47/SmallVee put
dup 48/LTriangle put
dup 49/LTriangleEqual put
dup 50/LTriangleBar put
dup 51/NotLTriangle put
dup 52/NotLTriangleEqual put
dup 53/NotLTriangleBar put
dup 54/Bar put
dup 55/Parallel put
dup 56/TrplBar put
dup 57/ShortBar put
dup 58/ShortParallelBar put
dup 59/BarExt put
dup 60/Tangental put
dup 61/NotTildeFullEqual put
dup 62/TildeEqual put
dup 63/EqualTilde put
dup 64/RTriangle put
dup 65/RTriangleEqual put
dup 66/RTriangleBar put
dup 67/NotRTriangle put
dup 68/NotRTriangleEqual put
dup 69/NotRTriangleBar put
dup 70/NotBar put
dup 71/NotParallelBar put
dup 72/NotTrplBar put
dup 73/ShortNotBar put
dup 74/ShortNotParallelBar put
dup 75/ParallelExt put
dup 76/NotTilde put
dup 77/NotTildeTilde put
dup 78/NotTildeEqual put
dup 79/NotEqualTilde put
dup 80/HumpEqual put
dup 81/HumpDownHump put
dup 82/NotHumpEqual put
dup 83/NotHumpDownHump put
dup 84/NotCongruent put
dup 85/DotEqual put
dup 86/RoundImplies put
dup 87/Equal put
dup 88/TrplEqual put
dup 89/FrownEqual put
dup 90/SmileEqual put
dup 91/Frown put
dup 92/Smile put
dup 93/FrownSmile put
dup 94/SmileFrown put
dup 95/NotSmileFrown put
dup 96/LessLess put
dup 97/DblLess put
dup 98/LessSlantEqual put
dup 99/LessFullEqual put
dup 100/LessTilde put
dup 101/NotLess put
dup 102/NotLessLess put
dup 103/NotDblLess put
dup 104/NotLessEqual put
dup 105/NotLessSlantEqual put
dup 106/NotLessFullEqual put
dup 107/NotLessTilde put
dup 108/SquareSubset put
dup 109/SquareSubsetEqual put
dup 110/NotSquareSubset put
dup 111/NotSqrSubsetEqual put
dup 112/GrtrGrtr put
dup 113/DblGrtr put
dup 114/GrtrSlantEqual put
dup 115/GrtrFullEqual put
dup 116/GrtrTilde put
dup 117/NotGrtr put
dup 118/NotGrtrGrtr put
dup 119/NotDblGrtr put
dup 120/NotGrtrEqual put
dup 121/NotGrtrSlantEqual put
dup 122/NotGrtrFullEqual put
dup 123/NotGrtrTilde put
dup 124/SquareSprset put
dup 125/SquareSprsetEqual put
dup 126/NotSquareSprset put
dup 127/DEL put
dup 128/Prcds put
dup 129/PrcdsEqual put
dup 130/PrcdsSlantEqual put
dup 131/PrcdsTilde put
dup 132/NotPrcds put
dup 133/NotPrcdsEqual put
dup 134/NotPrcdsSlantEqual put
dup 135/NotPrcdsTilde put
dup 136/LessGrtr put
dup 137/NotLessGrtr put
dup 138/LessEqualGrtr put
dup 139/Because put
dup 140/NotSqrSprsetEqual put
dup 141/NotSubsetEqual put
dup 142/NotSprset put
dup 143/NotSprsetEqual put
dup 144/Succds put
dup 145/SuccdsEqual put
dup 146/SuccdsSlantEqual put
dup 147/SuccdsTilde put
dup 148/NotSuccds put
dup 149/NotSuccdsEqual put
dup 150/NotSuccdsSlantEqual put
dup 151/NotSuccdsTilde put
dup 152/GrtrLess put
dup 153/NotGrtrLess put
dup 154/GrtrEqualLess put
dup 155/Proportion put
dup 156/Member put
dup 157/NotMember put
dup 158/CircleMinus put
dup 159/CircleDot put
dup 160/EqualParallel put
dup 161/MinusPlus put
dup 162/RTack put
dup 163/DblRTack put
dup 164/LTack put
dup 165/DblLTack put
dup 166/UpTee put
dup 167/DownTee put
dup 168/DownTeeLong put
dup 169/Between put
dup 170/VertTilde put
dup 171/HorizHourglass put
dup 172/RightAngle put
dup 173/MeasuredAngle put
dup 174/SphericalAngle put
dup 175/Diameter put
dup 176/Mho put
dup 177/NotExists put
dup 178/CapDigamma put
dup 179/CapStigma put
dup 180/CapKoppa put
dup 181/CapSampi put
dup 182/CurlyEpsilon put
dup 183/CurlyRho put
dup 184/Digamma put
dup 185/Stigma put
dup 186/Koppa put
dup 187/Sampi put
dup 188/Bet put
dup 189/Gimel put
dup 190/Dalet put
dup 191/Aleph put
dup 192/CapYAcute put
dup 193/YAcute put
dup 194/CapEth put
dup 195/Eth put
dup 196/CapThorn put
dup 197/Thorn put
dup 198/CapLSlash put
dup 199/LSlash put
dup 200/DownQuestion put
dup 201/Cent put
dup 202/Nbspace put
dup 203/Sterling put
dup 204/DownExclamation put
dup 205/Section put
dup 206/Paragraph put
dup 207/Yen put
dup 208/Micro put
dup 209/HBar put
dup 210/DotlessI put
dup 211/ItalDotlessI put
dup 212/DotlessJ put
dup 213/ItalDotlessJ put
dup 214/Dagger put
dup 215/DoubleDagger put
dup 216/Degree put
dup 217/Flat put
dup 218/Natural put
dup 219/Sharp put
dup 220/QuestionBox put
dup 221/ErrorIndicator put
dup 222/Angstrom put
dup 223/LamdaSlash put
dup 224/SolidSquare put
dup 225/Square put
dup 226/MedSolidSquare put
dup 227/MedEmptySquare put
dup 228/SmallSolidSquare put
dup 229/SmallEmptySquare put
dup 230/SolidCircle put
dup 231/Circle put
dup 232/Bullet put
dup 233/SmallCircle put
dup 234/Dot put
dup 235/OpenDot put
dup 236/MedSolidDiamond put
dup 237/MedEmptyDiamond put
dup 238/SmallSolidDiamond put
dup 239/SmallEmptyDiamond put
dup 240/SolidRectangle put
dup 241/Rectangle put
dup 242/SolidUpTriangle put
dup 243/UpTriangle put
dup 244/SolidDownTriangle put
dup 245/DownTriangle put
dup 246/SolidDiamond put
dup 247/Star6 put
dup 248/Star put
dup 249/Bullet3 put
dup 250/Bullet4 put
dup 251/Euler put
dup 252/ImagJ put
dup 253/DoubleGamma put
dup 254/DoublePi put
dup 255/DoubleScriptCapD put
 readonly def
/PaintType 0 def
/FontType 1 def
/StrokeWidth 0 def
/FontMatrix[0.001 0 0 0.001 0 0]readonly def
/UniqueID 5095645 def
/FontBBox{-109 -231 966 869}readonly def
currentdict end
currentfile eexec
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