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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 68, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/68\hfil Similarity solutions]
{Similarity solutions to evolution equations in one-dimensional interfaces}

\author[M. Benlahsen, A. Eldoussouki, M. Guedda, M. Jazar\hfil EJDE-2011/68\hfilneg]
{Mohammed Benlahsen, Ayman Eldoussouki, Mohammed Guedda, Mustapha Jazar}
% in alphabetical order

\address{Mohammed Benlahsen \newline
 LPMC, Department of Physic, Universit\'e de Picardie Jules Verne,
 33, rue saint-Leu,  Amiens, France}
\email{mohammed.benlahsen@u-picardie.fr}

\address{Ayman Eldoussouki \newline
 LAMFA, CNRS UMR 6140, Department of Mathematics\\
 Universit\'e de Picardie Jules Verne,
 33,  rue Saint-Leu, Amiens, France}
\email{ayman.eldoussouki@u-picardie.fr}

\address{Mohammed Guedda \newline
 LAMFA, CNRS UMR 6140, Department of Mathematics\\
 Universit\'e de Picardie Jules Verne,
 33,  rue Saint-Leu, Amiens, France}
\email{guedda@u-picardie.fr}

\address{Mustapha Jazar \newline
 LaMA-Liban, Lebanese University, P.O. Box 37 Tripoli via Beirut,
 Lebanon}
\email{mjazar@laser-lb.org}

\thanks{Submitted April 15, 2011. Published May 20, 2011.}
\subjclass[2000]{70K42, 34A34, 35K55}
\keywords{Nonlinear dynamic; instability; similarity solution; coarsening}

\begin{abstract}
 In this note, we study  the  evolution equation
 \[
 \partial_t h = -\nu\partial^2_xh-K\partial^4_xh
 +\lambda_1(\partial_x h)^2-\lambda_2\partial^2_x(\partial_x h)^2.
 \]
 which was introduced by Mu\~{n}oz-Garcia \cite{MCC}
 in the context of erosion by ion beam sputtering.
 We obtain an analytic solution that has the similarity form,
 which is used in obtaining the coarsening behavior.
 This solution has amplitude and wavelength that increase
 like $\ln(t)$ and $ \sqrt{t\ln(t)}$, respectively.
\end{abstract}

\maketitle
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\newtheorem{remark}[theorem]{Remark}
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\section{Results}

Mu\~{n}oz-Garcia, Cuerno and Castro \cite{MCC} derived and
analyzed numerically a one-dimensional class of unstable surface
growth models in the context of erosion by ion beam sputtering.
They showed that the surface morphology is described by the
 ``interfacial height'' equation
\begin{equation}\label{eq:1}
\partial_t h = -\nu\partial^2_xh-K\partial^4_xh+\lambda_1(\partial_xh)^2-\lambda_2\partial^2_x(\partial_xh)^2,
\end{equation}
or, after some rescaling,
\begin{equation}\label{eq:2}
\partial_t h = -\partial^2_xh-\partial^4_xh
+(\partial_xh)^2-r\partial^2_x(\partial_xh)^2,
\end{equation}
where $\nu > 0$, $K > 0, \lambda_1$ and $\lambda_2 $ are real
parameters  and $r = \nu\lambda_2/(K\lambda_1)$.
The parameters  $\lambda_1 $ and $\lambda_2 $ have the same sign;
 i.e., $r > 0$, for mathematical well-posedeness and are positive.

 The above equations are referred to as the mixed
Kuramoto-Sivashinsky equation. Equation \eqref{eq:1} was also
proposed for amorphous thin films in the presence of potential
density variations \cite{RLH}. This equation is also referred to
as the snow equation. Under  some conditions on the parameters,
\eqref{eq:1} models a snow surface growth based on solar radiation
\cite{Tiedje}.

 The first linear term on the right-hand side of \eqref{eq:1}
is the instability term. This term is balanced by the classical
stabilizing linear term $\partial_x^4h$. The nonlinear term
$\partial^2_x(\partial_xh)^2 $ is responsible for the coarsening
dynamics; i.e., the amplitude and wavelength, or the lateral
width, increase with time without bound (see below), while the
nonlinear term $(\partial_xh)^2$  interrupts the coarsening
process \cite{MCC,Tiedje}. For $\lambda_2 = 0$,
 or $ r= 0$  \eqref{eq:1} reduces to the
famous Kuaramoto-Sivashinsky equation  which  is known to produce
a spatio-temporal chaos and no coarsening. For $\lambda_1=0 $
($r\to\infty$), \eqref{eq:1} reads
\begin{equation}\label{CKS}
\partial_t h = -\nu\partial^2_xh-K\partial^4_xh
-\lambda_2\partial^2_x(\partial_xh)^2.
\end{equation}
Such an equation, referred to  as the Conserved
Kuaramoto-Sivashinsky (CKS) equation,  appears in different
physical contexts. Recently, the CKS equation has been derived and
used by Frisch and Verga \cite{FV} to study the step meandering
instability on a vicinal surface. The authors obtained a
$l(t)=t^{1/2}$ scaling, for large time  (uninterrupted
coarsening), and also demonstrated a linear time growth of the
characteristic meander amplitude ($A(t)=t$) (here the wavelength
$l(t) $ and  the amplitude $A(t)$  are  the mean lateral distance
between two consecutive local minima and  mean vertical distance
from a local minimum to the next local maximum, respectively).
Moreover it is shown that the solution of \eqref{CKS} is a
periodic juxtaposition of  parabolas of the form (see also
\cite{RLH2000})
\begin{equation}\label{parabola}
h(x) = a-\frac{4a}{b^2}x^2,
\end{equation}
where $ a $ and $b$ are real parameters. In fact, \eqref{parabola}
is a stationary solution to the CKS equation  for
$a/b^2=\nu/(16\lambda_2)$; $h(x)=a-\big(\nu x^2/(16\lambda_2)\big)$ 
for any $ a$, irrespective of $K$
\cite{FV}.

 Recently, using \eqref{parabola}, a class of
exact solutions to \eqref{CKS} are derived by Guedda et al.
\cite{Guedda}.  To be more precise the authors obtained a family
of exact solutions to the CKS equation having the form
\begin{equation}\label{GTPB}
h(x,t)= -\frac{\nu}{4\lambda_2}x^2,\quad \text{for } \vert x\vert
\leq y(t),
\end{equation}
and zero elsewhere, where  the parabola edge
$y=l(t)/2$ satisfies a nonlinear ordinary differential equation.
Different scenarios are found: the lateral coarsening $l(t)$
(i) grows with time like $\sqrt{t}$, (ii) disappears,
or (iii) does not  change
for all time, depending on the initial lateral coarsening $l(0)$.

 Let us return to \eqref{eq:2} which may present a transition
or interpolation between the chaotic behavior of the KS equation
(as $r\to 0$) and the coarsening behavior of the CKS equation
(as $r\to\infty$). From a mathematical point of view, since
\eqref{eq:1} can be considered as a nonlinear perturbation of the
CKS equation, we are concerned wit the effect of the KPZ term
$\lambda_1(\partial_xh)^2$  on the coarsening property of
\eqref{eq:1}, or  \eqref{eq:2}. To be more precise, we will
present an exact non stationary solution to \eqref{eq:2}  with  a
logarithmic growth of the amplitude and unbounded wavelength
(perpetual coarsening) which indicates, in particular, that
equation \eqref{eq:2} may have different behaviors.

As in \cite{MCC} we take $r=50$. The main line of argument used
here is similar to the one used  in \cite{FV} and \cite{Guedda}.
In fact we are looking for  an exact non stationary solution of
\eqref{eq:2} having the  form
\begin{equation}\label{exact}
 h(x,t)=A(t)\big[1-\frac{4}{l^2(t)}x^2\big],
\end{equation}
for $\vert x\vert \leq l(t)/2 $ and zero elsewhere, presenting
one individual cell, where $A$ and $l$ are unknown functions
to be determined explicitly.
The form of the relation  \eqref{exact} is instructive. Equation
\eqref{eq:2} may have a cell or a mound centered at the origin
with an amplitude or a height scale  $A(t) $ and  a lateral width
or a linear scale $l(t)$.

Ansatz \eqref{exact} is typical. This means that the ratio
$h(x,t)/A(t) $ depends only on a single variable $\eta = x/l(t)$;
\begin{equation}\label{class}
\frac{h(x,t)}{A(t)}=\varphi(\eta),\quad
\eta = x/l(t).
\end{equation}
A class of solutions \eqref{class} are
called invariant or similarity solutions and are commonly used for
extracting the coarsening behavior from a nonlinear continuum
equation
 (see for example \cite{Krug02,Pim5,PGPM,Pim2}).


For a useful comparison, we first consider equation \eqref{eq:2}
where the KPZ term is absent (i.e. CKS equation);
\begin{equation}\label{r-CKS}
\partial_t h = -\partial^2_xh-\partial^4_xh
-r\partial^2_x(\partial_xh)^2.
\end{equation}
If in \eqref{exact} we set
\[
A(t)=1+\frac{t}{400},\quad l(t)=2\sqrt{400+t},
\]
we obtain an exact similarity solution to \eqref{r-CKS} with
$r=50$. Clearly $A(t) \sim t $ and $l(t) \sim \sqrt{t}$, as $t\to
\infty$, which agree with the result of \cite{FV}. \par Next, we
study equation \eqref{eq:2}. Defining
\begin{equation}\label{A}
   A(t) = 7-\frac{640q^2t}{1+16qt} + \frac{1}{2}\ln(1+16qt),
 \end{equation}
and
\begin{equation}\label{L}
l(t) = \sqrt{1200+(112-640q)t +
\frac{1}{2q}(1+16qt)\ln(1+16qt)},
\end{equation}
where $q = 7/1200$, it can be verified that
 \eqref{exact}-\eqref{L} is an
exact solution to \eqref{eq:2}, where $r=50$, which begins with
the cell defined by the truncated parabola
\begin{equation}\label{exact-sat}
 h(x,0)=7[1-\frac{1}{300}x^2],
                \end{equation}
for $\vert x\vert \leq 10\sqrt{3}$, and zero elsewhere. The
initial amplitude and lateral width are $A(0) = 7 $ and $l(0)
=20\sqrt{3}$.

 We may deduce from this that equation
\eqref{eq:1}, or \eqref{eq:2}, has  solutions with amplitude and
wavelength  that  tend to infinity with $t$, and behave like, as
$t \to \infty$,
\begin{equation}\label{A-lambda}
 A(t) \sim \frac{1}{2}\ln(t),\quad l(t) \sim 2\sqrt{2t\ln(t)}.
\end{equation}
Interestingly, the above explicit solution reveals that the
structure may undergo logarithm-law coarsening.
Similar behavior is obtained for the one dimensional
convective Cahn-Hilliard equation \cite{Gol}.
The dynamic is controlled not only by the KPZ
term, but also by the Conserved KPZ term
$\partial^2_{x}(\partial_x h)^2$ (recall that the KS equation
produces  a chaotic behavior without coarsening and the CKS
equation exhibits a coarsening behavior with the power-law
$l(t) \sim \sqrt{t})$. Note that $\max{\vert \partial_x h\vert^2}$ and
$\max{\vert\partial^2_x(\partial_x h)^2\vert}$ go to zero as $t$
tends to infinity and that
\begin{equation}
\frac{\max{\vert \partial_x h\vert^2}}
{\max{\vert\partial^2_x(\partial_x h)^2\vert}}
= l^2(t)/8,
\end{equation}
which approaches infinity with $t$. This indicates that
$ \partial^2_{x}(\partial_x h)^2$ is small compared to
$(\partial_x h)^2$, for large $t$.

In summary,  a solution  which begins with the cell defined by
\eqref{exact-sat} is found in a closed form. This solution
displays unbounded coarsening.  An important and interesting task
for future investigation is to exhibit,  for large $r$,
different behaviors (if any) and to understand their dependence on
the initial lateral coarsening or initial amplitude.

\subsection*{Acknowledgments}
The present work was partially done during a visit
of the authors (M. B., A. E. and M. G. )
to the LaMA-Liban and the LASeR (Lebanese Association for
Scientific Research).
This work was supported in part by Project
MeRSI and by Direction des Affaires Internationales,
UPJV, Amiens France.
This work is dedicate to the memory of Professor Bernard Caroli.


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\end{thebibliography}

\end{document}