Large amplitude behavior of the Grinfeld instability\\Part I: High-order weakly nonlinear analysis
Peter Kohlert
Amplitude expansions are used to determine steady states of a semi-infinite solid subject to the Grinfeld instability in systems with a fixed (wave)length. We present two methods to obtain high-order weakly nonlinear results. Using the system size as a control parameter, we circumvent the problem that there is no instability threshold for an extended system in the absence of gravity. This way, the case without gravity becomes accessible to a weakly nonlinear treatment. The dependence of the branch structure of solution space on the level of gravity (or density difference) is exhibited. In the zero-gravity limit, we recover the solution branch obtained by Spencer and Meiron. A transition from a supercritical to a subcritical bifurcation is observed as gravity is increased or the nonhydrostatic stress is decreased at fixed gravity. At given values of the system parameters, we find a discrete, possibly infinite, set of solution branches. This is reminiscent of dendritic or eutectic growth, where similar solution sets exist, of which only a particular one is linearly stable. Despite the high order of our expansions, the approach is restricted to relatively small nondimensional amplitudes ($\lesssim 0.2$), a disadvantage we can overcome by a variational approach that will be discussed in a companion paper. At the critical point, we find that not only the first Landau coefficient is negative but all of them up to the highest amplitude order (15) we could compute so far.
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\begin{document}

\title{Large amplitude behavior of the Grinfeld instability\\Part I:
High-order weakly nonlinear analysis}

\author{Peter Kohlert}
 \email[]{physics@peter-kohlert.de}
\author{Klaus Kassner}
\affiliation{Institut f\"ur Theoretische Physik, Otto-von-Guericke-Universit\"at Magdeburg,\\Postfach 4120, D-39016 Magdeburg, Germany}
\author{Chaouqi Misbah}
\affiliation{Groupe de recherche sur les Ph{\'e}nom{\`e}nes hors de l'Equilibre, LSP, Universit{\'e} Joseph Fourier (CNRS), Grenoble I, B.P.\,87, Saint-Martin d'H{\`e}res, 38402 Cedex, France}

\date{\today}

\begin{abstract}
Amplitude expansions are used to determine steady states of a
semi-infinite solid subject to the Grinfeld instability in systems with a
fixed (wave)length. We present two methods to obtain high-order weakly
nonlinear results. Using the system size  as a control parameter, we
circumvent the problem that there is no instability threshold for an
extended system in the absence of gravity. This way, the case without
gravity becomes accessible to a weakly nonlinear treatment. The dependence
of the branch structure of solution space  on the level of gravity (or
density difference) is exhibited. In the zero-gravity limit, we recover the
solution branch obtained by Spencer and Meiron. A transition from a
supercritical to a subcritical bifurcation is observed as gravity is
increased or the nonhydrostatic stress is decreased at fixed gravity.
At given values of the system parameters, we find a discrete, possibly
infinite, set of solution branches. This is reminiscent of dendritic
or eutectic growth, where similar solution sets exist, of which only
a particular one is linearly stable.
Despite the high order of our expansions, the approach is restricted to
relatively small nondimensional amplitudes ($\lesssim 0.2$), a
disadvantage we can overcome by a variational approach that will be
discussed in a companion paper. At the critical point, we find that
not only the first Landau coefficient is negative but all of them up to
the highest amplitude order (15) we could compute so far.
\end{abstract}

\pacs{47.20.Hw,05.70.Ln,46.25.-y,81.10.Aj}
%05.70.Ln Nonequilibrium and irreversible thermodynamics
%81.10.Aj Theory and models of crystal growth; physics of crystal growth, crystal morphology and orientation
%81.30.Fb Solidification
%46.25.-y Static elasticity
%47.20.Hw Morphological instability; phase changes

\maketitle

\section{\label{sec:introduction}Introduction}

In the processing of solids, it is often inevitable that they are
submitted to elastic stresses. During solidification, for example,
solids contract or expand according to their thermal treatment, which
in a confined geometry will generally lead to the creation of strain
\cite{minari:94,grange:94,voelkl:94,fabietti:97,grange:97} and the
formation of defects.
Epitaxial growth, a technique of utmost importance in the fabrication
of novel electronic materials, necessarily produces strain in
neighboring layers of different composition whenever there is
a nonvanishing lattice mismatch of the substances involved
\cite{eaglesham:90,noetzel:94}.

The appearance of stresses is not restricted to environments dealing
with technical applications. A beautiful natural example is the
Giant's causeway in Northern Ireland, where cooling lava has turned
into hexagonal basalt columns, most likely under the influence
of stress. Very similar structures but on the mm scale instead of the m one,
were obtained in the drying of starch \cite{mueller:98}.
It is therefore interesting to study the influence of stress on the
growth, melting, or evaporation of solids.


As has been known for some time,
the initially flat interface of a solid subject to an elastic
field tends to become unstable against morphological perturbations
if an active transport mechanism is present. %provided.
This phenomenon is often called the Grinfeld instability after its
 rediscovery by
Grinfeld, who recognized its universal importance
\cite{grinfeld:86}. A description for a special system had
been given earlier \cite{asaro:72}.

The dynamics of the Grinfeld instability is energetically driven.
Therefore, the problem is variational and amplitude equations can be
derived from an appropriate free-energy functional. Since the basic modes
appearing as eigenfunctions in a linear stability analysis are plane waves
for a planar interface \cite{nozieres:93}, the natural approach is to
decompose the interface in terms of a Fourier expansion and to determine
the amplitudes of the modes by minimisation of the free energy. This is
equivalent to the calculation of steady states from weakly nonlinear
amplitude equations with the amplitude expansion taken to one order less
than the energy expansion. Consistency considerations as discussed in
\cite{kassner:94a} apply here, too.

%Due to the existence of an energy functional, the expansion is
%valid, regardless of the question of
%sub- or supercriticality of the instability, at all
%amplitudes where it converges.
%Of course, its convergence may be very slow for larger amplitudes
%and it may even become divergent. The point to be made here is
%only that negative Landau coefficients do not necessarily
%imply, as is usually assumed,  the expansion to become meaningless.

As we shall see later, an approach that yields much faster convergence
towards a steady-state solution can
be based on an expansion into a different set of generalized functions in
the spirit of a variational approach. This is the multicycloid method to
be presented in a second paper \cite{kohlert:02b},
henceforth referred to as II.
Its utility goes farther than improving convergence.
Certain interfaces with overhangs are tractable within that approach,
which extends its applicability beyond that of the amplitude method.

In the model, there is competition of at least two energetic terms: the
strain energy stored in the bulk of the stressed material which tends to
destabilize the surface  at any given wave number, and the surface energy
which is stabilizing. In many systems, more physics is added by the
presence of potential energy in a gravitational field which acts
stabilizing as well, if the density stratification corresponds to high
densities (solid) below the interface  and low densities (melt) above.

The main goal of this paper is to find steady state solutions
to the dynamical equations arising from the Grinfeld instability.
Transport can be, for example, via ``melting-crystallization''
waves, when there is contact with the melt, constituting a
particle reservoir, or by surface diffusion, for the case of a solid
in vacuum. In the first situation, the dynamics is nonconservative, in
the second, it is conservative.
However, kinetic aspects of the transport mechanism do not
play a role in the final steady state. Therefore, our results
will be applicable to both types of systems.

What is known about steady states for the Grinfeld instability so far?
Spencer and Meiron \cite{spencer:94} have shown by time dependent
simulations using a highly accurate spectral code that for a solid
submitted to stress in the absence of gravity there is a branch of stable
steady-state solutions only for a very limited set of wavelengths.
Moreover, they  showed that the
generic behavior for large wavelengths is the evolution of cusp
singularities. These singularities occur for interface shapes which are
very close to the shapes of certain cycloids, a result that is suggested
also by the analytic work of Chiu and Gao \cite{gao:93}.

We will present results on the solution
space of steady states including gravity.
As we will discuss, there
appears to be a ladder of solution branches reminiscent of the situation
in dendritic and eutectic growth.
In the former case, there exists a discrete infinite set of needle
solutions for any given undercooling, of which however only
(the fastest) one is stable.
In the latter case, a whole set of solution branches exists in a certain
wavelength range for given temperature gradient and pulling velocity.
In the present approach, different solution branches
correspond to different
wavelengths of the pattern. Similar morphology changes occur
between branches as in eutectic growth \cite{kassner:91a}.

Not including a full stability analysis using
Floquet-Bloch theory, our approach allows us to make stability statements
only for periodic perturbations. With respect to these perturbations,
there is, in the absence of gravity, one branch that is stable
along its full length, which  is the one found already
by Spencer and Meiron. With gravity, there is a critical
wave number, below which small amplitude solutions become unstable.
Only a part of the branch at finite amplitude is stable rendering the
bifurcation from the planar solution subcritical in a sense specified more
precisely below.

In principle, knowing the coefficients of the mode expansion, we could
extend our approach to time dependent calculations. This will however
not be pursued here.

The paper is organized as follows: Section~\ref{sec:energies} describes
the basic method and gives expressions for the relevant energies of the
system. In Sec.~\ref{sec:scaling}, we rescale the problem for
nondimensionalization and then formulate it more technically, adapted to
the Fourier expansion. The next section, Sec.~\ref{sec:methodology},
compares the results of two methods to obtain the minima of the energy
functional. One of them solves all the equations but one analytically,
with the second, we have to solve the final set of nonlinear equations
for the amplitudes by a numerical scheme (Newton-Raphson method). In
Sec.~\ref{sec:gravity}, we add gravity leading to the understanding of the
small-amplitude behavior down to the critical stress case described by
Nozi\`{e}res \cite{nozieres:93}.
Gravity plays an essential role for the solid-liquid interface close
enough to the instability threshold \cite{balibar:92}.

Finally, in section \ref{sec:summary}, we
will summarize the conclusions that can be drawn from our
high-order expansions.

\section{\label{sec:energies}Thermodynamical considerations}

In general, in order to find steady solutions, we have to equilibrate the
contributions of elastic, surface and gravitational energy terms.
Basically we have to solve
\begin{equation}
\label{eq:variation1}
\delta E=\delta\left[ E_{\rm elastic}+
E_{\rm surface}+ E_{\rm gravity}\right]=0\>,
\end{equation}
where $E$ stands for energy or free energy and the subscripts describe the
origin of the respective energy. Next, we have to verify the stability of
our solutions.

One way to express variations of the surface is to rewrite its position in
terms of some amplitudes $\alpha_i$ related to a given system of functions
and to vary this discrete set of amplitudes. In the first part of this
article, we will expand the interface shape into a Fourier series,
assuming periodicity. It can be restricted to even (i.e., cosine)
modes  [Eq.~(\ref{eq:zeta})] because of the reflection symmetry of the
system about $x\!=\!0$:
\begin{equation}
\label{eq:zeta} \zeta(x)=\sum_{n=1}^{\infty}\alpha_n \cos{(nkx)} \>.
\end{equation}

In a subsequent paper, we will consider an expansion in terms of
multicycloids, which will prove to constitute a more appropriate tool for
capturing cusp singularities at leading order.

Our model is a two-dimensional system possessing an interface located
about $z\!=\!0$ so that the mean interface position is zero. This
condition is automatically satisfied if $z$ is given by
equation~(\ref{eq:zeta}). The uniaxial prestress $\sigma_0$ is taken
parallel to the direction of the abscissa, $x$, and the elastic domain is
reaching downwards to minus infinity (whence our freedom of choice of the
average interface position).

\subsection{Strain Energy}

For linearly elastic solids, the strain energy density $w$ is given by
\begin{equation}
\label{eq:strainenergydensity}
w=\frac{1}{2}\sigma_{ij}u_{ij}=\frac{1}{2}S_{ijkl}\sigma_{ij}\sigma_{kl}
\>,
\end{equation}
 where $\sigma_{ij}$ is the stress, $u_{ij}$
the strain tensor and $S_{ijkl}$ is  the compliance moduli
tensor. As we assume an isotropic system and
plane strain (i.e., we restrict ourselves to two-dimensional
deformations), $S_{ijkl}$ becomes simply
\begin{equation}
\label{eq:compliance}
S_{ijkl}=\frac{1+\nu}{E_Y}\left(\delta_{ik}\delta_{jl}-\nu\delta_{ij}\delta_{kl}\right)
\end{equation}
(see for instance \cite{gao:91}). Herein, $\nu$ is Poisson's ratio, $E_Y$
is Young's modulus, and summation over repeated indices is implied. The
total amount of elastic energy stored is obtained by
\begin{eqnarray}
\label{eq:elastic}  E_{\rm elastic}&=&\frac{1-\nu^2}{2E_Y}\int_{\cal
D}\sigma_{ij}\sigma_{ij}\; d\tau \nonumber\\
&&-\frac{(1+\nu)\nu}{E_Y}\int_{\cal D} {\rm det} \>  \sigma \; d\tau \>,
\end{eqnarray}
where the domain $\cal D$ is
bounded by the surface and extends to minus infinity in
the $z$ direction, whereas it can be restricted to one
period of the basic mode in the $x$ direction. It turns out that the determinant term
is zero at each order of the mode expansion. Therefore, from here on, it will not
be explicitly written anymore.
The
detailed method of calculating the strain energy will be given in section
\ref{sec:scaling}.

\subsection{Surface Tension}

Surface tension is assumed to be isotropic. The corresponding total energy
of the surface then is the length of the surface multiplied by the surface
tension $\gamma$. If we take the zero of this energy to correspond to the
planar interface, $E_{\rm surface}$ of a deflected interface is:
\begin{equation}
\label{eq:surface}
 E_{\rm surface}=\gamma\left[\int_{{\cal
B}}\left(\sqrt{1+\zeta'(x)^2}-1\right)dx\right] \>.
\end{equation}
Here, as in Eq.~(\ref{eq:gravity}), we have to take a period of the
surface for the integration interval $\cal B$.

\subsection{Potential Energy}

Finally, the potential energy density, if referred to the planar interface
as base state, becomes proportional to the mean square amplitude of the
surface shape. This can be verified very easily by writing down the
potential energy of an infinitesimal slice, considering that the center of
gravity is at $\zeta(x)/2$, and by integrating then. We  obtain
\begin{equation}
\label{eq:gravity}
 E_{\rm gravity}=\frac{g \Delta\rho}{2}\int_{{\cal B}} \zeta(x)^2dx,
\end{equation}
where $\Delta\rho$ stands for the
density contrast and $g$ is the gravitational acceleration.

\section{\label{sec:scaling}Nondimensionalization and the derivation of
higher-order equations}

\subsection{Nondimensionalization}
As long as gravity is not considered, all physical parameters can be
scaled out of the equations of motion for the
Grinfeld instability, which renders the problem scale invariant.
%With this regard, it is just a geometrical problem.
The length scale $l_1$ to be used in this procedure, is, up to a factor,
equal to the Griffith length \cite{griffith:20}, a measure for the
competition between capillarity and elasticity. Setting
$w_0=\sigma_0^2(1-\nu^2)/2E_Y$, which is the elastic energy density of the
prestressed planar  state, it reads \cite{kassner:01}
\begin{equation}
l_1=\frac{\gamma}{2w_0} \>.
\end{equation}

When gravity is present, a second length scale $l_2$ appears, given by
\begin{equation}
l_2=\frac{w_0}{g \Delta\rho}\>.
\end{equation}
An important parameter of the nondimensionalized equations is the ratio of
these length scales
\begin{equation}
l_{12}:=\frac{l_1}{2l_2}.
\end{equation}
Note that in the absence of gravity, this ratio becomes zero as the
gravity length $l_2$ diverges.

Using these quantities, we nondimensionalize equations (\ref{eq:elastic}),
(\ref{eq:surface}) and (\ref{eq:gravity}), carrying out a formal
transformation $x\rightarrow l_1 x$, $z\rightarrow l_1 z$,
$\kappa\rightarrow l_1^{-1}\kappa$, $\zeta\rightarrow l_1\zeta$  and
$\sigma_{ij}\rightarrow\sigma_0\sigma_{ij}$. Thus, in equation
(\ref{eq:elastic}), we get a $\sigma_0^2$ and an additional $l_1$ coming
from the integration in the $z$~direction -- resulting in a new prefactor
$\gamma/2$. Equation (\ref{eq:surface}) just keeps the prefactor $\gamma$,
and Eq.~(\ref{eq:gravity}) can now be written with the prefactor $\gamma
l_{12}/2$, accounting for an ${l_1}^2$ coming from the integrand. The $x$
integration produces a pre\-factor common  to all integrals that can be
dropped.


Dropping the common prefactor $\gamma$ as well, our nondimensional
equation reads
\begin{equation}
\label{eq:nondim}
\frac{\delta}{\delta{\bf x}}\left[\int_{\cal D}\frac{\sigma_{ij}^2}{2}\; d\tau
+\int_{{\cal B}}\left(\sqrt{1+\zeta'(x)^2}+\frac{l_{12}}{2}\zeta(x)^2\right)dx\right]=0 \>.
\end{equation}

This equation is still exact, of course only within the framework of our
basic assumptions (no cross-effects between stresses, capillarity and
gravity, i.e., no capillary overpressure, stress tensor
divergence free, etc., see \cite{kassner:01}). The following calculations
are however done assuming that the square root can be expanded in powers
of $\zeta'(x)^2$. Essentially, this restricts the validity of results to
interfaces with $\vert\zeta'(x)\vert< 1$. If we do not want to consider
gravity, we can just set $l_{12}=0$. It should be kept in mind that from
now on we always consider nondimensional quantities.

\subsection{Expansion of the stress components into Fourier modes}
To compute the elastic energy, we obviously first have to solve the
elastic problem, i.e., the Lam{\'e} equations, with appropriate
boundary conditions. Next the stress tensor is expanded into a
trigonometric series, exploiting periodicity in the $x$ direction. The
elastic partial differential equations can then be reduced to ordinary
differential equations for the $z$ dependent coefficients. After solving
for the $z$ dependence analytically, we can express the stress tensor in
terms of expansion coefficients for arbitrary surface shape $\zeta(x)$.
This is a useful step, rendering feasible the minimization procedure
expressed by Eq.~(\ref{eq:nondim}).

Starting with an undisturbed plane half space, the nondimensional
prestress takes the form:
\begin{equation}
\Sigma_0=\left(\begin{array}{cc}1&0\\0&0\end{array}\right),
\end{equation}
and we impose the boundary condition that for $z\to-\infty$ the total
stress tensor approaches this quantity.


The changes of the stress tensor due to a surface corrugation will result
in a relaxational stress $\Sigma_a$. As the surface of the solid is free
-- the solid is in contact with a liquid -- the total stress tensor
$\Sigma=\Sigma_0+\Sigma_a$ has to fulfill the boundary condition
\begin{equation}
\label{eq:ggw} \left(\Sigma_0+\Sigma_a\right){\bf n}=\Sigma\,{\bf
n}=-p_\ell{\bf n}=0
\end{equation}
at the interface, where we have, for convenience, set the  pressure
$p_\ell$ of the liquid equal to zero. This is possible, because the
dynamics of the system is unaffected by an additive constant pressure
\cite{kassner:01}. It is here, where we neglect capillary overpressure,
which would impose a (small) jump discontinuity of the component
$\Sigma_{nn}$.

Calling the components of ${\bf\Sigma}_a$ $\sigma_{xx}$,
$\sigma_{xz}\!=\!\sigma_{zx}$ and $\sigma_{zz}$ and leaving out a common
prefactor in the normal vector, Eq.~(\ref{eq:ggw}) reads
\begin{equation}
\label{eq:ggw2}
\begin{split}
-\left(1+\sigma_{xx}\right)\zeta'(x)+\sigma_{xz}&=0\\
-\sigma_{xz}\;\zeta'(x)+\sigma_{zz}&=0\>,
\end{split}
\end{equation}
where stress components have to be taken at the interface and $\zeta(x)$
is the free surface, expressed according to Eq.~(\ref{eq:zeta}). We
introduce an expansion parameter $\varepsilon$ and assume
$\alpha_n\propto\varepsilon^n$. As is well-known, in two dimensions
calculations can be facilitated by introduction of an Airy stress function
$\chi(x,z)$,  connected with the stresses via
\begin{equation}
\sigma_{xx}=\frac{\partial^2\chi}{\partial z^2},\quad
\sigma_{zz}=\frac{\partial^2\chi}{\partial x^2},\quad
\sigma_{xz}=-\frac{\partial^2\chi}{\partial x\partial z}\>,
\end{equation}
i.e., our stress function refers only to the deviations from the
constant prestress.

Using the boundary conditions at $z=-\infty$ and solving for the $z$
dependence, we obtain
\begin{equation}
\label{eq:airy}
\chi=\sum_{n=0}^{\infty}\varepsilon^n\left(a_n+b_nz\right)
\cos{(nkx)}e^{-nkz},
\end{equation}
where we anticipated the leading $\varepsilon$-dependence of the
coefficients. Now, $a_n$ and $b_n$ have to be computed by use of
the boundary conditions (\ref{eq:ggw2}).
To this end, we first  expand $a_n$ and $b_n$:
\begin{equation}
\label{eq:expand_airy}
a_n=\sum_{m=0}^{\infty}\varepsilon^{2m}a_{n,2m},\quad
b_n=\sum_{m=0}^{\infty}\varepsilon^{2m}b_{n,2m}
\end{equation}

Note that odd expansion coefficients disappear, which is proven in
appendix (\ref{app:odd_disappear}). Furthermore, equations (\ref{eq:ggw2})
are given at the interface. So, the exponentials in the stress components
have to be expanded as power series in $z$, and then $z$ has to be
replaced by the interface position $\zeta(x)$.

After this has been done, equations (\ref{eq:ggw2}) will be two power
series in $\varepsilon$, and for every order $\varepsilon^n$,
 there exists a
set of equations provided by the contributions of different cosine and
sine modes, respectively, which have to be set equal to zero
separately \footnote{Technical note: The calculation of these coefficients
can be carried out very efficiently by a CAS like \texttt{Maple}. Yet, as
most calculations are conversions of Sines and Cosines, which are
time-consuming if they appear in large numbers it becomes necessary to
implement a symbolic conversion and integration scheme.}.

\subsection{\label{sec:linear} Second and fourth-order results}

To second order in the amplitude, the sum of the three energies is
\begin{equation}
\label{eq:linear}
\Delta{E}=\left(-2k+k^2+l_{12}\right)\frac{\alpha_1^2}{4}\>.
\end{equation}
We write $\Delta E$ from now on as a reminder that we have taken the
planar interface as the reference state with energy zero.

From Eq.~(\ref{eq:linear}), we can recover two well-known results:
\begin{itemize}
\item When neglecting gravity we get marginal wave numbers $k=\{0,2\}$.
\item Under gravity the critical wave number is $k=1$, and the critical
length ratio $l_{12}=1$.
\end{itemize}

More generally, we can conclude that for arbitrary $l_{12}$ in the interval
$[0,1]$ the band of unstable wave numbers ranges from $1-\sqrt{1-l_{12}}$
to $1+\sqrt{1-l_{12}}$. If $l_{12}>1$, which corresponds to low stress
or high gravity, the flat surface is linearly stable. If we now add
fourth-order terms we obtain
\begin{multline}
\label{eq:secondorder}
\Delta{E}=\frac{1}{k}\left(-\frac{1}{2}A_1^2
+\frac{3}{8}A_1^4-A_2^2+A_2A_1^2\right)\\
+\frac{1}{4}A_1^2+A_2^2-\frac{3}{64}A_1^4+\frac{l_{12}}{4k^2}\left
(A_1^2+A_2^2\right ),
\end{multline}
where we have set $A_n=k\alpha_n$ for short. The term having the prefactor
$1/k$ is the elastic contribution to the free-energy change, the term with
$l_{12}$ the gravity one, while the remaining three terms describe the
surface energy contribution. Minimizing this result with respect to $A_2$,
the weakly nonlinear expansion of Nozi\`{e}res \cite{nozieres:93} is
recovered. The main result is that the instability is subcritical: Even
for $l_{12}$ slightly larger than one a finite perturbation may trigger
the instability.

\section{\label{sec:methodology}Expansion Method}

\subsection{\label{sec:princ}Principles}
Having calculated the change of energy to a desired order $2n$, we can now
devise methods for solving the system
\begin{equation}
\label{eq:minimize}
 \Delta_j:=\frac{\partial\Delta E}{\partial \alpha_j}=0;\quad j=1\dots n.
\end{equation}
A scheme that goes a long way analytically is what we call the expansion
method. Here we expand, besides the coefficients $a_n$, $b_n$, also the
amplitudes $\alpha_2\dots \alpha_n$, either in powers of $\varepsilon$ or,
more straight-forwardly, in powers of $\alpha_1$:
\begin{equation}
\label{eq:amps_entw}
 \alpha_m=\alpha_1^m\sum_{j=0}^{\infty}c_{m,j}\;\alpha_1^j.
\end{equation}
If we calculate $\Delta E$ using the first $n$ amplitudes, then it seems
at first sight that we should actually expand $\Delta_j$ up to
$\alpha_1^{2n-1}$. But then we would have to take into account terms up to
order $\alpha_1^{2n+j-1}$ in the expression for $\Delta E$ \textit{before}
taking the derivative (which reduces the order by $j$).

However, the structure of the equations is such that obtaining $\alpha_n$
to order $\varepsilon^n$ from setting $\Delta_n=0$, we get $\alpha_{n-1}$
to order $\varepsilon^{n+1}$ from the equation $\Delta_{n-1}=0$ (in which
each $\alpha_n$ is multiplied by at least a factor $\alpha_1$), then
$\alpha_{n-2}$ to order $\varepsilon^{n+2}$, and so on, until the first
equation ($\Delta_1=0$) yields $\alpha_1$ accurate to order
$\varepsilon^{2n-1}$, as desired.

The coefficients of the powers of $\alpha_1$ in $\Delta_2\dots\Delta_n$
provide a linear system of equations for
the calculation of all $c_{m,j}$ with
$m+j\leq n$. Interestingly, similar to the Airy coefficients, all
$c_{m,j}$ where $j$ is odd are zero, a formal fact allowing us to
considerably increase the speed of a calculation.

Finally, $\Delta_1$ becomes a function of $\alpha_1$ and $k$ only. So
we can numerically solve $\Delta_1=0$ for any given wave number.

In order to provide a flavor of the shape of these terms, we give the
general solution for $n=4$, using the short-hand expressions from equation
(\ref{eq:secondorder}) again:

\begin{widetext}
\begin{eqnarray}
\label{eq:zweibisvier}
\alpha_2&=&-2\frac{\alpha_1^2}{l_{12}-4k+4k^2}
+\frac{1}{3}\frac{-294k^3+234k^4-94kl_{12}
+121k^2l_{12}+10l_{12}^2-252k^2}{\left(l_{12}-4k+4k^2\right)^2
\left(l_{12}-6k+9k^2\right)}\alpha_1^4\nonumber\\
\alpha_3&=&-\frac{1}{8}\frac {-60k^2-80k-12l_{12}+12k^3+3kl_{12}}
{\left(l_{12}-4k+4k^2\right)\left(l_{12}-6k+9k^2\right)}\alpha_1^3\\
\alpha_4&=&-\frac{1}{2}\frac{-288k^6+644k^3l_{12}+44kl_{12}^2
+2112k^5-72k^2l_{12}-672k^3+13k^2l_{12}^2+2l_{12}^3-960k^4-52k^4l_{12}}
{\left(l_{12}-4k+4k^2\right)^2\left(l_{12}-6k+9k^2\right)
\left(l_{12}-8k+16k^2\right)}\alpha_1^4\nonumber
\end{eqnarray}
\end{widetext}

We recognize a restriction of our method: because of the
denominators originating from the solution of our system for $a_{m,j}$ the
functions representing $\alpha_n$ %(eq.\,\ref{eq:zweibisvier})
will contain
a number of poles, increasing with order. Their locations depend on the
value of $l_{12}$. Generally, the positions of poles are solutions of
\begin{equation}
l_{12}-2jk+j^2k^2=0;\quad j=2\dots n,
\end{equation}
that is
\begin{equation}
k_{\rm pole}=\frac{1\pm\sqrt{1-l_{12}}}{j}.
\end{equation}

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\caption{\label{fig:margpole}Location of the
marginal wave numbers (bold) and the first pole coming from the
denominators of the amplitude solutions.}
\end{figure}

For the no-gravity case $l_{12}=0$, we find poles at $\nicefrac{2}{j}$
($j=2\dots n$). So the
method will be reliable for all $k$ values sufficiently far above
$k=(1+\sqrt{1-l_{12}})/2$ (unless the solution of the final nonlinear
equation leads to a new singularity).
If $l_{12}>0$ the zeros move to the left, thereby
increasing the range of validity of our method.

Hence, at the critical wave number $k=1$ of the model with gravity, the
method is usable because the largest $k$ value with a pole is located at
$k=\nicefrac{1}{2}$ for $l_{12}=1$. In Fig.~\ref{fig:margpole},
 we have plotted the positions of the marginal wave numbers
and the first poles, depending on $l_{12}$. The range of $k$ values,
in which the expansion method should work properly, is bounded from below
by the upper part of the thin line.

Another point we have to stress concerning the validity of solutions
results from the necessity of expanding the square root in the arc length
formula [Eq.~(\ref{eq:surface})], giving the surface energy. This expansion
does not converge for $\zeta'(x)^2 > 1$. As long as the integral of the
resulting divergent series is still convergent, the total energy will not
be strongly influenced by a nonconvergence of the integrand. Therefore, a
violation of this criterion within a small $x$ interval may not immediately
render the calculation useless. We find that in practice numerical solutions
start to diverge when the criterion
\begin{equation}
\left<\zeta'(x)^2 \right>\!\leq \frac{1}{2}
\end{equation}
is not satisfied. The angular brackets denote a spatial average over one
periodicity unit.


A third factor which might become problematic will be pointed out briefly:
The solution of $\Delta_1=0$, which has to be carried out numerically has
to deal with very large natural numbers appearing in the problem when we
go to high order. For a reliable solution the computer algebra system
(CAS) \texttt{Maple} is an appropriate tool for two reasons: The
engine works with fractions of natural numbers as long as possible,
thereby avoiding truncation errors, and for the genuine floating point
operations we can adjust the accuracy as necessary, which is essential for
the results to follow.

\subsection{\label{sec:no_g_sec}Solutions with gravity neglected}

The method introduced in section \ref{sec:princ} will now be
applied to the simplest variant of the model where only
elasticity and surface tension are present. Therefore we set
$l_{12}=0$. A first check is a comparison with the numerical
result of Spencer and Meiron \cite{spencer:94}.

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\end{picture}
\caption{\label{fig:00_sm}Approximation of the numerical solution by the
expansion method with accuracy $n\!=\!2$ to $n\!=\!8$. The bold line is
the (digitized and scaled) numerical solution from \cite{spencer:94}.
Numbers in circles correspond to curves in Fig.~\ref{fig:example}.}
\end{figure}

Figure \ref{fig:00_sm} shows how the method approaches the numerical
results.
%The curves in the right part of Fig.~\ref{fig:00_sm} approach
%the numerical solution with enhancing accuracy. The two solutions pointing
%upwards correspond to $n=2$ and $n=3$.
%For higher order we encounter a bending of the branch to the right, a
%behavior that we will recover again in the cycloid model later.
The shapes corresponding to each data point grow from a shallow cosine to
a sharply grooved structure, converging to the cycloid-like behavior
proposed by Chiu and Gao \cite{gao:93}, see Fig.~\ref{fig:example}.

With increasing order the solutions get closer to the numerical branch.
Remember that
every solution point corresponds to a whole shape. The shapes
corresponding to marked points in Fig.~\ref{fig:00_sm} are shown
 in Fig.~\ref{fig:example}.
The more the solution approaches a cusped structure,
the more inaccurate it becomes due to the lack of higher modes
necessary to describe the cusps correctly. A sign of decreasing accuracy
is the existence of nonphysical undulations in the shapes, as observable
in the outer parts of curve \ding{200}.

Moreover, we point out that all solutions are stable which can be checked
by calculating the second derivatives with respect to the amplitudes
contained.

%In the left part of Fig.~\ref{fig:00_sm} another branch appears tending
%to become  bent more and more sharply and approaching the termination
%point of the solution of Spencer and Meiron from above. We regard this as
%evidence for the continuation of the solution branch  by shapes like the
%one shown in Fig.~(\ref{fig:example}). \textit{Was hier gebraucht wird,
%ist ein Vergleich der beiden Zweige (auch der Formen) und eine Aussage zur
%Wellenl�ngenverdopplung, falls eine stattfindet}

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\caption{\label{fig:example}Shapes of some
particular solutions of the no-gravity model calculated using eight
amplitudes, according to Fig.~\ref{fig:00_sm}. Curves are scaled in the
$x$ direction to map them onto an interval of length $2\pi$.}
\end{figure}

\subsection{Secondary solutions in the absence of gravity}
Besides the solution which has already been found  numerically \cite{spencer:94}, our
method finds several types of secondary solutions whose physical relevance
has to be discussed.
\subsubsection{Solutions at large $k$ values}
First, there appears to be an infinite number of subcritical ''bubbles''
at increasing wave numbers. To convey an idea of what they look like we
present the first of these branches at different orders of the calculation
in Fig.~\ref{fig:bubbles}. Obviously, to the order of accuracy available
to us, namely $8$, it is impossible to decide with certainty what will be
the behavior of these solutions at infinite order.

At order two there is not a complete bubble but a branch decaying to zero
at infinity. This can be seen by inspecting the second-order energy,
Eq.~(\ref{eq:secondorder}),
replacing $A_2$ by the minimized $A_2$ calculated
in Eq.~(\ref{eq:a2}),
taking the derivative with respect to $A_1$ and setting
$l_{12}=0$. Apart from the trivial solution $A_1=0$ we find
\[A_1^2=8\frac {2-3k+k^2}{3k^2-27k+40},\] which goes to
$\nicefrac{8}{3}$ at large $k$. Calculating the mean square amplitude
$\overline\alpha$ is connected with a division by $k$, hence the decay to
zero.

At order three, this branch bends up again, diverging at $k=22.45$ with no
real solutions for higher $k$ and at order four it moves to lower $k$
(this case is shown in part \textbf{A} of Fig.\,\ref{fig:bubbles}), and an
additional branch (not shown) at higher $k$ emerges, also bending up at
order five and so on.


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\put(.079375,.162557){\makebox(0,0){$\circ$}}
\put(.099375,.142557){\makebox(0,0)[tl]{7}}
\put(.040686,.140087){\makebox(0,0){$\circ$}}
\put(.040686,.120087){\makebox(0,0)[tl]{8}}
\put(.106250,.520625){\makebox(0,0)[tl]{D}}
\end{picture}
\end{minipage}
\caption{\label{fig:bubbles}Additional solutions at different orders. In
each case, \textsl{not all} additional branches are shown. The $\bigodot$
symbols denote the change of stability. For comparison, we have included
the numerical solution from \cite{spencer:94} in each plot. \textbf{A}:
First additional branch at order four. The
 second, decaying one starts above $k \approx 60$ here.
 \textbf{B}: First two additional branches at order six. The
 third, decaying one starts above $k \approx 329$ in this approximation.
 \textbf{C}: Only the first additional branch at order eight. There are
 three more solutions not shown in this diagram. \textbf{D}: Onset
 points
 of the first additional stable branch to orders four through eight.}
\end{figure}

\begin{figure}
%\input{addsol.pic}
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\end{picture}
\caption{\label{fig:addsol} An example shape of the additional solution at
the onset of the stable branch in Fig.~\ref{fig:bubbles}\,\textbf{C},
stretched to an interval $2\pi$. }
\end{figure}
%The question of physical relevance is quickly decided if one looks at the
%energy formulae.
Let us have a look at the energy formulas. As we have expanded them in
powers of $A_1$, they are polynomes. Consequently, the large amplitude
behavior is dominated by the coefficient of the largest power of $A_1$.
Moreover, it turns out that for $k>2$, the lower powers in the solutions
do not produce additional extrema.

So it is easy to decide about the stability of the solutions. As we know
the flat surface to be linearly stable for $k>2$, the next solution
towards higher amplitudes at any fixed $k$ should be unstable and the next
one, if it exists, stable again, because energy minima and maxima must
alternate as a function of the amplitude. So all branches
in Fig.~\ref{fig:bubbles} are stable above the $\bigodot$
symbols, since the energy is
minimal there.

An interesting conclusion is suggested by figure
\ref{fig:bubbles}\,\textbf{D}. The onset point of the first additional
branch seems to converge to $k=2$.
As with no method available another stable
solution at this point is found, the branch might become
infinitely steep at high order, which would mean that the exact energy
landscape becomes independent of amplitude at $k=2$.

A less spectacular interpretation would be that the additional branches
just get very closely spaced without collapsing onto  single lines.

Figure \ref{fig:addsol} gives an interface shape for
the first additional branch,
showing explicitly that interface morphologies from this branch are
significantly different from those of the original  branch
(compare with Fig.~\ref{fig:example}). While this solution is
stable with respect to amplitude perturbations of the same periodicity,
we expect the whole branch to be unstable with respect to large
wavelength perturbations. A Floquet-Bloch analysis would presumably
show that each new branch has one more unstable eigenmode, a situation
that is also encountered in dendritic growth when analysing steady-state
needle crystal solutions.

\subsubsection{Small $k$ values}
First of all, we should notice that for a curve $\zeta(x)$ given by its
Fourier series, which is a solution, there exists an infinite number of
alternative representations
\begin{equation}
\label{eq:ghosts} \zeta(x)=\sum_{j=1}^{\infty}\alpha_j
\cos{\left(nj\left(\frac{k}{n}\right)x\right)}\>,\qquad n\in{\mathbbm N}.
\end{equation}
So, if we for instance choose $n=2$, then we should expect a ''ghost
branch'' starting at $k=1$ instead of $2$ which contains only modes $2, 4,
6\dots$\,. As our expansion method requires the first amplitude, $A_1$, to
be nonzero, we did not expect those solutions to be found by it.

Yet, notwithstanding these considerations, our algorithm
finds solution branches at $k=1$, $k=\nicefrac23$, etc., with a
shape of the mean square amplitude plot similar but not identical
to that of the branch at $k=2$. To understand this, let us consider
the case $k\approx1$.
Numerically, $\alpha_1$ can become very small while still being nonzero.
Then $\alpha_2$ will be finite
-- the first denominator in Eq.~(\ref{eq:zweibisvier})
is close to zero --, $\alpha_3$ will be small again, because the
denominator is $O(\alpha_1^2)$ and the numerator $O(\alpha_1^3)$,
$\alpha_4$ will be finite, and so on.
Hence, we just obtain the ghost branch at $n=2$ from
 Eq.~(\ref{eq:ghosts}). Clearly, the numerical quality of approximation
will not be good in these circumstances.
At larger amplitudes, the ghost branches are not
exact images of their original anymore, because the Fourier expansion
will contain increasingly large contributions from the ``forbidden''
modes $\cos{(l{k}x/{n})}$, $l\ne nj$.

By use of the numerical scheme described in section \ref{sec:numeric},
one avoids the zeros of the denominators, stemming from the expansion
of the amplitudes in powers of $\alpha_1$. Instead, the denominators
will be renormalized by the amplitudes themselves. Hence, it is
possible to reproduce the exact image of the $k=2$ branch as a ghost
at $k=1$ and at lower $k$ values. Moreover, one may decide whether there
are new truly different solution branches near the ghosts. It turns out
that this is not the case, i.e., the
only true solution branch
in the lower vicinity of $k=1$ is the $n=2$ ghost according to
Eq.~(\ref{eq:ghosts}).

\begin{figure}[htb]
%\input{00_add_3.pic}
\setlength{\unitlength}{\columnwidth}
\begin{picture}(1.000000,.590000)(-.150000,-.030000)
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\caption{\label{fig:00_add_3}$n=2$, $3$ and $4$ ghost branches, emerging
at $k=1$, $\nicefrac{2}{3}$ and $\nicefrac{1}{2}$, respectively.}
\end{figure}

We conclude that in the case without gravity, the Spencer-Meiron branch
is presumably
the only one that is stable in finite systems. All additional branches are
either ``ghost images'' of this one or are unstable.

It must be kept in mind, however, that the existence of steady-state
solutions, even though they may be unstable, could play a decisive role in
dynamics. An example is given by the Kuramoto-Sivashinski equation
\cite{misbah:91}, where there exists an infinite set of unstable
steady-state solutions for which it has been attempted to connect them
with spatio-temporal chaos \cite{manneville:87}.

\section{\label{sec:gravity}The case with gravity}
\subsection{The critical case}
The classical linear analysis shows that at a critical stress
\[\sigma_0^2=\frac{1-\nu^2}{E}\sqrt{\Delta\rho\,g\gamma}\] a
single wave number \[k_{\rm c}=\sqrt{\frac{\Delta\rho\,g}{\gamma}}\]
becomes neutral. In our reduced units, this critical point is at
\[k=1\>,\quad l_{12}=1\>.\] Nozi{\`e}res calculated the change of the free
energy up to order $\alpha_1^4$ \cite{nozieres:93} at this point. The
negative sign of the coefficient of the $\alpha_1^4$ term lead to the
conclusion that the Grinfeld instability is subcritical. As a first
example, we calculate this quantity up to $\alpha_1^{16}$ using our
method. As usual, the prefactor $\gamma$ which is also contained in the
elastic energy due to the rescaling has been divided off, and so, at
$k=1$, the change of the free energy is
\begin{multline}
\label{eq:energy_16}
\Delta E=-{\frac {43}{64}}\,{\alpha_{{1}}}^{4}-{\frac
{11661}{1024}}\,{\alpha_{ {1}}}^{6}-{\frac
{3446989}{12288}}\,{\alpha_{{1}}}^{8}\\-{\frac {
317790953065}{37748736}}\,{\alpha_{{1}}}^{10}-{\frac {
154676894913120211}{543581798400}}\,{\alpha_{{1}}}^{12}\\-{\frac {
81526198232945992800847}{7827577896960000}}\,{\alpha_{{1}}}^{14}\\-{
\frac
{2229374504089574965742093912371}{5523138964094976000000}}\,{
\alpha_{{1}}}^{16}\>.
\end{multline}
The prefactors are negative at all orders, and their absolute value
increases exponentially. This behavior shows that a restabilization  does
not occur, at least not at the basic wave number $k=1$. For the elastic
energy can be reduced indefinitely by increasing the amplitude of the
system \footnote{This is of course possible only, because the system is
infinitely extended in the $z$ direction and the homogeneously prestrained
state contains an infinite amount of energy in each periodicity unit. It
is therefore important to take energy differences that are finite in all
the calculations.}.

\subsection{Analysis in the full $l_{12}$ range}
We will now consider nonzero values of $l_{12}$. Solution branches will
start at wave numbers smaller than 2 as has already been pointed out in
discussing Fig.~\ref{fig:margpole}.

In Fig.~\ref{fig:multi_l12} we show the result of the highest-order
expansion available so far, including eight amplitudes. In doing so, we
restrict ourselves to the branches emerging at upper marginal wave numbers.
There are more  solutions for $0<k\leq 1$.

\begin{figure}[htb]
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\caption{\label{fig:multi_l12}Solution branches for different $l_{12}$
values (small numbers), obtained using the expansion method. Solutions
above the dashed line are probably less reliable because at least in the
no-gravity case the analytical result starts to considerably differ from
the numerics (bold again). Other symbols are explained in the text.}
\end{figure}


We will now consider the weakly nonlinear analysis for arbitrary
wave numbers taking gravity into account. The small-amplitude behavior can be
entirely derived from the simplest weakly nonlinear model, namely
Eq.~(\ref{eq:secondorder}). For steady solutions we must take $\Delta
E$ from this equation and require \[\frac{\partial\Delta E}{\partial
A_1}=0\>,\qquad\frac{\partial\Delta E}{\partial A_2}=0\>,\] which leads to
the condition
\begin{equation}
\label{eq:a2}A_2=\frac {2A_1^2 k}{4k\left(1-k\right)-l_{12}}\>.
\end{equation}
Inserting this into the second equation results in a rather lengthy
expression from which at any $(k,l_{12})$ the $A_1$ part of steady
solutions may be obtained. $A_2$ is then calculated by equation
(\ref{eq:a2}), and the quantity plotted in Fig.~\ref{fig:multi_l12} is
the mean square amplitude of the resulting pattern, which is defined by
\begin{equation}
\label{eq:meanamp}\overline{\alpha}=\sqrt{\int\zeta(x)^2 {\rm d}x
}\>.
\end{equation}
In the special case considered, we simply
have
\begin{equation}
\label{eq:meanamp2}\overline{\alpha}k=\overline{A}=
\sqrt{\frac{A_1^2+A_2^2}{2}}\>.
\end{equation}
Now we would like to discuss the stability at the onset of the branches.
Remember that the flat interface is stable against wave numbers larger than
the upper marginal one. The nontrivial steady branch emerges right at the
marginal wave number, and for sufficiently low values of $l_{12}$ it
continues at finite amplitudes towards smaller $k$ values. For $l_{12}=0$
and small amplitudes we find a square root behavior.

This branch pattern, which we may call supercritical (although the wave number
$k$ is not
what one would usually talk about as an order parameter) changes to
subcritical at a value which may be determined by considering the branch
%growing straight up from zero at constant $k$,
emerging uncurved from the bifurcation point.
Hence, to obtain this tricritical point, we just
have to calculate the parameters, where  the second derivative of $\Delta
E$ with respect to the amplitude $A_1$ vanishes at the onset of the
nontrivial branch, i.e., we have to solve the system:
\[k=1+\sqrt{1-l_{12}}\>,\qquad\frac{\partial^2\Delta E}{\partial
A_1^2}=0\] The solution is
\begin{subequations}
\begin{equation}
l_{12,{\rm t}}=-{\frac {148}{9}}+{\frac {20}{9}}\sqrt {57}\approx
0.333
\end{equation}
\begin{equation}
k_{\rm t}=\frac{13}{3}-\frac{1}{3}\sqrt {57}\approx 1.817\>.
\end{equation}
\end{subequations}
Consequently, all branches emerging left of $k_{\rm t}$ (bullet symbol) in
Fig.~\ref{fig:multi_l12} are unstable at the bifurcation point.

Another remark has to be made:
This simple weakly nonlinear model is accurate for amplitudes up to
$\approx 0.2$. So, solutions obtained at
larger amplitudes should not be trusted.
In II, we will present a method for getting reliable solutions
 at  larger amplitudes.

Now we consider the stability of branches emerging to the left of $k_t$. It
can be examined by differentiating $\Delta{E}(A_1,k)$ by $A_1$
twice and computing its sign at all the data points.

For a parameter range up to $l_{12}=0.6$ this yields onset points of
stability, which are located, where the inclination of our solution
branches changes sign (open circles in Fig.~\ref{fig:multi_l12}). At even
higher $l_{12}$ values, a possible onset of stability cannot be calculated
anymore as it would lie in amplitude regions which are not correctly
treated by our expansion.

As a first conclusion we would like to point out that the results in
Fig.~\ref{fig:multi_l12} lead to the analog of the Spencer-Meiron branch in a
system with gravity, showing it to be stable  for small gravity
(with respect to
fixed-periodicity perturbations) at small amplitudes, whereas for larger
gravity stability begins only at a finite amplitude.

Figure \ref{fig:multi_l12_lower} shows the corresponding branches emerging
from the lower marginal wave numbers, for different $l_{12}$ again. At the
onset they are all unstable,
and the large-amplitude behavior is difficult to
assess because there is a rapid change in the amplitudes at almost the
same $k$ value. The shape of solution branches is very sensitive to
changes of order, and for the larger $l_{12}$ parameters our method stops
finding solutions at all.

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\caption{\label{fig:multi_l12_lower}Branches emerging from lower marginal
wave numbers at different $l_{12}$, to highest accuracy available. Comments
see text.}
\end{figure}

\section{\label{sec:numeric}Numerical Investigation}
Having calculated the change of energy due to a corrugation up to some
amplitude we can apply an alternative method to calculate our sets of
amplitudes. This is to take the energy formula and calculate its derivatives
[Eq.~(\ref{eq:minimize})] without any expansion.
Then fix $k$ and try to find a
solution vector ${\mathbf A}=(A_1,A_2,\dots)$ numerically.

When applying this method, a number of nonphysical solution branches
emerge. This happens because of the restricted set of equations which is
used: if we take, for instance, the energy to sixth order, then it
contains the first three amplitudes, and there will be a term like
$A_1^4A_2$, but not $A_1^6A_2$. So, artificial intersection points in the
parameter space $\{A_1, A_2, A_3\}$ are produced which are smeared out and
replaced by other artificial solutions at higher orders.

We therefore have to start searching at $k$ slightly below $2$, thereby
restricting the solution space to small absolute values. This ensures that
the solution found is on the branch already known from \cite{spencer:94}
and section \ref{sec:no_g_sec}. By slowly varying $k$ and taking the
previous solution vector as a guess for the new one we can scan a whole
$k$ interval, until the method breaks down. The procedure can be
programmed with a few lines of \texttt{Maple} code, although the values of
the different solution amplitudes differ by several orders of magnitude
and the system of equations is highly nonlinear.

We have calculated solutions again for the simplest model, at $l_{12}=0$.
The results are shown in Fig.~\ref{fig:nrsols}.

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\end{picture}
\caption{\label{fig:nrsols}Numerical solution of the system
(\ref{eq:minimize}) to different accuracy (increasing from top) at
$l_{12}=0$.}
\end{figure}

The solutions possess a good qualitative agreement with the numerics up to
$k$ slightly below $1.9$ where, at least for order $3$ and higher, the
numerics suddenly terminates by jumping to one of the nonphysical
branches which are not shown here. This behavior can be explained by
regarding Fig.~\ref{fig:bubbles}\,\textbf{C} again: at $k=1.9$ the
unstable solution branch diverges, and the method presented in this
section obviously produces a nonphysical connection between the two
branches.

\section{\label{sec:summary}Summary}
We have extended the weakly nonlinear analysis of the Grinfeld instability
to higher order in the amplitude than previous approaches
\cite{nozieres:93}. %This makes sense as the bifurcation is subcritical at
%the critical point ($k_c$, $l_{12,c}=1$) and the existence of an energy
%functional makes expansions meaningful for finite amplitudes, as long as
%the series converges at the amplitude considered \footnote{When the system
%is nonvariational, we cannot expect more than asymptotic validity of an
%expansion, so finite-amplitude expansions are more doubtful.}.

An interesting fact we found here is that at the critical point
($k_c$, $l_{12,c}=1$) all studied
orders in the amplitude expansion (up to the 15th)
showed subcriticality [see Eq.~(\ref{eq:energy_16}), which gives the
energy at threshold to 16th order].
We conjecture that this phenomenon continues to infinite order.
This feature is rarely encountered in nonlinear systems and interesting in
itself.

For the Grinfeld instability in the absence of gravity, this is the first time
such an expansion is performed. Since in that case, there is no instability
threshold for the infinite system, the system size/wavelength or else the
wave number may be taken as a control parameter. We recover the Spencer-Meiron
result - a supercritically bifurcating branch of steady-state solutions,
which end in a cusp singularity (that we cannot quite reach with our approach).

As gravity (i.e., $l_{12}$) is increased, the bifurcation moves to
smaller wave numbers and changes from supercritical to subcritical. Stable
finite-amplitude solutions are found for intermediate values of  $l_{12}$
but not for all of them. For example, between $l_{12}=0.6$ and $l_{12}=1.0$,
there are no stable solutions with amplitudes smaller than 0.2, and for larger
amplitudes, the methods presented here are not really
useful. In II, we will be
able to make more precise statements about the behavior at larger amplitudes,
as the variational approach used there is more powerful.

There is a second bifurcation in wave number space at small wave numbers,
where the dispersion relation crosses zero again (for nonvanishing
gravity, the $k=0$ mode must always be linearly stable). All branches
emerging there are however unstable. Also we cannot interpret this
bifurcation as being controlled by the system size, because small
wave numbers correspond to large systems, which already contain unstable
modes of smaller wavelength. So this second branch is of no physical
relevance.

With the mode expansion, one can find more branches of solutions, which do not
bifurcate from the planar state but lead a detached existence. They might have
relevance in very small systems which are linearly stable but forced into a new state
by a strong perturbation. However, we cannot make clear-cut statements about
the stability of these states,
since the expansion converges very slowly.
As we have pointed out, even if they are unstable, these states may
by their very existence play an important role in the dynamics.

To summarize, this approach is useful for identifying possible stable steady
states (which are only metastable in an extended system), but despite the high
order accessible in the expansion using computer algebra, it is limited in the
size of the amplitudes about which reliable assertions can be made. The variational
approach to be discussed in II goes beyond this limitation and is even capable of
producing cusped solutions analytically. Moreover, it is not restricted to
interfaces without overhangs.

\begin{acknowledgments}
This work was supported by the {\em Deutsche Forschungsgemeinschaft} under
Grant No. Ka 672/4-2 and FOR 301/2-1, which is gratefully acknowledged. In
addition, we acknowledge travel grants by PROCOPE, Grant No. 9619897
(DAAD, Germany) and 97176 (APAPE, France), enabling a closer collaboration
between the two groups involved in this work.
\end{acknowledgments}

\appendix
\section{\label{app:show_the second}Example for the calculation of the
energy terms to fourth order in the amplitudes}

In order to provide a taste of the algorithm, we present the method of
calculating the necessary energy terms up to order four now. Basically this
is a shorter (in terms of writing effort) but generalized version of
\cite{nozieres:93}. Presuming $\alpha_n\sim\varepsilon^n$, using the
reduced amplitudes $A_n$ and carrying out the transformation
$xk\rightarrow\xi$ we may write
\begin{equation}\label{app:zeta}\zeta(\xi)=k^{-1}\sum_{n=1}^2\varepsilon^n
A_n\cos{\left(n\xi\right)}\end{equation}
and use the parameter $\varepsilon$ for bookkeeping. $\alpha_3$ and
$\alpha_4$ will not play any role for the order 4, because any
combinations of $\alpha_3$, $\alpha_4$, as well as $a_{3,n}, a_{4,n},
b_{3,n}, b_{4,n}$ up to order 4 in the energy terms disappear due to
translational invariance. The derivative is then given by:
\begin{equation}\label{app:derivative}\zeta'(\xi)=-\sum_{n=1}^2\varepsilon^n n
A_n\sin{\left(n\xi\right)}\>.\end{equation}
The simplest term is gravity, and the nondimensionalized and
transformed integral (\ref{eq:gravity}) is given by
\begin{equation}
\label{app:grav}
\frac{l_{12}}{2}\frac{1}{\pi}\int_{\xi=0}^{\pi}\zeta(\xi)^2 {\rm
d}\xi=\frac{l_{12}}{4k^2}\left( \varepsilon^2 A_1^2+\varepsilon^4
A_2^2\right)\>.
\end{equation}

Next we expand the arc length standing for the reduced surface energy.
It is expressible as
\begin{multline}
\label{app:ofl} \frac{1}{\pi}\int_{\xi=0}^{\pi}\sqrt{1+\zeta'(\xi)^2} {\rm
d}\xi-1\\=\varepsilon^2\frac{1}{4}A_1^2+\varepsilon^4\left(A_2^2-
\frac{3}{64}A_1^4\right)\>.
\end{multline}

The most difficult part is the elastic energy. We start with the setup of
the Airy stress function. Again, it is sufficient to go to order 2 in the
ansatz known from Eq.~(\ref{eq:airy}), which we have slightly
modified in order to facilitate later calculations,
\begin{equation}
\chi=\sum_{n=1}^2\varepsilon^n\left(\frac{a_n}{n^2k^2}+\frac{b_n}{nk}
z\right)\cos{(nkx)}e^{nkz} \>.
\end{equation}

First, we write down the components of the additional stress tensor
originating from the derivatives of $\chi$,
in doing so assuming the orders of
$a_n$ and $b_n$ to be $\varepsilon^n$. Additionally we carry out the usual
transformations $kx\rightarrow x$ and $kz\rightarrow z$:
\begin{eqnarray}
\label{app:components} \sigma_{xx}&=&\sum _{n=1}^{2}\varepsilon^n\cos
\left( nx \right)
\left(a_n\!+\!2b_n\!+\!b_n zn \right)e^{nz}\>,\nonumber\\
\sigma_{xz}&=&\sum _{n=1}^{2}\varepsilon^n\sin \left( nx \right)
\left(a_n\!+\!b_n\!+\!b_n zn \right)e^{nz}\>,\\
\sigma_{zz}&=&-\sum _{n=1}^{2}\varepsilon^n\cos\left(nx
\right)\left(a_n\!+\!b_n zn\right) e^{nz}\>.\nonumber
\end{eqnarray}
We now have a scheme for the definite calculation of as many Airy
coefficients $a_{i,j}, b_{i,j}$ as necessary. We have to keep in mind
that, according to our ansatz, the \textit{sum} of subscripts of an Airy
coefficient is identical with its order. For instance, $a_{3,0}$ and
$a_{1,2}$ are both of order $\varepsilon^3$.

Our scheme yields  a set of equations at every order $\varepsilon^n$
allowing us to calculate the coefficients whose subscript
 sum is $n$, respectively.
The following steps have to be carried out:

\begin{enumerate}
\item Write down the equations (\ref{eq:ggw2}) using the stress
components (\ref{app:components}) and the transformed derivative
(\ref{app:derivative}).
\item Expand those equations up to order $z^3$. (A $z^4$ term would produce at least $\varepsilon^4$ terms in the equations. These are not necessary for the calculation of the energy to order $\varepsilon^4$ as can be seen from table \ref{tab:scheme}.)
\item Replace $z$ by the interface (\ref{app:zeta}).
\item Expand the Airy coefficients $a_n$ and $b_n$ to sufficiently
high order according
to Eq.~(\ref{eq:expand_airy}). This means we have to consider all $a_{n,m}$ and $b_{n,m}$ where $n + m \leq 3$, omitting odd values of $m$.
\item Keep every term in the equations up to order $\varepsilon^3$.
\end{enumerate}

According to \textsl{every} order $\varepsilon^n$ we get two equations
(\ref{app:vielegleichungen}), in which we have replaced $\sin(nkx)$ and
$\cos(nkx)$ by $S_n$ and $C_n$ for brevity. Moreover we have used the theorem
to be proven in Appendix \ref{app:odd_disappear} by setting all
coefficients whose second subscript is odd to zero.

\begin{table}[htb]
\caption{\label{tab:scheme}Schematic view of the expansion method. It shows, which pairs of coefficients come out of which mode and order. Subscripts denote the energy order they are necessary for. We already took into account that coefficients with an odd second coefficient are zero.}
\begin{tabular}[t]{|c|cccc|}
Order &\multicolumn{4}{|c|}{Mode}\\
\hline
$\varepsilon^6$ & $\uparrow$     &$\nearrow$&&\rule{12mm}{0mm}\\
$\varepsilon^5$ & $\left[a_{1,4}, b_{1,4}\right]_{6}$ &&$\nearrow$&\\
$\varepsilon^4$ &  &$\left[a_{2,2}, b_{2,2}\right]_{6}$&&$\nearrow$\\
$\varepsilon^3$ & $\left[a_{1,2}, b_{1,2}\right]_{4}$ &&$\left[a_{3,0}, b_{3,0}\right]_{6}$&\\
$\varepsilon^2$ &&$\left[a_{2,0}, b_{2,0}\right]_{4}$&&\\
$\varepsilon^1$ & $\left[a_{1,0}, b_{1,0}\right]_{2}$ &&&\\
\hline
& $(S_1,C_1)$& $(S_2,C_2)$& $(S_3,C_3)$&$\longrightarrow$\\
\end{tabular}
\end{table}


\begin{widetext}
\begin{eqnarray}
\label{app:vielegleichungen}
\left( A_1+b_{1,0}+a_{1,0}\right)S_1&=&0\nonumber\\
-a_{1,0}\,C_1&=&0\nonumber\\
\left( 2A_1b_{1,0}+A_1a_{1,0}+b_{2,0}+a_{2,0}+2A_2\right)S_2&=&0\nonumber\\
\left(A_1b_{1,0}+A_1a_{1,0}+a_{2,0}\right)C_2&=&0\nonumber\\
\left( a_{1,2}+b_{1,2}+\frac{9}{8}b_{1,0}A_1^2+
\frac{3}{8}a_{1,0}A_1^2+\frac{1}{2}a_{1,0}A_2+
b_{1,0}A_2+b_{2,0}A_1+\frac{1}{2}a_{2,0}A_1\right)S_1 &&\nonumber\\
+\left({\rm equation\ for\ } b_{3,0}\right)S_3&=&0\nonumber\\
\left(\frac{1}{2}a_{1,0}A_2+\frac{1}{2}b_{1,0}A_2-\frac{1}{2}a_{2,0}
A_1-\frac{1}{2}b_{2,0}A_1-a_{1,2}-\frac{1}{8}a_{1,0}A_1^2-
\frac{1}{4}b_{1,0}A_1^2\right)C_1 &&\nonumber\\
+\left({\rm equation\ for\ } a_{3,0}\right)C_3&=&0
\end{eqnarray}
\end{widetext}
Now, the coefficients of every function
$S_n$ and $C_n$ have to be
zero separately, which finally provides a system of six equations for the
coefficients $a_{1,0}$, $a_{1,2}$, $a_{2,0}$, $b_{1,0}$, $b_{1,2}$ and
$b_{2,0}$ ($a_{3,0}$ and $b_{3,0}$, which we would get from the
coefficients of $S_3$ and $C_3$ are not necessary for order 4). The
solution is
\begin{eqnarray}
a_{1,0}=0\>,&&b_{1,0}=-A_1\>,
\nonumber\\
a_{2,0}=A_1^2\>,&&b_{2,0}=A_1^2-2A_2\>,
\\
a_{1,2}=-\frac{3}{4}A_1^3+\frac{1}{2}A_1A_2\>,&&
b_{1,2}=\frac{3}{8}A_1^3+\frac{5}{2}A_1A_2\>.\nonumber
\end{eqnarray}

Finally, the elastic energy is calculated by taking the reduced elastic
energy density
$\nicefrac{1}{2}((1+\sigma_{xx})^2+\sigma_{zz}^2+2\sigma_{xz}^2)$,
integrating first over $z$ from $-\infty$ to $\zeta$, expanding the result to
sufficient order in $\zeta$, replacing $\zeta$ by $\zeta(x)$, integrating over
$x$ and keeping
everything up to order $\varepsilon^4$. The result is the first line of
Eq.~(\ref{eq:secondorder}).

\section{\label{app:odd_disappear}Proof for the vanishing
of odd expansion coefficients in the Airy constants}

The basis of this consideration are equations (\ref{eq:ggw2}) which we
write down here again. We will call the l.h.s. of these equations $t_1$
and $t_2$.
\begin{equation}
\label{app:t1t2}
\begin{split}
t_1&:=-\left(1+\sigma_{xx}\right)\zeta'(x)+\sigma_{xz}\>,\\
t_2&:=-\sigma_{xz}\;\zeta'(x)+\sigma_{zz}\>.
\end{split}
\end{equation}
The principle we have to apply is the generalized version of the one
exhibited in appendix (\ref{app:show_the second}): Use the terms from
Eqns.~(\ref{app:derivative}) and (\ref{app:components}) in (\ref{app:t1t2}),
expand the Airy coefficients $a_n$ and $b_n$ as well as the exponential functions,
replace $z$ by the surface (\ref{app:zeta}), and then first sort by orders
of $\varepsilon$ and then by modes. This will provide a system for all
expansion coefficients of $a_n$ and $b_n$. Note that in \textsl{this}
calculation no square root is included. So the elastic field calculated in
this way will be correct for all  $\zeta$, regardless of the value of the derivative,
for which the power series converges. Since the functions to be expanded are
exponentials, for which the radius of convergence is infinite, it is expected
that the expansion of the elastic fields has a much larger radius of convergence
than that of the square root.
% \textsl{any} $\zeta$, regardless of the value of the derivative.

\begin{definition}
The n-th order part of a term $t$ will be denoted by $t^{[n]}$. The
selection of a mode $m$ from such a term will be called
$\left.t^{[n]}\right|_{m}$.
\end{definition}

Obviously, $t_1$ and $t_2$ do not contain zeroth order parts. So,
\begin{subequations}
\label{app:t1t2_b}
\begin{equation}
\label{app:t1t2_b1}
t_1^{[n]}=-\sum_{m=1}^{n-1}\sigma_{xx}^{[m]}\zeta'(x)^{[n-m]}
+\sigma_{xz}^{[n]}-\zeta'(x)^{[n]}\>,
\end{equation}
\begin{equation}
\label{app:t1t2_b2}
t_2^{[n]}=-\sum_{m=1}^{n-1}\sigma_{xz}^{[m]}\zeta'(x)^{[n-m]}
+\sigma_{zz}^{[n]}\>.
\end{equation}
\end{subequations}
Consider any one of the stress terms. First take a look at the
exponential function. In
the course of our algorithm it will be expanded, i.e., every $z^j$ with
$j\geq0$ will appear in the sum. Provided $\zeta(\xi)$ is given as in
(\ref{app:zeta}), we note that $\zeta(x)^{[2m]}$ corresponds to even modes
and $\zeta(x)^{[2m+1]}$ to odd modes. Consequently, when constructing
${\zeta(\xi)^2}^{[j]}$ using the well-known formula
\[\cos \left( nx \right) \cos \left( mx \right) =\frac{1}{2}\left[
\cos \left((n-m)x\right) +\cos \left((n+m)x\right)\right]
\]
any even (odd) order $j$ contains only even (odd) modes again. We can
conclude by induction that this holds for any ${\zeta(\xi)^n}^{[j]}$, and
hence we formulate

\begin{lemma}
Provided $\zeta(\xi)$ is given in the form of Eq.~(\ref{app:zeta}),
the expansion of the term ${\exp\left(n\zeta(\xi)\right)}^{[j]}$ will
contain only even (odd) modes, if $j$ is even (odd).
\end{lemma}
Without proof we additionally state
\begin{lemma}
${\exp\left(n\zeta(\xi)\right)}^{[j]}$ contains the cosine modes $0\dots
j$.
\end{lemma}
For shorthand notations we now define
\begin{eqnarray*}
{\exp\left(n\zeta(\xi)\right)}^{[j]}&:=&E_{n,j}\>,\\
\cos\left(n\xi\right)&:=&C_n\>,\\
\sin\left(n\xi\right)&:=&S_n\>.\\
\end{eqnarray*}
 Next we consider an
$\varepsilon$-order of a generalized stress-like term, call it
$\sigma_{\mu\nu}^{[n]}$. Let it contain some general Airy coefficient
$g_j$, so that
\[\sigma_{\mu\nu}=\sum_{j=1}^{\infty}\varepsilon^j g_j
C_je^{j\zeta(\xi)}\>.\] Of course, for the general form, we could have
used sines instead of cosines as well. We expand the Airy coefficients and
the exponential functions:
\[g_j=\sum_{i=0}^{\infty}\varepsilon^ig_{j,i}\>,\qquad
e^{j\zeta(\xi)}=\sum_{i=0}^{\infty}\varepsilon^iE_{j,i}\>.\] Now,
$\sigma_{\mu\nu}^{[n]}$ has to be composed from parts of the $n$ lowermost
summands in this series:
\begin{equation}
\label{app:s_terms}
\sigma_{\mu\nu}^{[n]}=\sum_{j=1}^{n}C_j\sum_{i=0}^{n-j}
g_{j,n-i-j}E_{j,i}\>,\quad n=1\dots\infty \>.
\end{equation}
It is useful to subdivide the terms (\ref{app:s_terms}) into those with
even and odd $n$:
\begin{subequations}
\begin{equation}
\label{app:s_terms2a}
\sigma_{\mu\nu}^{[2n-1]}=\sum_{j=1}^{2n-1}C_j\sum_{i=0}^{2n-1-j}
g_{j,2n-1-i-j}E_{j,i}\>,
\end{equation}
\begin{equation}
\label{app:s_terms2b}
\sigma_{\mu\nu}^{[2n]}=\sum_{j=1}^{2n}C_j\sum_{i=0}^{2n-j}
g_{j,2n-i-j}E_{j,i}\>.
\end{equation}
\end{subequations}
To be able to give brief descriptions we introduce
\begin{definition}
Airy coefficients whose second index is even (odd) will be called even
(odd).
\end{definition}
By inspection we can derive now

\begin{lemma}
\label{app:secondlemma} $\sigma_{\mu\nu}^{[2n-1]}|_{2l}$ and
$\sigma_{\mu\nu}^{[2n]}|_{2l-1}$ contain odd Airy coefficients only.
$\sigma_{\mu\nu}^{[2n-1]}|_{2l-1}$ and $\sigma_{\mu\nu}^{[2n]}|_{2l}$
contain even Airy coefficients only.
\end{lemma}

\begin{lemma}
\label{app:thirdlemma} Lemma \ref{app:secondlemma} also holds for terms of
the extended form $\sigma_{\mu\nu}^{[m]}\zeta'(\xi)^{[2n-1-m]}$ and
$\sigma_{\mu\nu}^{[m]}\zeta'(\xi)^{[2n-m]}$, respectively.
\end{lemma}

Consequently, we can subdivide our set of equations for the Airy
coefficients $\{t_1^{[n]}|_{m},t_2^{[n]}|_{m}\}$ into
\begin{eqnarray*}
s_1&:=&\{t_1^{[2n-1]}|_{2m},t_1^{[2n]}|_{2m-1},t_2^{[2n-1]}|_{2m},t_2^{[2n]}|_{2m-1}\}\>,\nonumber\\
s_2&:=&\{t_1^{[2n]}|_{2m},t_1^{[2n-1]}|_{2m-1},t_2^{[2n]}|_{2m},t_2^{[2n-1]}|_{2m-1}\}\>.
\end{eqnarray*}

The set $s_1$ contains the equations which determine the odd Airy
coefficients and $s_2$ stands for the even ones.

The only source of terms that are \textsl{not} products including Airy
coefficients is the single $\zeta'(\xi)^{[n]}$ in Eq.~(\ref{app:t1t2_b1}).
 Now, $\zeta'(\xi)^{[n]}$ is simply equal to $-n A_n
\sin {n \xi}$, and consequently $\left.\zeta'(\xi)^{[n]}\right|_{m}=-n
A_n\delta_{n,m}$.

So, this term does not contribute to the $t_1$-equations in $s_1$ but only
in $s_2$. Hence $s_1$ is a homogenous system for the determination of
$\{a_{n,2m-1}, b_{n,2m-1}\}$. It is solved by $\{a_{n,2m-1}=0,
b_{n,2m-1}=0\}$, which had to be proven.

\textbf{Remark}: It is possible but lengthy to show that this is the
\textsl{only} solution. In principle one takes advantage of the fact that
the determination of the odd coefficients can be carried out uniquely by a
bottom-up scheme using the empty entries in table \ref{tab:scheme}.


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\end{thebibliography}