\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 118, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/118\hfil Kuramoto-Sivashinsky equation]
{A spatially periodic Kuramoto-Sivashinsky equation as a
model problem for inclined film flow over wavy bottom}

\author[H. Uecker, A. Wierschem\hfil EJDE-2007/118\hfilneg]
{Hannes Uecker, Andreas Wierschem}  % in alphabetical order

\address{Hannes Uecker \newline
Institut f\"ur Analysis, Dynamik und Modellierung, Universit\"at
Stuttgart,\newline D-70569 Stuttgart, Germany}
\email{hannes.uecker@mathematik.uni-stuttgart.de}

\address{Andreas Wierschem \newline
Fluid Mechanics and Process Automation, Technical University of Munich,
D-85350 Freising, Germany}
\email{wiersche@wzw.tum.de}

\thanks{Submitted May 15, 2007. Published September 6, 2007.}
\subjclass[2000]{35B40, 35Q53}
\keywords{Inclined film flow; wavy bottom; Burgers equation; \hfill\break\indent
  stability; renormalization}

\begin{abstract}
 The spatially periodic Kuramoto-Sivashinsky equation (pKS)
 $$
 \partial_t u=-\partial_x^4 u-c_3\partial_x^3 u-c_2\partial_x^2 u+2
 \delta\partial_x(\cos( x)u)-\partial_x(u^2),
 $$
 with $u(t,x)\in\mathbb{R}$, $t\geq 0$, $x\in\mathbb{R}$,
 is a model problem for inclined film flow
 over wavy bottoms and other spatially periodic systems
 with a long wave instability.
 For given $c_2,c_3\in\mathbb{R}$ and small $\delta\geq 0$ it has
 a one dimensional family of spatially periodic stationary solutions
 $u_s(\cdot;c_2,c_3,\delta,u_m)$,  parameterized by the mass
 $u_m=\frac 1 {2\pi}\int_0^{2\pi} u_s(x) \,{\rm d} x$.
 Depending on the parameters these stationary solutions
 can be linearly stable or unstable.
 We show that in the stable case localized perturbations decay with a
 polynomial rate and in a universal nonlinear self-similar way: the limiting
 profile is determined by a Burgers equation in Bloch wave space.
 We also discuss linearly unstable $u_s$, in which case we approximate
 the pKS by a constant coefficient KS-equation. The analysis is based on
 Bloch wave transform and renormalization group methods.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}\label{i-sec}

The inclined film problem concerns the flow of a viscous
liquid film down an inclined plane, driven by gravity. This has various
engineering applications, where often the bottom plate is not flat
but has a wavy profile, for instance  $y=\delta\cos( x)$. Over an
infinitely long flat ($\delta=0$) bottom  the problem
can be reduced in a hierarchy of (formal) reductions to a variety of simpler
equations, such as Boundary Layer equations,
Integral Boundary Layer equations (IBL), also called Shkadov models,
and KdV and Kuramoto--Sivashinsky (KS) type
of equations, see \cite{cd02} and the references therein.
Moreover, there exist approximation
results \cite{ue03ibl} concerning the validity of some of
these simplified equations, and results on special nontrivial
solutions such as pulse trains and their stability; see
\cite{cd02}, and \cite{psu03b}. Finally, for the problem over a flat
incline it is shown in \cite{ue04nsb} that
in the linearly stable case localized perturbation of the trivial
(Nusselt) solution decay in a universal way to zero, with limiting
profile determined by the Burgers equation.

Over wavy bottoms the problem becomes much more complicated.
For experimental results  we refer to
\cite{wla05,avb06}.
Analytically, first of all, the basic spatially periodic stationary
solution $U_s$ (Nusselt solution) is not known in closed form.
In \cite{wla05} an expansion of $U_s$
in the film thickness is given,
and the associated critical Reynolds number ${\rm R}_c$
is calculated, such that $U_s$ is linearly stable for ${\rm R}{\le}{\rm R}_c$
and unstable for ${\rm R}{>}{\rm R}_c$.
However, for thicker films  over wavy bottoms no analytical results are known.
Therefore, simplifications of the full Navier--Stokes problem
are much desired. In \cite{wbhua06} the problem is reduced to a
two dimensional quasilinear parabolic system with spatially
periodic coefficients,
the periodic Integral Boundary Layer equation (pIBL), and stationary
solutions $U_S$ of the pIBL are calculated which
show good agreement with experiments.
Moreover, preliminary numerical simulations of the dynamic pIBL
yield a variety of interesting regimes, two of which are:
\begin{itemize}
\item[(i)]
self--similar decay of localized
perturbations of $U_s$ to zero in the case of linearly stable $U_s$;
\item[(ii)] modulated pulses in the linearly unstable case.
\end{itemize}
Here we prove, for a model problem,
a rigorous nonlinear stability result which explains the behaviour
in (i). Moreover, we remark on (formal) explanations for (ii).
Our model problem is an extension of the KS equation as a model
problem for inclined film flow over flat bottom. In particular,
it has dynamics similar to (i), (ii) above, see Fig.~\ref{f1}.
In detail, our model problem is
\begin{equation}
\partial_t u=-\partial_x^4 u-c_3\partial_x^3u-c_2
\partial_x^2 u+2\delta\partial_x(\cos( x)u)-\partial_x(u^2),
\label{pks}
\end{equation}
with $t\geq 0$, $x\in\mathbb{R}$, $u(t,x)\in\mathbb{R}$;
i.e., a spatially periodic Kuramoto--Sivashinsky
equation (pKS) over the infinite line.
In context with the inclined film problem,
$u$ should be interpretated as the film height, while
the parameters $c_2,c_3,\delta,$ have the following meaning:
$c_2\in\mathbb{R}$
corresponds to ${\rm R}-{\rm R}_c$; i.e., to the distance from criticality,
and $\delta$ models the amplitude of the bottom, thus, w.l.o.g $\delta\geq 0$.
The linear terms $-\partial_x^4 u-c_2\partial_x^2 u$ model
a long wave instability (for $c_2>0$) with short wave saturation,
while $-c_3\partial_x^3$ models 3rd order dispersion.
The nonlinearity $-\partial_x(u^2)$ is the standard convective one.
An important feature of the inclined film problem is
the conservation of mass; i.e., $ \partial_t\int_\mathbb{R} 
h(t,x){-}h_0\,{\rm d} x = 0$, where $h$ is the film height and
$h_0$ the reference film height.
This also holds for \eqref{pks}: the right hand side
is a total derivative. Consequently the mass (or average film height)
$$
u_m:=\lim_{M\to\infty}\frac 1 {2M}\int_{-M}^{M} u(t,x)\,{\rm d} x
$$
can be seen as a $4^{th}$ parameter.
Given $c_2,c_3,u_m\in\mathbb{R}$ and small
$\delta>0$, \eqref{pks} has a unique spatially
$2\pi$ periodic stationary solution
$$
u_s(x)=u_s(x;c_2,\delta,u_m)=u_m+\delta u_1(x)+\mathcal{O}(\delta^2)
$$
which can be calculated by expansion in $\delta$ (see sec.\ref{s-sec}).

\begin{figure}[htbp]
\begin{center}
\hspace{6mm}(a)\hspace{56mm} (b)\\
\includegraphics[width=0.45\textwidth]{fig1a} \quad
\includegraphics[width=0.45\textwidth]{fig1b}
\end{center}
\caption{Numerical simulations of \eqref{pks} with periodic boundary
conditions on large domains.
(a) $(\delta,c_2,c_3)=(0.1,-1,0)$ (stable case), $x\in[0,80\pi]$,
 initial data $u_0(x)=2$ for
$x\in[39\pi,41\pi]$, $u_0(x)=1$ else. The solution decays
to $u_s$ in a self--similar way determined by
the Burgers equation \eqref{bfp} below.
(b) $(\delta,c_2,c_3)=(0.1,0.2,-1)$ (unstable case),
$x\in[0,40\pi]$,
initial data $u(0,x){=}2$ for $19\pi\leq x\leq 21\pi$ and
$u(0,x){=}1$ else (full line). Modulated pulses
emerge and travel forever. The dotted line shows the solution of
the associated amplitude equation \eqref{ccks}, see Appendix \ref{a-sec}.} 
\label{f1}
\end{figure}


\begin{remark}\label{mrem} \rm
From the modelling point of view it appears reasonable to also include
a parameter $\gamma$ for the bottom wave number; i.e., to consider
\begin{equation}
\partial_t u=-\partial_x^4 u-c_3\partial_x^3 u
-c_2\partial_x^2 u+2\delta\partial_x(\cos(\gamma x)u)-\partial_x(u^2).
\label{pks2}
\end{equation}
However, rescaling $v(\tau,y)=\alpha u(\beta\tau,\gamma y)$ with
$\beta=\gamma^{4}$ and $\alpha=\gamma^{3}$ yields that $v$ fulfills
\eqref{pks} with $c_3,c_2,\delta$ replaced by $\tilde{c}_3=\gamma^2c_3,
\tilde{c}_2=\gamma^2c_2$
and $\tilde{\delta}=\gamma^3\delta$. Hence $\gamma$ is not independent.

Moreover, the KdV term $-c_3\partial_x^3$ plays no essential role in case (i)
and hence for the main results of
our paper; however, in the unstable case (ii), the KdV term becomes
important, in particular in the limit of large $c_3$,
see \cite{psu03b} for the constant coefficient case. Therefore
we also include it in \eqref{pks}.
\end{remark}


\subsection{Linear and Nonlinear diffusive stability}
Setting $u(t,x)=u_s(x)+v(t,x)$ we obtain
\begin{equation}\label{ppks}
\partial_t v=\mathcal{L}(x)v-\partial_x(v^2)
\end{equation}
with $2\pi$ periodic linear operator
\begin{equation}
\mathcal{L}(x)v=-\partial_x^4 v-c_3\partial_x^3v-c_2\partial_x^2 v+2\delta\partial_x(\cos( x)v)
-2(u_s'(x)v+u_s(x) \partial_x v),
\label{pksl}
\end{equation}
where $u_s'=\partial_x u_s$.
To calculate the eigenfunctions of the linearization
$\partial_t v=\mathcal{L}(x)v$
we make a Bloch wave ansatz \cite{scar99}
$$
v(t,x)={\rm e}^{\lambda(\ell)t+{\rm i} \ell x}\tilde{v}(\ell,x).
$$
Here $\ell\in [-1/2,1/2)$, which is called the first Brillouin zone,
and $\tilde{v}(\ell,x+2\pi)=\tilde{v}(\ell,x)$ and
$\tilde{v}(\ell+1,x)={\rm e}^{{\rm i} x}\tilde{v}(\ell,x)$.
Then $\lambda(\ell)$ and $\tilde{v}(\ell,\cdot)$ are determined from the
linear eigenvalue problem
\begin{equation*}
\begin{aligned}
\lambda(\ell)\tilde{v}(\ell,x)&=\tilde{\mathcal{L}}(\ell,x)\tilde{v}(\ell,x)
\\
:=&\bigl[-(\partial_x{+}{\rm i}\ell)^4-c_3(\partial_x{+}{\rm i}\ell)^3
 -c_2(\partial_x{+}{\rm i}\ell)^2+
2\delta( \cos( x)(\partial_x{+}{\rm i} \ell){-}\sin( x))\bigr]\tilde{v} \\
&{-}2\bigl[u_s'(x){+}u_s(x)(\partial_x{+}{\rm i}\ell)\bigr]\tilde{v}
\end{aligned}
\end{equation*}
over the bounded domain $x\in (0,2\pi)$.
Thus we obtain
curves of eigenvalues $\ell\mapsto\lambda_n(\ell)$, $n\in\mathbb{N}$,
which we sort by $\mathop{\rm Re}\lambda_n(\ell)\geq \mathop{\rm Re}\lambda_{n+1}(\ell)$,
with associated eigenfunctions $\tilde{v}_n(\ell,x)$.
The $\lambda_n(\ell)$ can again be
calculated by perturbation analysis in $\delta$,
again see sec.\ref{s-sec} for details.
Clearly, $\mathop{\rm Re}\lambda_n(\ell)\to-\infty$ as
$n\to\infty$, and $u_s$ is linearly stable if $\mathop{\rm Re}\lambda_1(\ell)\leq 0$
for all $\ell\in [-1/2,1/2)$. Since, for given $c_2,c_3,\delta$,
we have a 1--parameter family $u_s(x;c_2,c_3,\delta,u_m)$ of
stationary solutions, parametrized by $u_m$,
we always have $\lambda_1(0)=0$ with
$$
\tilde{v}_1(0,x)=\partial_{u_m}u_s(x;c_2,c_3,\delta,u_m).
$$
This corresponds to conservation of mass in the inclined film problem.
Next writing
\begin{equation}
\lambda_1(\ell)=-{\rm i} d_1\ell-d_2\ell^2+\mathcal{O}(\ell^3)\label{lam1e}
\end{equation}
and assuming that $\mathop{\rm Re}\lambda_1(\ell)<0$ outside some
neighborhood of
$\ell=0$ we find that $u_s$ is linearly stable for small $\delta$
if $d_2>0$. In this case, the continuous spectrum up to the
imaginary axis yields diffusive decay of
localized perturbations to zero; i.e., for solutions $v$
of $\partial_t v(t,x)=\mathcal{L}(x)v(t,x)$ with $v_0\in L^1(\mathbb{R})$
we have
\begin{equation}
v(t,x)=\frac z{\sqrt{4\pi d_2 t}}\exp(-(x-d_1t)^2/4d_2t)
\tilde{v}_1(0,x)+\mathcal{O}(t^{-1}),
\label{lindecay}
\end{equation}
where $z=\int_\mathbb{R} v_0(x)\,{\rm d} x$ is the mass of the perturbation
and where $d_1=2u_m+\mathcal{O}(\delta)$ is the speed of the comoving frame.

In contrast to exponential decay rates, the algebraic decay
\eqref{lindecay} is too weak to control arbitrary nonlinear terms.
For instance,
solutions to $\partial_tv=\partial_x^2 v+v^2$ on the real line may blow
up in finite
time \cite{wei81}, even for arbitrary small initial data.
On the other hand, for
$\partial_tv=\partial_x^2 v+v^{p_1}(\partial_x v)^{p_2}$
with $p_1+2p_2>3$ it is well known that
solution to small localized initial data decay asymptotically
as for the linear problem $\partial_t v=\partial_x^2 v$
(cf.\,\eqref{lindecay} with
$d_1=0, d_2=1$). This is called nonlinear diffusive
stability, and the nonlinearity is called asymptotically irrelevant.
A very robust method to prove such results is the
renormalization group \cite{bkl94}, which uses an iterative
rescaling argument and has been
applied to a variety of diffusive stability problems
\cite{gs96a,gs98,ue04nsb}.

The case $p_1+2p_2=3$ is called marginal.
In fact, we show that the asymptotics of solutions
of \eqref{ppks} to small localized initial conditions are not given by
Gaussian decay as in \eqref{lindecay} but are determined by a non Gaussian
profile related to the Burgers equation
 \begin{equation}
\partial_t v=d_2\partial_x^2 v+b\partial_x(v^2)\quad\text{with}\quad
b=-1+\mathcal{O}(\delta^2).
\label{b1}
\end{equation}
This profile is obtained by
Cole Hopf transformation. Setting
$$
\psi(t,x)=\exp\Big(\frac b{d_2}\int_{-\infty}^{\sqrt{d_2}x}
  v(t,\xi)\,{\rm d}\xi\Big),\quad
v(t,x)=\frac{\sqrt{d_2}}{b}\frac{\psi_y(t,y)}{\psi(t,y)},\quad
y=x/\sqrt{d_2}
$$
the Burgers equation is transformed into
the linear diffusion equation
$\partial_t\psi=\partial_x^2\psi$, $\psi|_{t=0}=\psi_0$.
For $\lim_{x\to-\infty}\psi_0(x)=1$ and setting $\lim_{x\to\infty}
\psi_0(x){=}z{+}1$; i.e.,
\[
\ln(z{+}1){=}\frac b{d_2}\int_\mathbb{R} v_0(t,\xi)\,{\rm d}\xi,
\]
it is well known that
$1+z\mathop{\rm erf}(x/\sqrt{t})$ with 
${\rm erf}(x)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^x
{\rm e}^{-\xi^2/4}\,{\rm d} \xi$
is an exact solution of $\partial_t\psi=\partial_x^2\psi$.  It follows that
\begin{equation}
  v^{(z)}(t,x)= t^{-1/2}f_z(x/\sqrt{t}) \quad\text{with}\quad f_z(y)=
  \frac{\sqrt{d_2}}{b}\frac {z\mathop{\rm erf}'(y)} {1+z\mathop{\rm erf}(y)}
\label{bfp}
\end{equation}
is an exact solution of the Burgers equation.  Moreover,
$$
\psi(t,x)=\frac{1}{\sqrt{4\pi t}}\int {\rm e}^{-(x-y)^2/(4t)} \psi_0(y)\,{\rm d} y
=1+z\mathop{\rm erf}(x/\sqrt{t})+\mathcal{O}(t^{-1/2})
$$
 as  $t\to\infty$, for initial conditions $\psi_0\in L^\infty(\mathbb{R})$
with $\lim_{\xi\to-\infty}\psi(\xi)=1$ and with $\lim_{\xi\to\infty}
\psi(\xi)=1+z$.  Therefore the so called renormalized
solution of \eqref{b1} satisfies
\begin{equation}
  t^{1/2}v(t,t^{1/2}x)=f_z(x)+\mathcal{O}(t^{-1/2});
\label{bua}
\end{equation}
i.e., it converges towards a non-Gaussian limit.
It has been shown in
\cite{bkl94} that the dynamics \eqref{bua} in the
Burgers equation is stable under addition of higher order terms.
Similarly, our basic idea is that after
a suitable transform (see \eqref{wtidef} below), \eqref{ppks}
in the linearly stable case ($d_2>0$ in \eqref{lam1e}) can be interpretated
as a higher order perturbation of the Burgers equation \eqref{b1}.

\subsection{The nonlinear stability result}

Throughout this paper we denote many different constants that
are independent of $\delta$ and the rescaling parameter $L{>}0$ (see below) by
the same symbol $C$.  For $m,n\in\mathbb{N}$ we define the weighted spaces
$H^m(n){=}\{u\in L^2(\mathbb{R}) : \| u \|_{H^m(n)}{<}\infty\}$ with $\| u
\|_{H^m(n)} = \| u \rho^n \|_{H^m(\mathbb{R})}$, where $ \rho(x) = (1+|x|^2)^{1/2}$
and $H^m(\mathbb{R})$ is the Sobolev space of functions with derivatives up to order
$m$ in $L^2(\mathbb{R})$. With an abuse of notation we sometimes write, e.g.,
$\|u(t,x)\|_{H^m(n)}$ for the $H^m(n)$ norm of the function $x\mapsto
u(t,x)$. For the bounded domain $(0,2\pi)$ with periodic boundary
conditions we also write $\mathcal{T}_{2\pi}$; i.e.,
$\int_{\mathcal{T}_{2\pi}} u(x)\,{\rm d} x:=\int_0^{2\pi} u(x)\,{\rm d} x$.
Fourier transform is denoted by $\mathcal{F}$, e.g., if $u\in L^2(\mathbb{R})$, then
$\hat{u}(k):=\mathcal{F}(u)(k)=\frac 1{2\pi}\int {\rm e}^{-{\rm i} k x} u(x) \,{\rm d} x$.  From
$\mathcal{F}(\partial_x u)(k)={\rm i} k\hat{u}(k)$ and Parseval's identity we
have that $\mathcal{F}$ is an
isomorphism between $H^m(n)$ and $H^n(m)$; i.e., the weight in
$x$--space yields smoothness in Fourier space and vice versa. This smoothness
in $k$ is essential for the proof of the following theorem, where for
convenience we take initial conditions at $t=1$.

\begin{theorem}\label{mth}
Assume that the parameters $c_2,c_3,u_m\in \mathbb{R}$ and $\delta>0$ small
are chosen in such a way that $d_2>0$ in the expansion \eqref{lam1e},
and $\mathop{\rm Re}\lambda_n(\ell)<0$ for all $n\in\mathbb{N}$ and all
$\ell\in[-1/2,1/2)$, except for $\lambda_1(0)=0$.
Let $p\in(0,1/2)$. There exist $C_1,C_2>0$ such that the following
  holds. If $ \| v_0 \|_{H^2(2)}\leq C_1 $, then there exists a
unique global solution $v$ of \eqref{ppks} with $v|_{t=1}=v_0$, and
\begin{equation}
\sup\limits_{x\in\mathbb{R}} \left
|v(t,x)-t^{-1/2}f_z(t^{-1/2}(x-d_1t))\tilde{v}_1(0,x)\right|
\le C_2 t^{-1+p}, \quad t\in[1,\infty),
\label{mest}
\end{equation}
with $d_1=2u_m+\mathcal{O}(\delta)$ from \eqref{lam1e},
$f_z(y)=\frac{\sqrt{d_2}}{b}\frac{z\mathop{\rm erf}'(y)} {1+z\mathop{\rm erf}(y)}$
from \eqref{bfp}, and $\ln(1+z)=\frac {b}{d_2}\int_\mathbb{R} v_0(x)\,{\rm d} x$.
\end{theorem}

Thus we have the asymptotic
$(H^2(2),L^\infty)$--stability
of $v=0$; i.e., for all $\varepsilon > 0$ there
exists a $\nu{>}0$ such that
$\|v_0\|_{H^r(2)}\le \nu$ implies
$\|v (t) \|_{L^\infty}\leq \varepsilon$, for all $t{\ge}1$, and
$\|v (t) \|_{L^\infty}\to 0$ with rate $t^{-1/2}$.
The perturbations decay in an universal manner determined
by the decay of localized initial data in the Burgers equation.
Theorem \ref{mth} is in fact a corollary to the more detailed
Theorem \ref{mthb} stated in Bloch space in \S\ref{bl-sec}.
In \S\ref{s-sec} we give examples such that the
assumption $d_2>0$ holds. In particular, we shall see that
$d_2>0$ may hold for $c_2>0$; i.e., the critical ``Reynolds number''
may be larger than $0$ which is the critical Reynolds number
in the spatially homogeneous case. A similar
effect is also known in the full inclined film problem \cite{wla05}.

The remainder of this paper is organized as follows. In \S\ref{s-sec}
we briefly review the properties of stationary
solutions to \eqref{pks}, explain the set--up of Bloch waves,
and give examples for $\lambda_1(\ell)$ from \eqref{lam1e}
for some chosen
parameter values. In \S\ref{d-sec} we review the
concept of irrelevant nonlinearities and the idea of renormalization,
give a formal
derivation of the Burgers equation as the amplitude equation for
the critical mode $\tilde{v}_1(0,x)$ for \eqref{ppks} in the linearly
stable case, and introduce Bloch spaces with weights to formulate
our precise result Theorem \ref{mthb}.
In \S\ref{p-sec} we set up a renormalization
process to prove Theorem \ref{mthb}.
In Appendix \ref{a-sec} we give some remarks on the unstable case (ii).


\section{Spectral analysis}\label{s-sec}

\subsection{Expansion of the stationary solutions}
To calculate $u_s$ we expand in $\delta$. We set
$$
u_s(x)=u_m+\delta u_1(x)+\delta^2 u_2(x)+\mathcal{O}(\delta^3),
$$
where $u_j$ for $j\geq 1$ is $2\pi$--periodic and has zero mean; i.e., $u_m$ is considered as an additional parameter.
Thus we write $u_s(x)=u_s(x;c_2,c_3,\delta,u_m)$.
We obtain a hierarchy of linear inhomogeneous equations of the form
$$
\mathcal{L}_0 u_j(x)=g(x),\quad
\mathcal{L}_0 u=-\partial_x^4 u-c_3\partial_x^3 u -c_2\partial_x^2 u-2u_m\partial_x u,
$$
where $g(x)$ comes from the previous step. At $\mathcal{O}(\delta)$ we have
$g(x)=2u_m\sin( x)$, hence we use the ansatz $u_1(x)=\alpha_1\cos( x)
+\beta_1\sin( x)$ to obtain the linear system
\begin{equation}
\begin{pmatrix} \mu_1&\nu_1\\
-\nu_1&\mu_1\end{pmatrix}
\begin{pmatrix}\alpha_1\\
\beta_1\end{pmatrix}=\begin{pmatrix} 0\\2u_m\end{pmatrix}, \quad
\mu_j=-j^4+c_2 j^2,\quad
\nu_j=c_3j^3-2j u_m,
\label{ls1}
\end{equation}
with solution
$$
\begin{pmatrix} \alpha_1\\\beta_1\end{pmatrix}=\frac 1{d_1}
\begin{pmatrix} -(c_3-2u_m)2u_m\\
(-1+c_2)2u_m\end{pmatrix},
\quad d_j=(-j^4+c_2j^2)^2+(c_3j^3-2u_mj)^2.
$$
At $\mathcal{O}(\delta^2)$ we have $g(x)=2u_1'u_1-2\partial_x(\cos( x)u_1)$,
hence $u_1=\alpha_2\cos(2 x)+\beta_2\sin(2 x)$, which
again yields a $2\times 2$ linear system for $(\alpha_2,\beta_2)$, while
at $\mathcal{O}(\delta^3)$, the right hand side contains harmonics
${\rm e}^{{\rm i} j x}$
with $j=1,2,3$. Thus we need the ansatz
$$
u_3(x)=\alpha_{31}\cos( x)+\alpha_{32}\cos(2 x)+\alpha_{33}\cos(3 x)
+\beta_{31}\sin( x)+\beta_{32}\sin(2 x)+\beta_{33}\sin(3 x),
$$
and have to solve a $6\times 6$ linear system.
This can be continued to any order in $\delta$ and the resulting
systems can conveniently be solved using some symbolic
algebra package. Moreover, from
the diagonals of the linear systems we obtain
the convergence of the Fourier series for $u_s$.


The maximum amplitude of $u_s$ and the phase--shift
with the ``bottom profile'' $\cos( x)$ depend on the parameters
$c_2,c_3,\delta,u_m$ in a rather complicated way. Here, instead of giving
explicit formulas we plot some solutions $u_s$ in fig.\ref{sfig}
on page \pageref{sfig},
together with eigenvalue curves for the associated linearizations.

\subsection{Bloch wave analysis}\label{bt-sec}
To calculate the spectrum of the linearization
$\mathcal{L}$ of \eqref{pks} around $u_s$ we use the Bloch wave transform.
The basic idea is to write
\begin{equation}
\begin{aligned}
v(x)&=\int_\mathbb{R} {\rm e}^{{\rm i} kx}\hat{v}(k)\,{\rm d}k\\
&=\sum_{j\in\mathbb{Z}}\int_{-1/2+j}^{1/2+j}
{\rm e}^{{\rm i} kx}\hat{v}(k)\,{\rm d} k\\
&=\int_{-1/2}^{1/2} \sum_{j\in\mathbb{Z}}{\rm e}^{{\rm i} (\ell+j)x}
\hat{v}(\ell+j)\,{\rm d} \ell
 \\
&=\int_{-1/2}^{1/2}{\rm e}^{{\rm i} \ell x}\tilde{v}(\ell,x)\,{\rm d} \ell
=:(\mathcal{J}^{-1} \tilde{v})(x)
\end{aligned} \label{bt0}
\end{equation}
where $\tilde{v}(\ell,x)=(\mathcal{J} v)(\ell,x)
=\sum_{j\in\mathbb{Z}}{\rm e}^{{\rm i} j x}\hat{v}(\ell+j)$.
By construction we have
\begin{equation}
\tilde{v}(\ell,x) = \tilde{v}(\ell,x + 2 \pi)   \quad
\mbox{and} \quad
\tilde{v}(\ell,x) = \tilde{v}(\ell + 1,x) {\rm e}^{{\rm i}  x}.
\label{bt2}
\end{equation}
Bloch transform is an isomorphism between
$H^s(\mathbb{R},\mathbb{C})$ and
$L^2((-1/2,1/2],H^s(\mathcal{T}_{2\pi}))$  \cite{scar99}, where
$$
\| \tilde{v} \|_{ L^2((-1/2,1/2],H^s(\mathcal{T}_{2\pi}))}
=  \Big(\int_{-1/2}^{1/2}
 \|\tilde{v}(\ell,\cdot )\|_{H^s(\mathcal{T}_{2\pi})}^2 d\ell\Big)^{1/2}.
 $$
Multiplication
$u(x) v(x)$ in $x$-space corresponds in Bloch space to the ``convolution''
\begin{equation}
(\tilde{u}* \tilde{v}) (\ell, x) =
\int_{-1/2}^{1/2} \tilde{u} (\ell - m,x)
\tilde{v}(m,x) \,{\rm d} m,
\label{bconv1}
\end{equation}
where \eqref{bt2} has to be used for $|\ell-m|>1/2$. However, if
$\chi:\mathbb{R}\to\mathbb{R}$ is $2\pi$ periodic, then
$\mathcal{J}(\chi u)(\ell,x)=\chi(x)(\mathcal{J} u)(\ell,x)$.

In Bloch space the linear eigenvalue problem for $\mathcal{L}(x)$ thus becomes
\begin{equation}
\begin{aligned}
\lambda(\ell)\tilde{v}(\ell,x)
&\stackrel{!}{=}\tilde{\mathcal{L}}(\ell,x)\tilde{v}(\ell,x)
:={\rm e}^{-{\rm i} \ell x} 
\bigl[\mathcal{L}(x){\rm e}^{{\rm i}\ell x}\tilde{v}(\ell,x)\bigr] \\
&=\bigl[-(\partial_x{+}{\rm i}\ell)^4-c_3(\partial_x{+}{\rm i}\ell)^3
 -c_2(\partial_x{+}{\rm i}\ell)^2 \\
&\quad + 2\delta( \cos( x)(\partial_x{+}{\rm i} \ell)-\sin( x))\bigr]\tilde{v}
-2\bigl[u_s'(x)+u_s(x)(\partial_x{+}{\rm i}\ell)\bigr]\tilde{v}
\end{aligned}\label{blevp}
\end{equation}
over the bounded domain $\mathcal{T}_{2\pi}$. This yields
curves of eigenvalues $\lambda_n(\ell)$, with $\ell$ in $(-1/2,1/2)$
and $n\in\mathbb{N}$.
To calculate $\lambda_n(\ell)$ we let
$$
\tilde{v}(\ell,x)=\sum_{j\in\mathbb{Z}} b_j(\ell){\rm e}^{{\rm i} j x},
$$
which yields the infinite coupled system
\begin{equation}
{\rm i}\delta(j{+}\ell)(1-2a_1)b_{j-1}
+m_j(\ell)b_j+{\rm i}\delta(j{+}\ell)(1-2\overline{a}_1)b_{j+1}
 +\mathcal{O}(\delta^2)b_k
=\lambda(\ell)b_j,
\label{infevp}
\end{equation}
$ j,k\in\mathbb{Z}$, where
$m_j(\ell)=-(j+\ell)^4+c_3{\rm i}(j+\ell)^3+c_2(j+\ell)^2-2{\rm i} u_m(j+\ell)$
and $a_1=\frac 1 2 (\alpha_1-{\rm i}\beta_1)$ from $u_s=u_m
+\delta(\alpha_1\cos( x)+\beta_1\sin( x))+\mathcal{O}(\delta^2)$.
For $\delta=0$ we have eigenvalues
$m_j(\ell)$ with
$\phi_j(\ell,x)={\rm e}^{{\rm i} j x}$. In particular, $m_0(0)=0$
with $\phi_0(x)\equiv 1$. For $\delta>0$, and returning to counting
$\lambda_n(\ell)$ with $n\in\mathbb{N}$, we still always have $\lambda_1(0)=0$,
with
\begin{equation}
\tilde{v}_1(0,x)=\partial_{u_m}u_s(x;c_2,c_3,\delta,u_m)
=1+\delta[b_1(0){\rm e}^{{\rm i} x}+b_{-1}(0){\rm e}^{-{\rm i} x}]
+\mathcal{O}(\delta^2).\label{v1ti}
\end{equation}
To order $\delta^2$ the eigenvalue problem \eqref{infevp} yields
\begin{equation}
{\rm det}(A(\ell)-\lambda(\ell))=0
\label{3evp}
\end{equation}
with
\[
A=\begin{pmatrix} m_{-1}(\ell)
&{\rm i}\delta(\ell-1)(1-2\overline{a}_1)&0\\
{\rm i}\delta\ell(1-2a_1)&m_0(\ell)&{\rm i}\delta\ell(1-2\overline{a}_1)\\
0&{\rm i}\delta(1+\ell)(1-2a_1)&m_1(\ell)
\end{pmatrix}.
\]
The truncated eigenvalue problem \eqref{3evp} can again be solved
explicitly using some
algebra package, but as mentioned above,
instead of giving the explicit formulas, in fig.\,\ref{sfig}
we plot $u_s$ and $\lambda_1(\ell)$ for some parameters values.\\[0mm] 

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.37\textwidth]{fig2a} \qquad\qquad
\includegraphics[width=0.37\textwidth]{fig2b}
\end{center}
\caption{Left:
parametric dependence of stationary solutions $u_s(x;c_2,c_3,\delta,u_m)$ on
$c_2$ for $(c_3,\delta,u_m)=(0,0.5,1)$.
Top down at $x=0$: $c_2=0.5,0,-0.5$,  and $\partial_{u_m}u_s(x;0,0,0.5,1)$
(dotted curve).
Right: $\mathop{\rm Re}\lambda_1(\ell)$ as obtained from \eqref{3evp} for
$(\delta,u_m)=(0.5,1)$; left to right at $\mathop{\rm Re}\lambda=-0.00125$:
$c_2=-0.5,0,0.1,0.2$, and $\mathop{\rm Re} m_0(\ell)$ for
$(u_m,c_2,c_3)=(1,0.1,0)$ 
(dotted curve);
$u_s(\cdot,0.1,0.5,1)$ is still spectrally stable, while
clearly in the homogeneous case ($\delta=0$) we have $u_s\equiv u_m$
unstable for $c_2>0$. The imaginary part of $\lambda_1(\ell)$ only
depends very weakly on $c_2$ and $\delta$ and is given by
$\mathop{\rm Im}\lambda_1(\ell)=-2u_m\ell-{\rm i} d_3\ell^3+\mathcal{O}(\ell^5)$ with
$d_3=-c_3+\mathcal{O}(\delta)$.}
\label{sfig}
\end{figure}

\section{Nonlinear analysis in Bloch wave space}\label{d-sec}

\subsection{The idea of renormalization}
\label{rgi-sec}
To explain the idea of irrelevant nonlinearities and the renormalization
group we consider
\begin{equation}
\partial_t v=\partial_x^2 v-\partial_x(v^2)+\alpha h(v),\quad
v|_{t=1}=v_0, \quad h(v)=v^{p_1}(\partial_x v)^{p_2}, \label{b17}
\end{equation}
with $p_1{+}2p_2\geq 4$. For $\alpha{=}0$ we know
that $\|v(t,x)-t^{-1/2}f_z(x/t^{1/2})\|_{L^\infty} = \mathcal{O}(t^{-1})$
as $t\to\infty$ with $\ln(1+z)=-\int v_0 \,{\rm d} x$. To show a similar
behaviour for $\alpha\neq 0$ we may use the renormalization group.
These calculations are well documented in the literature,
but we briefly repeat them here as the template
for treating \eqref{ppks}.

For $L>0$ we define the rescaling operators
$\mathcal{R}_L$ with $\mathcal{R}_Lv(x)=v(Lx)$, and for $L>1$ chosen sufficiently large
we let
\begin{equation}
v_n(\tau,\xi):=L^n v(L^{2n}\tau,L^n\xi)=L^n\mathcal{R}_{L^n}v(L^{2n}\tau,\xi).
\label{rge1}
\end{equation}
Then $v_n$ satisfies
\begin{equation}
  \partial_\tau v_n=\partial_\xi^2v_n{+}\partial_\xi(v_n^2){+}\alpha h_n(v_n)
\quad\text{with}\quad h_n(v_n)=L^{(3-p_1-2p_2)n} v_n^{p_1}
\partial_\xi (v_n^{p_2}),
\label{fnd}
\end{equation}
and solving \eqref{b17} for $t\in[1,\infty)$ is equivalent to
iterating
\begin{equation}
  \text{solve \eqref{fnd} on } \tau\in[L^{-2},1]\text{ with initial data }
  v_n(L^{-2},\xi)=L\mathcal{R}_Lv_{n-1}(1,\xi)\in X,
\label{r0}
\end{equation}
where $X$ is a suitable Banach space. For $p_1+2p_2\geq 4$ the
term $h_n$ in \eqref{fnd}
formally goes to zero. Thus, in the limit $n\to \infty$ we recover the Burgers
equation for $v_n$, with family of exact solutions
$\{v_z(\tau,\xi)=\tau^{-1/2}f_z(\xi/\sqrt{t}): z>-1\}$. In particular,
these solutions are fixed points of the renormalization map
$v(1/L^2,\cdot)\mapsto L v(1,L\cdot)$ where $v$ solves the Burgers
equation.

It turns out that this line of fixed points is
attractive in suitable spaces $X$, for instance
$X=H^2(2)$, cf.\,the definition on p.\,\pageref{mth}.
Moreover, this also holds for the flow of the perturbed
Burgers equation. For more details concerning problems of type
\eqref{b17} we refer to \cite{bkl94,ue04nsb}.
However, two observations are most important:
(a) In \eqref{fnd} we see that derivatives in $x$,
corresponding to factors ${\rm i} k$ in Fourier space
according to $\mathcal{F}(\partial_x u)(k)={\rm i} k \hat{u}(k)$,
give additional factors $L^{-1}$ in the rescaling;
(b) The diffusive spreading in $x$ space corresponds to
concentration at $k=0$ in Fourier space according to
$\mathcal{F}(L\mathcal{R}_L u)(k)=\hat{u}(k/L)$. Therefore, only the parabolic shape
of the spectrum $\lambda(k)=-k^2$ of the operator $\partial_x^2$ locally near
$k=0$ is relevant, as well as only the local behaviour of the nonlinearity
near $k=0$. For \eqref{pks} Fourier analysis has to be replaced
by Bloch wave analysis, where similar ideas apply: a factor ${\rm i} \ell$
corresponds to a derivative in $x$, and spreading in $x$
corresponds to localization at $\ell=0$. This is made rigorous
in Lemma \ref{dxlem} below.

\subsection{Formal derivation of the Burgers equation}\label{fd-sec}

In order to (formally) derive the Burgers equation as the amplitude equation
 for the critical mode $\tilde{v}_1(0,x)$ for \eqref{ppks} in the linearly
stable case -- and in order to later justify this and rigorously
prove Theorem \ref{mth} -- we consider \eqref{ppks} in Bloch space, i.e.
\begin{equation}
\partial_t\tilde{v}(t,\ell,x)=\tilde{\mathcal{L}}(l,x)\tilde{v}(t,\ell,x)
+N(\tilde{v}(t))(\ell,x),
\label{nlb0}
\end{equation}
with
$$
N(\tilde{v}(t))(\ell,x)=\mathcal{J}(-\partial_x v^2(t))(\ell,x)
=-(\partial_x+{\rm i} l)(\tilde{v}(t)*\tilde{v}(t))(\ell,x).
$$
To motivate the next transform we recall that the curve
$\lambda_1(\ell)={-}{\rm i} d_1 \ell{-}d_2\ell^2{+}\mathcal{O}(\ell^3)$ with
critical mode $\tilde{v}_1(\ell,\cdot)$ corresponds to
 $\partial_t v=(-d_1\partial_x+d_2\partial_x^2)v$; i.e.,
the linear diffusion equation
in the comoving frame $y=x-d_1 t$.
Thus, it is  tempting to simply go into this comoving
frame in \eqref{ppks}. However, this would give a
space and time periodic operator $\mathcal{L}(y+d_1 t)$ in \eqref{ppks}
which would make
the subsequent analysis more complicated. Instead we introduce
\begin{equation}
\tilde{u}(t,\ell,x)={\rm e}^{{\rm i}\ell d_1 t} \tilde{v}(t,\ell,x)
\label{wtidef}
\end{equation}
which fulfills
\begin{equation}
\partial_t\tilde{u}(t,\ell,x)=\tilde{\mathcal{M}}(l,x)\tilde{v}(t,\ell,x)+N(\tilde{v}(t))(\ell,x),
\quad \tilde{\mathcal{M}}(l,x)=\tilde{\mathcal{L}}(l,x)+{\rm i} d_1\ell.
\label{nlb}
\end{equation}
Clearly, $\tilde{\mathcal{M}}$ has the same eigenfunctions $\tilde{v}_j$
as $\tilde{\mathcal{L}}$ with eigenvalues
$\mu_j(\ell)=\lambda_j(\ell)+{\rm i} d_1\ell$. In particular
$$
\mu_1(\ell)=-d_2\ell^2+\mathcal{O}(\ell^3).
$$
In general, \eqref{wtidef} does not correspond to a simple transform
in $x$--space. However, if $\tilde{u}$ has the special form
$\tilde{u}(t,\ell,x)=\tilde{\alpha}(t,l)g(x)$ then
$$
v(t,x)=\int_{\mathcal{T}_{2\pi}} {\rm e}^{{\rm i}\ell(x-d_1 t)}\tilde{\alpha}(t,\ell)g(x)\,{\rm d}\ell
=\alpha(t,x-d_1t)g(x),
$$
which we will exploit to prove Theorem \ref{mth}.


Next we introduce mode filters to extract the critical mode
$\tilde{v}_1(\cdot,\cdot)$ from $\tilde{u}$.
Let $\rho>0$ be sufficiently small such that
$\mu_1(\ell)$ is isolated from the rest of the spectrum
of $\tilde{\mathcal{L}}(\ell,\partial_x)$
for $|\ell|\leq \rho$, and let $\chi:\mathbb{R}\to\mathbb{R}$
 be a smooth cut-off function
with $\chi(\ell)=1$ for $|\ell|\leq \rho/2$ and $\chi(\ell)=0$
for $|\ell|>\rho$. Then define
$$
\tilde{E}_c(\ell)\tilde{v}(\ell,x)=
\chi(\ell)\langle \tilde{v}(\ell,\cdot),\tilde{u}_1(\ell,\cdot)\rangle\tilde{v}_1(\ell,x).
$$
Here $\langle v,w\rangle=\int_{\mathcal{T}_{2\pi}} v(x)\overline{w}(x)\,{\rm d} x$, and
$\tilde{u}_1$ is the critical eigenfunction of the
$L^2(\mathcal{T}_{2\pi})$-adjoint operator
\begin{equation}
\tilde{\mathcal{M}}^*(\ell,x)
=-(\partial_x{+}{\rm i}\ell)^4{+}c_3(\partial_x{+}{\rm i}\ell)^3
-c_2(\partial_x{+}{\rm i}\ell)^2
-2(\partial_x{+}{\rm i}\ell)(\delta\cos(x)\tilde{u}{-}u_s\tilde{u})
 -{\rm i} d_1\ell,
\label{ad1}
\end{equation}
normalized such that $\langle \tilde{v}_1(\ell),\tilde{u}_1(\ell)\rangle=1$.
Let $\tilde{E}_s=\mathop{\rm Id}-\tilde{E}_c$. Moreover, define auxiliary mode filters
$\tilde{E}_c^h(\ell)\tilde{u}(\ell,x)=\chi(2\ell)
\langle \tilde{u}(\ell,\cdot),\tilde{u}_1(\ell,\cdot)\rangle\tilde{v}_1(\ell,x)$
and $\tilde{E}_s^h(\ell)\tilde{u}(\ell,x)=\tilde{v}(\ell,x)-\chi(\ell/2)
\langle \tilde{u}(\ell,\cdot),\tilde{u}_1(\ell,\cdot)\rangle\tilde{v}_1(\ell,x)$.
Thus $\tilde{E}_c^h\tilde{E}_c=\tilde{E}_c$ and
$\tilde{E}_s^h\tilde{E}_s=\tilde{E}_s$, which
will be used to substitute for missing projection
properties of $\tilde{E}_c$ and $\tilde{E}_s$. Finally, define the scalar
 mode filter
$\tilde{E}^*_c$ by $\tilde{E}_c(\ell)\tilde{u}(\ell,x)
=(\tilde{E}_c^*(\ell)\tilde{u}(\ell,\cdot))\tilde{v}_1(\ell,x)$.

Thus, if $(\tilde{\alpha},\tilde{u}_s)$ satisfies
\begin{equation}
\begin{gathered}
\partial_t\tilde{\alpha}(t,\ell)=\mu_1(\ell)\tilde{\alpha}(t,\ell)
+\tilde{E}_c^*N(\tilde{v}(t))(\ell),\\
\partial_t\tilde{u}_s(t,\ell,x)=\tilde{\mathcal{M}}_s\tilde{v}(t,\ell,x)+\tilde{E}_sN(\tilde{v}(t))(\ell,x),
\end{gathered}
\label{cevo}
\end{equation}
then $\tilde{u}(t,\ell,x)=\tilde{v}_c(t,\ell,x)+\tilde{u}_s(t,\ell,x)$
satisfies equation \eqref{nlb}, where $\tilde{u}_c(t,\ell,x)=\tilde{\alpha}(t,\ell)\tilde{v}_1(\ell,x)$.
The idea of this splitting is that $\tilde{u}_s$ is linearly exponentially
damped. Thus we may expect that the dynamics of \eqref{nlb} and
hence of \eqref{ppks} are dominated by the dynamics of $\tilde{\alpha}$.
This will be made rigorous in \S\ref{p-sec}. Here
we first formally derive the Burgers equation for $\tilde{\alpha}$,
ignoring $\tilde{u}_s$.
Then the nonlinearity in \eqref{cevo} is given by (suppressing $t$ for now)
\begin{align*}
&\tilde{E}_c^*N(\tilde{u}_c)(\ell)\\
&=-\chi(\ell) \int_{\mathcal{T}_{2\pi}}(\partial_x
+{\rm i}\ell)\int_{-1/2}^{1/2}\tilde{\alpha}(\ell{-}m)
\tilde{v}_1(\ell{-}m,x)\tilde{\alpha}(m)\tilde{v}_1(m,x)
\,{\rm d} m\,\overline{\tilde{u}}_1(\ell,x)\,{\rm d} x \\
&=\int_{-1/2}^{1/2} K(\ell,\ell-m,m)\tilde{\alpha}(\ell-m)\tilde{\alpha}(m)
\,{\rm d} m
\end{align*}
with
\begin{equation}
K(\ell,\ell-m,m)=\chi(\ell)\int_{0}^{2\pi} \tilde{v}_1(\ell-m,x)\tilde{v}_1(m,x)
({\rm i}\ell\overline{\tilde{u}}_1(\ell,x)+\partial_x\overline{\tilde{u}}_1(\ell,x))\,{\rm d} x.
\label{kdef}
\end{equation}
Automatically we have $K(0,0,0)$. This is due to the
following abstract argument \cite{gs96a,gs98,ue99}.
If we consider \eqref{ppks} over $\mathcal{T}_{2\pi}$, then
there exists a one dimensional center manifold
$\mathcal{W}_c=\{v(x)=\gamma \tilde{v}_1(0,x)+h(\gamma)(x):
\gamma\in(-\gamma_0,\gamma_0)\}$, and the flow
on $\mathcal{W}_c$ is given by the reduced equation
$$
\frac{\rm d}{{\rm d}t}\gamma=P_c\bigl[\mathcal{M}(\gamma \tilde{v}_1+h(\gamma))+N(\gamma \tilde{v}_1+h(\gamma))\bigr]
=\langle -\partial_x(\tilde{v}_1^2),w_1\rangle \gamma^2
+{\rm h.o.t.},
$$
where ${\rm h.o.t.}$ denotes higher order terms and the projection $P_c$ is
$\tilde{E}_c^*(0)$.
However, $\mathcal{W}_c$ coincides with the one-dimensional family
of stationary solutions $\{u_s(\cdot,c_2,\delta,m): m\approx u_m\}$, hence
$\frac{\rm d}{{\rm d}t}\gamma{=}0$ and the projection vanishes. Alternatively,
we can inspect \eqref{ad1} to see that $\tilde{u}_1(0,x)\equiv const$,
which implies $K(0,0,0)$ as well.

We expand $K(\ell,\ell-m,m)=\partial_1K(\mathbf{0})\ell
+\partial_2 K(\mathbf{0})(\ell-m)
+\partial_3K(\mathbf{0})m+\mathcal{O}((\ell+m)^2)$.
Ignoring for now the $\mathcal{O}((\ell+m)^2)$ terms we obtain
\[
\tilde{E}_c^*N(\tilde{u}_c)(\ell)
=\chi(\ell){\rm i} b\ell(\tilde{\alpha}^{*2})(\ell)
\]
with
\begin{equation}
\begin{aligned}
b&=-{\rm i}\Big(\partial_1 K(0)+\frac 1 2\partial_2 K(0)
+\frac 1 2\partial_3 K(0)\Big) \\
&=-{\rm i}\Big(\int_{\mathcal{T}_{2\pi}}\tilde{v}_1^2(0,x)(-{\rm i}\overline{\tilde{u}}_1(0,x)+\partial_\ell\partial_x
\overline{\tilde{u}}(0,x))+(\partial_\ell\tilde{v}_1(0,x))
\tilde{v}_1(0,x)\partial_x\overline{\tilde{u}}_1(0,x)\,{\rm d} x\Big) \\
&=-1+\mathcal{O}(\delta^2)\in{\rm i}\mathbb{R},
\end{aligned}
\end{equation}
where we used the facts that $\tilde{v}_1(0,x)=1+\mathcal{O}(\delta)\in\mathbb{R}$,
that $\tilde{u}_1(0,x)=1/2\pi$ due to the normalization
$\langle \tilde{v}_1(0,\cdot),\tilde{u}_1(0,\cdot)\rangle=1$, and that
${\rm i}\partial_\ell\tilde{u}_1(0,x)\in\mathbb{R}$, see \eqref{ad1}.

The result of these calculations (ignoring $\tilde{u}_s$ and the
$\mathcal{O}((l+m)^2)$
terms in $K(\ell,\ell-m,m)$) is that $\tilde{\alpha}$ fulfills
\begin{equation}
\partial_t\tilde{\alpha}(t,\ell)=\mu_1(\ell)\tilde{\alpha}(t,\ell)+{\rm i} b\ell\tilde{\alpha}^{*2}(t,\ell),
\label{atif}
\end{equation}
Motivated by \S\ref{rgi-sec} we may for now also discard the
$\mathcal{O}(\ell^3)$ terms in $\mu_1(\ell)$
to see that $\alpha(t,x)=(\mathcal{J}^{-1}\tilde{\alpha}(t))(x)$
fulfills the Burgers equation
$$
\partial_t\alpha=d_2\partial_x^2\alpha+b\partial_x(\alpha^2).
$$
In a nutshell, this, combined with \S\ref{rgi-sec}, explains
why the ``comoving frame Burgers profile''
$t^{-1/2}f_z((x-d_1t)/\sqrt{t})\tilde{v}_1(0,x)$
gives the lowest order asymptotics for \eqref{ppks}.


\subsection{The result in Bloch wave space}\label{bl-sec}

To make the formal calculations from \S\ref{fd-sec}
rigorous and thus prove Theorem \ref{mth} we need scaled Bloch spaces
with regularity and weights. We first collect a number of definitions
and basic properties. Let $\rho(\ell)=(1+|\ell|^2)^{1/2}$.
For $L>1$ and $m,n,b\ge 0$ define
\begin{gather*}
\mathcal{B}_L(n,m,b):=\{\tilde{v}\in H^n((-L/2,L/2),H^m(\mathcal{T}_{2\pi})
 : \|\tilde{v}\|_{\mathcal{B}_L(n,m,b)}<\infty\}, \\
\|\tilde{v}\|_{\mathcal{B}_L(n,m,b)}^2=\sum_{\alpha\leq n}\sum_{\beta\leq
    m} \|(\partial_\ell^\alpha \partial_x^\beta
  \tilde{v})\rho^{b}\|^2_{L^2((-L/2,L/2),L^2(\mathcal{T}_{2\pi}))}.
\end{gather*}
Let $\mathcal{B}(n,m,b):=\mathcal{B}_1(n,m,b)$. Based on Parseval's
identity we have that
$\mathcal{J}$ is an isomorphism between $H^m(n)$ and $\mathcal{B}(n,m,b)$,
with arbitrary $b\geq 0$, see, e.g.,
\cite[Lemma 5.4]{gs98}. Indeed, for fixed $L>0$
the weight $\rho$ is irrelevant since due to the bounded wave number domain
all norms $\|\cdot\|_{\mathcal{B}_L(n,m,b_1)}$ and
$\|\cdot\|_{\mathcal{B}_L(n,m,b_2)}$
are equivalent, but the constants depend on $b_1,b_2$ and
$L$, see \eqref{re1}, which will be crucial
in our analysis. Next we define the scaling operators
$$
\mathcal{R}_{1/L}:\mathcal{B}(n,m,b)\to\mathcal{B}_L(n,m,b),\quad
\mathcal{R}_{1/L}\tilde{v}(\ell,x)=\tilde{v}(\ell/L,x).
$$
Only $\ell$ is rescaled, and $x$ is not, and as in \eqref{wtidef} this
in general does not correspond to a simple rescaling in $x$--space.
In Bloch space our main result now reads as follows.

\begin{theorem}\label{mthb}
Assume that the parameters $c_2,c_3,u_m\in\mathbb{R}$ and $\delta>0$
sufficiently small are chosen in such a way that $d_2>0$ in the
expansion \eqref{lam1e}
and that $\mathop{\rm Re}\lambda_n(\ell)<0$ for all $n\in\mathbb{N}$ and all
$\ell\in[-1/2,1/2)$, except for $\lambda_1(0)=0$.
Let $p\in(0,1/2)$.
There exist $C_1,C_2>0$ such that the following holds. If
$\|v_0\|_{H^2(2)}\leq C_1$, then
\begin{equation}
\big\|(\ell,x)\mapsto
\bigl[\tilde{v}(t,\ell/\sqrt{t},x)-{\rm e}^{{\rm i}\ell d_1 t}\tilde{f}_z(\cdot)
\tilde{v}_1(0,x)\big]\big\|_{\mathcal{B}_{\sqrt{t}}(2,2,2)}\leq C_2t^{-1/2+p},
\label{mestb}
\end{equation}
with $d_1=2u_m+\mathcal{O}(\delta)$ from \eqref{lam1e},
$\tilde{f}_z(\ell)=\mathcal{F}(f_z)(\ell)$, where
$f_z(y)=\frac{\sqrt{d_2}}{b}\frac{z\mathop{\rm erf}'(y)}
 {1+z\mathop{\rm erf}(y)}$
from \eqref{bfp} with $z=\frac b{d_2}\int_\mathbb{R} v_0(x)\,{\rm d} x$.
\end{theorem}

Before proving this theorem we translate
\eqref{mestb} back into $x$-space. In $L^\infty(\mathbb{R})$ we have
\begin{align*}
v(t,x)&=\int_{-1/2}^{1/2} \exp({\rm i} \ell x)\tilde{v}(t,\ell,x)\,{\rm d} \ell
=\int_{-1/2}^{1/2} \exp({\rm i} \ell(x-d_1 t))\tilde{u}(t,\ell,x)\,{\rm d} \ell
\\
&=t^{-1/2}\int_{-\sqrt t/2}^{\sqrt t/2} \exp({\rm i}\ell t^{-1/2} (x-d_1 t))
\tilde{u}(t,t^{-1/2}\ell,x)\,{\rm d} \ell \\
&=t^{-1/2}\int_{-\sqrt t/2}^{\sqrt t/2} \exp({\rm i}\ell t^{-1/2} (x-d_1 t))
\tilde{f}_z(\ell)\,{\rm d}\ell \tilde{v}_1(0,x)+\mathcal{O}(t^{-1+p/2}) \\
&=t^{-1/2}f_z(t^{-1/2}(x-d_1t))\tilde{v}_1(0,x)+\mathcal{O}(t^{-1+p/2}).
\end{align*}
This proves Theorem \ref{mth}.


\section{Proof of Theorem \ref{mthb}}\label{p-sec}

\subsection{The rescaled systems}\label{rg-sec}

To prove Theorem \ref{mthb} we now start with the system \eqref{cevo}.
Similar to \eqref{rge1} we introduce scaled variables
\begin{equation}
\alpha_n(\tau,\kappa)
=\mathcal{R}_{L^{-n}}\tilde{\alpha}(L^{2n}\tau,\kappa)\quad\text{and}\quad
w_n(\tau,\kappa,x)=L^{n(1-p)}\mathcal{R}_{L^{-n}}\tilde{u}_s(L^{2n}\tau,\kappa,x).
\end{equation}
Here we ``blow up'' $w_n$ since by this we can more directly control
the terms involving $w_n$ in the equation for $\alpha_n$, see Lemma \ref{nllem}
below. We obtain
\begin{equation}
\begin{gathered}
\partial_\tau\alpha_n(t,\kappa)=L^{2n}\mu_1(\kappa/L^n)\alpha_n(\tau,\kappa)
+L^{2n}N^c_n(\alpha_n,w_n), \\
\partial_\tau w_n(\tau,\kappa,x)=L^{2n}\tilde{\mathcal{M}}_n^s w_n
+L^{(3-p)n}N_n^s(\alpha_n,w_n),\\
\end{gathered}
\label{cevo3}
\end{equation}
where $\tilde{\mathcal{M}}_n^s=L^{2n}\mathcal{R}_L^{-n}
\tilde{\mathcal{M}}_s\mathcal{R}_{L^n}$ and
\begin{equation}
\begin{gathered}
N^c_n(\alpha_n,w_n)(\kappa,x)=
\mathcal{R}_{L^{-n}}\tilde{E}_c N\left((\mathcal{R}_{L^n}\alpha_n)
\tilde{v}_1(\kappa,x)+L^{-n(1-p)}\mathcal{R}_{L^n}w_n\right), \\
N^s_n(\alpha_n,w_n)(\kappa,x)=
\mathcal{R}_{L^{-n}}\tilde{E}_s N\left((\mathcal{R}_{L^n}\alpha_n)
\tilde{v}_1(\kappa,x)+L^{-n(1-p)}\mathcal{R}_{L^n}w_n\right).
\end{gathered}
\end{equation}
Similar to \eqref{r0}, we consider the following iteration:
\begin{equation}
\begin{gathered}
\text{solve \eqref{cevo3} on } \tau\in[L^{-2},1]\text{ with initial data }\\
\begin{pmatrix} \alpha_n\\w_n\end{pmatrix}
(L^{-2},\kappa,\xi)=\mathcal{R}_{1/L}\begin{pmatrix}\alpha_{n-1}\\
L^{1-p}w_{n-1}\end{pmatrix}
(1,\kappa,\xi).
\end{gathered}
\label{r1}
\end{equation}
As phase space for \eqref{cevo3} we choose $\mathcal{X}_n\times\mathcal{X}_n$ with
$\mathcal{X}_n=\mathcal{B}_{L^n}(2,2,2)$, where for
$\alpha_n$ we can identify $\mathcal{X}_n$ with the
Fourier space $H^2(2)$ since
$\alpha_n$ is independent of $x$. Moreover,
$\mathop{\rm supp}\alpha_n\subset\{|\ell|\leq L^n\rho\}$.
To treat \eqref{r1} we note a number of estimates.

\begin{lemma}\label{linlem}
For $b_2\geq b_1\geq 0$ there exists a $C>0$ such that in the
critical part we have
\begin{equation}
\|{\rm e}^{L^{2n}\mu_1(\cdot/L^n)(\tau-\tau')}\alpha_n\|_{B_{L^n}(2,2,b_1)}
\leq C(\tau-\tau')^{(b_1-b_2)/2}\|\alpha_n\|_{B_{L^n}(2,2,b_1)}.
\label{linecrit}
\end{equation}
The stable part is linearly exponentially damped; i.e., there exists a
$\gamma_0>0$ such that
\begin{equation}
\|{\rm e}^{L^{2n}\tilde{\mathcal{M}}_n^s(\tau-\tau')}w_n\|_{\mathcal{B}_{L^n}(k,b,b)}
\leq C{\rm e}^{-\gamma_0L^{2n}(\tau-\tau')}(\tau-\tau')^{-1/2}
\|w_n\|_{\mathcal{B}_{L^n}(k,b-1,b-1)}.
\label{linestab}
\end{equation}
\end{lemma}

\begin{proof} Inequality \eqref{linecrit} follows from the locally parabolic
shape of $L^{2n}\mu_1(\kappa/L^n)=-d_2\kappa^2+\mathcal{O}(\kappa/L^n)$ near
$\kappa=0$. Inequality \eqref{linestab} follows from
$\mathop{\rm Re}\sigma(\tilde{\mathcal{M}}_s){\leq}-\gamma_0$.
In fact, $\tilde{\mathcal{M}}_s$ is a 4$^{th}$ order
operator and therefore has better smoothing properties than stated
in \eqref{linestab}, but this estimate is sufficient in the following.
\end{proof}

Next we note
\begin{equation}
\|\mathcal{R}_{1/L}\tilde{v}\|_{\mathcal{B}_L(2,2,b)}\leq CL^{b+1/2}\|\tilde{v}\|_{\mathcal{B}(2,2,b)},
\label{re1}
\end{equation}
and, for $\tilde{u},\tilde{v}\in \mathcal{B}_L(n,m,0)$ with
$n,m\geq 1/2$ and $\ell\in(-L/2,L/2)$,
\begin{align*}
&\mathcal{R}_{1/L}(\mathcal{R}_L\tilde{u}*\mathcal{R}_L\tilde{v})(\ell,x)\\
&=\int_{-1/2}^{1/2} \tilde{u}(\ell-Lm,x)v(Lm,x)\,{\rm d} m
 \\
&=L^{-1}\int_{-L/2}^{L/2}\tilde{u}(\ell-m,x)\tilde{v}(m,x)\,{\rm d} m
=:L^{-1}(\tilde{u}*_L\tilde{v})(\ell,x),
\end{align*}
which will be used to express the rescaled nonlinear terms.
Henceforth we will drop the subscript $_L$ in $*_L$.
To estimate the nonlinearity $\partial_x(v^2)$ in Bloch space we need to
exploit  the derivative using the following Lemma
\cite[Lemma 14]{gs96a}.


\begin{lemma}\label{dxlem}
Let $\tilde{K}\in C^2_{b}([-1/2,1/2)^2,H^2(\mathcal{T}_{2\pi}))$ with
\[
\|\tilde{K}(\kappa{-}\ell,\ell)\|_{H^2(\mathcal{T}_{2\pi})}
\leq C(|\kappa{-}\ell|{+}|\ell|)^\gamma.
\]
 Then
$$
(\tilde{v},\tilde{u})\mapsto (\mathcal{M}_{1/L} K)(\tilde{v},
\tilde{u})(\kappa):=
\int \bigl(\mathcal{R}_{1/L}\tilde{K}(\kappa-\ell,
\ell,x)\bigr)\tilde{v}(\kappa,x)\tilde{u}(\kappa-\ell,x)\,{\rm d}\ell
$$
defines a bilinear mapping
$(\mathcal{M}_{1/L} K):\mathcal{B}_L(2,2,2)\times \mathcal{B}_L(2,2,2)
\to \mathcal{B}_L(2,2,2)$.
There exists a $C>0$ such that for all $L>1$ we have
$$
\|(\mathcal{M}_{1/L} K)(\tilde{v},\tilde{u})\|_{\mathcal{B}_L(2,2,2-\gamma)}
\leq CL^{-\min\{\gamma,1\}}\|\tilde{v}\|_{\mathcal{B}_L(2,2,2)}
\|\tilde{u}\|_{\mathcal{B}_L(2,2,2)}.
$$
\end{lemma}

\begin{lemma}\label{nllem}
For $p\in(0,1/2)$ there exists a
$C>0$ such that for all $(\alpha_n,w_n)\in X_n$ we have
$L^{2n}N_n^c(\alpha_n,w_n)=s_1+s_2+s_3+s_4$ with
$s_1(\kappa)={\rm i} b \kappa\alpha_n^{*2}(\kappa)$ and
\begin{gather}
\|s_2\|_{B_{L^n}(2,2,2p)}\leq CL^{-n(1-2p)}\|\alpha_n\|_{\mathcal{X}_n}^2,\label{s2e}\\
\|s_3\|_{B_{L^n}(2,2,1)}\leq
CL^{-n(1-p)}\|\alpha_n\|_{\mathcal{X}_n}\|w_n\|_{\mathcal{X}_n},\label{s3e}\\
\|s_4\|_{B_{L^n}(2,2,1)}\leq
CL^{-2n(1-p)}\|w_n\|_{\mathcal{X}_n}^2\label{s4e}.
\end{gather}
Moreover,
\begin{equation}
\begin{aligned}
& L^{(3-p)n}\|N_n^s(\alpha_n,w_n)\|_{\mathcal{B}_{L^n}(2,1,1)}\\
& \leq C\big(L^{(1-p)n}\|\alpha_n\|_{\mathcal{X}_n}^2
 +\|\alpha_n\|_{\mathcal{X}_n}
\|w_n\|_{\mathcal{X}_n}+L^{-(1-p)n}\|w_n\|_{\mathcal{X}_n}^2\big).
\end{aligned} \label{s5e}
\end{equation}
\end{lemma}

\begin{proof} Explicitly we have
\begin{equation}
L^{2n}N_n^c(\kappa)=L^{2n}\chi(\frac\kappa{L^{2n}})\int_{x\in\mathcal{T}_{2\pi}}
\Big(\int_{m=-1/2}^{1/2}
(\partial_x+\frac{{\rm i}\kappa}{L^n})\Pi(\kappa,m,x)\,{\rm d} m\Big)
\overline{\tilde{u}}_1(\frac{\kappa}{L^n},x)\,{\rm d} x,\label{nncex}
\end{equation}
with
\begin{align*}
\Pi(\kappa,m,x)=&\left(\alpha_n(\kappa-L^nm)
\tilde{v}_1(\frac\kappa{L^n}-m,x)+L^{-n(1-p)n}w_n(\kappa-L^n m)\right) \\
&\times\left(\alpha_n(L^n m)\tilde{v}_1(m,x)+
L^{-n(1-p)n}w_n(L^nm)\right).
\end{align*}
Thus, substituting $L^nm\to m$ in \eqref{nncex} yields
$L^{2n}N_n^c(\kappa)=s_1+s_2+s_3+s_4$ with
\begin{align*}
s_1+s_2&=L^n\int_{-L^n/2}^{L^n/2} K(\frac \kappa{L^n},\frac{\kappa-m}{L^n},
\frac m{L^n})\alpha_n(\kappa-m)\alpha_n(m) \,{\rm d} m\\
&={\rm i} b\kappa\alpha_n^{*2}+L^n\int_{-L^n/2}^{L^n/2}(\mathcal{R}_{L^{-n}}M(\kappa,m))
\alpha_n(\kappa-m)\alpha_n(m)\,{\rm d} m,
\end{align*}
where $M\in C^2$ with $M(\ell,m)\leq C((\ell+m)^\gamma)$ with
$1<\gamma\leq 2$. Now using Lemma \ref{dxlem} with $\gamma=2-2p$ we obtain
\eqref{s2e}.

Similarly,
\begin{align*}
&\|s_3(\kappa)\|_{\mathcal{B}_{L^n}(2,2,1)}\\
&=L^nL^{-n(1-p)}
\big\|\chi(\frac\kappa{L^n})\langle (\partial_x+\frac{{\rm i}\kappa}{L^n})
(w_n *_{L^n}({\rm e}^{{\rm i} d_1 L^n\cdot \tau}\alpha_n\tilde{v}(\cdot/L^n))),
\tilde{u}_1(\frac\kappa{L^n})\rangle\big\|_{\mathcal{B}_{L^n}(2,2,1)}
\end{align*}
which shows \eqref{s3e} by again using that $\partial_x\tilde{u}_1(\frac \kappa{L^n})
=\mathcal{O}(\frac\kappa{L^n})$, and \eqref{s4e} follows in the same way, as
well as the estimate \eqref{s5e} in the stable part.
\end{proof}

The terms involving $\alpha_n$ in \eqref{s5e} do not decay.
However, combining \eqref{s5e} with the exponential
decay of the stable semigroup we
still get a local existence result for \eqref{cevo3} with
bounds independent of $n$.

\begin{lemma}\label{loc-lem}
There exist $C_1,C_2>0$ and $L_0>1$ such that for $L>L_0$ the following
holds.
Let
$$
\rho_{n-1}:=\|(\alpha_{n-1},w_{n-1})(1)\|_{\mathcal{X}_{n-1}}
\leq C_1L^{-5/2}.
$$
Then there exists a local solution
$(\alpha_n,w_n)\in C([1/L^2,1],\mathcal{X}_n)$
of \eqref{cevo3}, with
\begin{equation}
\sup_{\tau\in[1/L^2,1]}\|(\alpha_n,w_n)\|_{\mathcal{X}_n}\leq C_2L^{5/2}\rho_{n-1}.
\label{loces}
\end{equation}
\end{lemma}

\begin{proof}
 The variation of constant formula for \eqref{cevo3} yields
\begin{gather}
\begin{aligned}
\alpha_n(\tau)=&{\rm e}^{(\tau-L^{-2})L^{2n}\mu_1(\kappa/L^n)}
\mathcal{R}_{1/L}\alpha_{n-1}(1) \\
&+L^{2n}\int_{1/L^2}^\tau
{\rm e}^{(\tau-s)L^{2n}\mu_1(\kappa/L^n)}N_n^c(\alpha_n(s),w_n(s))\,{\rm d} s,
\end{aligned} \label{vdk1}
\\
w_n(\tau)={\rm e}^{(\tau-L^{-2})\mathcal{M}_n^s}\mathcal{R}_{1/L}w_{n-1}(1)
+L^{(3-p)n}\int_{1/L^2}^\tau {\rm e}^{(\tau-s)\mathcal{M}_n^s}
N_n^c(\alpha_n(s),w_n(s))\,{\rm d} s.\label{vdk2}
\end{gather}
Combining \eqref{re1} with Lemmas \ref{linlem} and \ref{nllem} and applying
the contraction mapping theorem yields the result.
\end{proof}

\subsection{Splitting and iteration}\label{rg-sec2}

Due to the loss of $L^{5/2}$ in Lemma \ref{loc-lem} we need
to refine our estimate of the solutions of \eqref{cevo3}. Therefore
let
\begin{equation}
\alpha_n(\tau,\kappa)=\alpha^{(z)}_n(\tau,\kappa)+\gamma_n(\tau,\kappa)
\label{bezdef}
\end{equation}
where
$\alpha^{(z)}_n(\tau,\kappa):=\chi(\kappa/L^n)\hat{v}_z(\tau,\kappa)$,
$\hat{v}_z(\tau,\kappa)=\hat{f}_z(\tau^{1/2}\kappa)$,
with $z$ defined by $\ln(z+1)=\frac b{d_2}\alpha_n(1/L^2,\kappa)|_{\kappa=0}$.
Since $N(\cdot)(\ell,x)$
in \eqref{nlb} vanishes at $\ell=0$, so do $N_n^c$ and $N_n^s$ at
$\kappa=0$, which corresponds to the conservation of mass by the
nonlinearity $-\partial_x(v^2)$ in \eqref{pks}.
Therefore $\gamma_n(\tau,0)=0$ for all $n\in\mathbb{N}$ and all $\tau\in [1/L^2,1]$.
We obtain
\begin{equation}
\partial_\tau\gamma_n=L^{2n}\mu_1(\cdot/L^n)\gamma_n+L^{2n}(N_n^c(\alpha_n,w_n)
-N_n^c(\alpha^{(z)}_n,0))+\mathop{\rm Res}_n,
\label{dtga}
\end{equation}
where
$$
\mathop{\rm Res}_n=-\partial_\tau\alpha^{(z)}_n+L^{2n}
(\mu_1(\cdot/L^n)\alpha^{(z)}_n
+N_n^c(\alpha^{(z)}_n,0)).
$$

\begin{lemma}\label{res-lem}
Let $|z|{<}1$. There exists a $C{>}0$ such that
\[
\sup_{\tau\in[L^{-2},1]}\|\mathop{\rm Res}_n\|_{\mathcal{X}_n}{\leq}
CL^{-n}|z|.
\]
\end{lemma}

\begin{proof}
By construction, $L^{2n}\mu_1(\kappa/L^n)=-d_2\kappa^2+\mathcal{O}(\kappa^3/L^n)$
and $L^{2n}N_n^c(\alpha^{(z)}_n,0))=({\rm i} b\kappa +\mathcal{O}((\kappa/L^n)^2))
(\alpha^{(z)}_n*\alpha^{(z)}_n)$.
Combining this with $\partial_\tau \hat{v}_z=-d_2 \ell^2 \hat{v}_z+{\rm i} b\kappa
(\hat{v}_z*\hat{v}_z)$ yields
$$
\mathop{\rm Res}_n=CL^{-n}(\mathcal{O}(\kappa^3)\alpha_n^{(z)}
+\mathcal{O}(\kappa^2(\alpha^{(z)}_n*\alpha^{(z)}_n))
$$
which can be estimated in $\mathcal{X}_n=\mathcal{B}_{L^n}(2,2,2)$
by $CL^{-n}|z|$ since $\hat{v}_z$ is an
analytic and exponentially decaying function.
\end{proof}

Next write
$$
\alpha_n(1,\kappa)=\alpha^{(z)}_n(1,\kappa)+g_{n,c}(\kappa), \quad
w_n(1,\kappa,x)=g_{n,s}(\kappa,x).
$$
By construction $g_{n,c}(0)=0$, and finally we use the
contraction properties of
the linear semigroup ${\rm e}^{L^{2n}\mu_1(\cdot/L^n)(1-L^2)}\mathcal{R}_{1/L}$
when acting on functions $g(\cdot)$ with $g(0)=0$, i.e.
\begin{equation}
\|{\rm e}^{L^{2n}\mu_1(\cdot/L^n)(1-L^2)}\mathcal{R}_{1/L}g\|_{\mathcal{B}_{L^n}(2,2,2)}
\leq CL^{-1}\|g\|_{\mathcal{B}_{L^{n-1}}(2,2,2)}.\label{ccontr}
\end{equation}
Here we need the smoothness in $\ell$.
Using $g(\ell/L,x)=g(0,x)+(\ell/L)\partial_\ell g(\tilde{l},x)=
(\ell/L)\partial_\ell g(\tilde{l},x)$ we have
\[
\|g(\ell/L,\cdot)\|_{H^2(\mathcal{T}_{2\pi})}
\leq \frac \ell L \|g\|_{C^1(\mathcal{T}_{L^n},H^2(\mathcal{T}_{2\pi}))}
\leq C\frac \ell L \|g\|_{B_{L^n}(2,2,2)},
\]
cf., e.g., \cite[Lemma 28]{ue99}.
Thus, combining \eqref{linecrit}, \eqref{re1}, \eqref{s2e}--\eqref{s4e} and
\eqref{ccontr} we obtain in the critical part
\begin{equation}
\rho_{n,c}:=\|g_{n,c}\|_{\mathcal{X}_n}\leq CL^{-1}\|g_{n-1,c}\|_{\mathcal{X}_{n-1}}+
C(|z|L^{5/2}\rho_{n-1}+(L^{5/2}\rho_{n-1})^2+L^{-n}|z|),
\label{cit}
\end{equation}
while in the stable part we have,  for $L$ sufficiently large,
\begin{equation}
\begin{aligned}
\rho_{n,s}&:=\|g_{n,s}\|_{\mathcal{X}_n}\\
&\leq C{\rm e}^{-\gamma_0L^{2n}(1-L^{-2})}
\bigl[L^{5/2}\|g_{n-1,s}\|_{\mathcal{X}_{n-1}}+L^{(1-p)n}
(L^{5/2}\rho_{n-1})^2\bigr]  \\
&\leq L^{-1}\rho_{n-1}
\end{aligned} \label{sit}
\end{equation}

\begin{proof}[Proof of Theorem \ref{mthb}]
This now follows from a simple iterative argument.
Let $\rho_0\leq L^{-m_0}=:\varepsilon$, hence also
$|z|\leq CL^{-m_0}$, and
let $L\geq L_0$ with $L_0$ sufficiently large such that
$CL^{-1}\leq L^{-(1-p)}$. Then \eqref{cit} implies
$\rho_{n,c}\leq L^{-(m_{n}-np)}+L^{-n(1-p)}|z|$
with
$$
m_{n}=\min\{m_{n-1}+1,m_0+m_{n-1}-5/2,2m_{n-1}-5\},
$$
while \eqref{sit} yields $\rho_{n,s}\leq L^{-n(1-p)}$.
Letting, e.g., $m_0=6$ yields $m_1=7$, $m_2=8,\ldots$, hence
$\rho_{n,c}\leq L^{-n(1-p)}(1+|z|)$ and $\rho_n\leq C|z|+L^{-m_n}$.
For $\tilde{v}_n(\kappa,x):=\tilde{v}(L^{2n},\kappa/L^n,x)$ this yields
\begin{align*}
\|v_n-\alpha^{(z)}_n\mathcal{R}_{1/L^n}\tilde{v}_1\|_{\mathcal{X}_n}
&=\left\|v_n(1)-\chi(\frac\cdot{L^n})\hat{f}_z(\cdot)\tilde{v}_1(\frac\cdot{L^n},\cdot)
\right\|_{\mathcal{X}_n} \\
&\leq \|g_{n,c}+L^{-n(1-p)}g_{n,s}\|_{\mathcal{X}_n}\leq 2L^{-n(1-p)}.
\end{align*}
This is \eqref{mestb} for $t=L^{2n}$, and the local existence Lemma
\ref{loc-lem} yields the result for all $t\in[L^{2n},L^{2(n+1)}]$.
\end{proof}

\section{Remarks on the unstable case}\label{a-sec}

In the unstable case $d_2<0$ in \eqref{lam1e}
we expand $\lambda_1(\ell)$ further to obtain
\begin{equation}
\lambda_1(\ell)=-{\rm i} d_1\ell-d_2\ell^2-{\rm i} d_3\ell^3-d_4\ell^4
+\mathcal{O}(\ell^5)\label{lam1e2}
\end{equation}
with $d_3=-c_3+\mathcal{O}(\delta)\in\mathbb{R}$ and $d_4=-1+\mathcal{O}(\delta)<0$. Then, with the
same ansatz as in \S\ref{fd-sec}; i.e.,
\begin{equation}
\tilde{v}(t,\ell,x)={\rm e}^{-{\rm i}\ell d_1 t}\tilde{\alpha}(t,\ell)\tilde{v}_1(\ell,x),
\label{wti2}
\end{equation}
we may formally derive the  constant coefficient Kuramoto-Sivashinsky
equation
\begin{equation}
\partial_t \alpha=(d_2\partial_x^2+d_3\partial_x^3-d_4\partial_x^4)
\alpha-b\partial_x(\alpha^2)\label{ccks}
\end{equation}
as the amplitude equation for the critical mode $\tilde{v}_1(\ell,x)$, which
gives
\begin{equation}
\partial_t \alpha=(-d_1\partial_x+d_2\partial_x^2+d_3\partial_x^3-d_4
\partial_x^4)\alpha-b\partial_x(\alpha^2)
\label{ccksl}
\end{equation}
as the amplitude equation in the laboratory frame.
Equation \eqref{ccks} (or \eqref{ccksl})
is only slightly simpler than \eqref{pks}, but,
importantly, \eqref{ccks} is a well known (at least with $d_3=0$),
much studied, generic
amplitude equation for long wave instabilities; see, e.g.,
\cite{mori-kura98,cd02} and the references therein,
 and  \cite{GO05} for recent progress.

However, it is not clear a priori
if \eqref{ccks} is a useful approximation in our problem,
in contrast to the stable case, where the Burgers equation \eqref{b1}
is used to ``guess'' the lowest order asymptotics of small
localized solutions of \eqref{pks} which is then proved rigorously a
posteriori.
In the unstable case no such behaviour can be expected: solutions
of \eqref{ccks} are $\mathcal{O}(1)$ in general
and do not decay but show complicated dynamical behaviour,
see the references above, and \cite{psu03b}.

Thus, first of all, already for the formal
derivation of \eqref{ccks} an amplitude parameter $\varepsilon$ should
be introduced. This can be done by defining
$$
\varepsilon:=\mathop{\rm Re}\lambda_1(\ell_c)\quad\text{with}\quad
\partial_\ell\mathop{\rm Re}\lambda_1|_{\ell=\ell_c}=0;
$$
i.e., $\varepsilon$ is defined as the maximum growth rate in \eqref{lam1e}.
Then the so called justification
of \eqref{ccks} as the amplitude equation for \eqref{pks} should be studied,
namely:  over what time--scales (relative to $\varepsilon$)
and in what spaces do solutions of \eqref{ccks}
via \eqref{wti2} approximate solutions of \eqref{pks}?
Here we refrain from this analysis, which would first require a
number of assumptions on the coefficients $c_2,c_3,\delta$ in \eqref{pks};
we refer to \cite{ue03ibl} and the references
therein for related work in this direction.

Instead, here we report some numerical simulations concerning the
approximation of \eqref{pks} by \eqref{wti2} and \eqref{ccksl}.
The full line in Fig.\ref{f1}b) shows
a numerical solution to \eqref{pks} with
\begin{equation}
(\delta,c_2,c_3)=(0.1,-0.2,-1)\quad\text{and mass}\quad u_m=1
\label{npar}
\end{equation}
domain and initial condition $u_m$ as noted. For the perturbation
analysis with parameters \eqref{npar}
we numerically find that a good 4$^{th}$ order approximation for
$\lambda_1$ is $\lambda_1(\ell)=-2{\rm i} \ell+0.186\ell^2-{\rm i}\ell^3-\ell^4$.
Next we approximate $b=-1+\mathcal{O}(\delta^2)$ by $-1$ and thus consider
\eqref{ccks} with
\begin{equation}
(d_1,d_2,d_3,d_4,b)=(2,-0.186,-1,-1,-1).
\label{npar2}
\end{equation}

The KS-equation \eqref{ccks} has
boost (or Galilean) invariance: if $\alpha(t,x)$ solves
\eqref{ccks} then $\beta(t,x)=\alpha(t,x)+c$ solves
$\partial_t \beta=(bc\partial_x+d_2\partial_x^2+d_3\partial_x^3-d_4\partial_x^4)\beta
-b\partial_x(\beta^2)$.
Therefore the amplitude $\alpha$ for the approximation
$u_\alpha$ can be calculated in two different ways:
First we may set $\alpha(0,x)=u(0,x)$ and integrate \eqref{ccks}.
This gives the dotted line in Fig.\,\ref{f1}b),
while Fig.\ref{nf1}a) compares the solutions $u(100,x)$ and
$\alpha(100,x)$ thus obtained.

\begin{figure}[ht]
\begin{center}
\hspace{8mm}(a)\hspace{58mm} (b) \\
\includegraphics[width=0.48\textwidth]{fig3a} \quad
\includegraphics[width=0.48\textwidth]{fig3b}
\end{center}
\caption{(a) the numerical solutions of \eqref{pks} (full line) and
\eqref{ccks} (dotted line) from Fig.\ref{f1}b) at $t=100$.
(b) the solution of
\eqref{pks} (a), of \eqref{ccksl} with $\alpha(0,x)=u(0,x)-1$ (b),
and of \eqref{ccksl} with $d_2=-0.2$ (c), all at $t=500$.}
\label{nf1}
\end{figure}

Equivalently, but more in the spirit of amplitude equations,
we may set $\alpha(0,x)=u(0,x){-}1$ such that $\alpha$ represents the
amplitude of the perturbation, and integrate \eqref{ccksl}.
The result at $t{=}500$ is shown in Fig.~\ref{nf1}b)
(curve b), together with $u|_{t=500}$ (curve a) and
finally compared with the solution of \eqref{ccks} with
$d_2=-c_2=-0.2$ (curve c).
This last solution corresponds to simply setting $\delta=0$ in \eqref{pks}.
Clearly, $d_2=-0.186$ gives a much
better approximation. The reason is that $d_2{=}-0.2$ gives a stronger
instability than the ``effective instability'' with $d_2{=}-0.186$;
therefore the humps in the amplitude curve c are larger
and hence travel
faster than in curve b, which over large times
in particular leads to the incorrect shift in curve c.
Similar results were obtained in all our simulations which
covered a variety of parameter-regimes and initial conditions,
in particular also for \mbox{larger $\delta$.}

In summary, we see that the formal derivation of \eqref{ccks}
gives a useful amplitude
equation, in contrast to just setting $\delta=0$ in \eqref{pks}
which can be seen as a (very) naive averaging.
Whether \eqref{ccks} allows to show interesting
rigorous results for \eqref{pks} in the unstable case remains to be seen.


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\end{document}