text:
\begin{document}

\title[Peaked periodic waves in the Camassa--Holm equation]{\bf Growth of perturbations
to the peaked periodic waves \\ in the Camassa--Holm equation}

\author{Aigerim Madiyeva}
\address[A. Madiyeva]{Department of Mathematics and Statistics, McMaster University,
Hamilton, Ontario, Canada, L8S 4K1}
\email{madiyeva@mcmaster.ca}

\author{Dmitry E. Pelinovsky}
\address[D. Pelinovsky]{Department of Mathematics and Statistics, McMaster University,
Hamilton, Ontario, Canada, L8S 4K1}

\keywords{Peaked periodic waves, Camassa--Holm equation, characteristics, stability, instability}

\begin{abstract}
Peaked periodic waves in the Camassa--Holm equation are revisited. Linearized evolution
equations are derived for perturbations to the peaked periodic waves
and linearized instability
is proven both in $H^1$ and $W^{1,\infty}$ norms. Dynamics of perturbations in $H^1$
is related to the existence of two conserved quantities and is bounded in the full nonlinear
system due to these conserved quantities. On the other hand,
perturbations
to the peaked periodic wave grow in $W^{1,\infty}$ norm
and may blow up in a finite time in the nonlinear evolution of the Camassa--Holm equation.
\end{abstract}

\date{\today}
\maketitle

\section{Introduction}

We address the Camassa-Holm (CH) equation given by
\begin{equation}
\label{CH}
u_t - u_{txx} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx},
\end{equation}
in the setting of $\mathbb{T} := [-\pi,\pi]$ subject to the periodic boundary conditions at $\pm \pi$. Following the original derivation in \cite{CH,CHH},
the Camassa--Holm equation arises in hydrodynamical applications as a model for propagation of unidirectional shallow water waves \cite{CL,Johnson}. A generalized version of this equation also models propagation of nonlinear waves inside a cylindrical hyper-elastic rod with a small diameter \cite{Dai}. The CH equation can be interpreted geometrically in terms of geodesic flows on the diffeomorphism group \cite{Kouranbaeva,Mis1}. Generalizations of the CH equation with multi-peakons were constructed in \cite{Anco,Anco2}.

Let $\varphi \in H^1_{\rm per}(\mathbb{T})$ be the Green function satisfying
\begin{equation}
(1 - \partial_x^2) \varphi = 2 \delta_0, \quad x \in \mathbb{T},
\label{Green}
\end{equation}
with $\delta_0$ being Dirac delta distribution centered at $x = 0$.
The CH equation (\ref{CH}) can be rewritten in the convolution form
\begin{equation}
\label{CHconv}
u_t + u u_x + \frac{1}{2} \varphi' \ast \left( u^2 + \frac{1}{2} u_x^2 \right) = 0,
\end{equation}
where $(f \ast g)(x) := \int_{\mathbb{T}} f(x-y) g(y) dy$ denotes the convolution operator
and $\varphi'$ denotes piecewise continuous derivative of $\varphi$ in $x$.
The Green function $\varphi$ can be expressed explicitly in the form
\begin{equation}
\label{Green-explicit}
\varphi(x) = \frac{\cosh(\pi - |x|)}{\sinh(\pi)}, \quad x \in \mathbb{T},
\end{equation}
which shows that $\varphi \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$
is a piecewise $C^1$ function with the maximum at
\begin{equation}
\label{maximum}
M := \varphi(0) = \coth(\pi)
\end{equation}
and the minima at
\begin{equation}
\label{minimum}
m := \varphi(\pm \pi) = {\rm csch}(\pi).
\end{equation}
The central peak of $\varphi$
is located at $x = 0$ with $\varphi'(0^{\pm}) = \mp 1$, from which $\varphi$ is
monotonically decreasing towards the turning points at $x = \pm \pi$ where $\varphi'(\pm \pi) = 0$. The graph of $\varphi$ versus $x$ is shown on Figure \ref{peaked-wave}.

\begin{figure}[htb!]
	\includegraphics[width=0.6\textwidth]{peakon}
	\caption{The graph of $\varphi$ on $[-2\pi,2\pi]$ with $M$ and $m$ given by (\ref{maximum}) and (\ref{minimum}).} \label{peaked-wave}
\end{figure}

The Green function $\varphi$ determines also the travelling periodic wave solution $u(t,x) = \varphi(x-ct)$
to the CH equation (\ref{CHconv}) rewritten in the weak form, where $c$ is the wave speed. The wave speed is uniquely determined by $c = M$ which gives the unique solution
\begin{equation}
\label{unique-wave}
u(t,x) = \varphi(x-Mt).
\end{equation}
By using the elementary scaling transformation, this unique solution generates a family of travelling periodic waves of the form:
\begin{equation}
\label{peaked-traveling-wave}
u(t,x) = \gamma \varphi(x-\gamma M t), \quad \gamma \in \mathbb{R}.
\end{equation}
Since the travelling periodic waves (\ref{peaked-traveling-wave}) have a peaked profile $\varphi$,
we call them {\em the peaked periodic waves}. Stability of such peaked periodic waves in $H^1_{\rm per}(\mathbb{T})$ was studied by J. Lenells in \cite{Len1,Len2}
after similar results on stability of peaked solitary waves in $H^1(\mathbb{R})$ in \cite{CM,CS}.

The initial-value problem for the periodic CH equation (\ref{CHconv}) is locally well-posed in the space $H^3_{\rm per}(\mathbb{T})$ \cite{CE,CE2000},
$H^s_{\rm per}(\mathbb{T})$ with $s > \frac{3}{2}$ \cite{D,HM2002}, $C^1_{\rm per}(\mathbb{T})$ \cite{Mis2}, and $H^1_{\rm per}(\mathbb{T}) \cap {\rm Lip}(\mathbb{T})$ \cite{DKT}, where ${\rm Lip}(\mathbb{T})$ stands
for Lipshitz continuous functions. Local well-posedness includes the existence, uniqueness, and continuous dependence of solutions from the initial data.
Peaked periodic waves and their perturbations can be considered in the space of lower regularity $H^1_{\rm per}(\mathbb{T}) \cap {\rm Lip}(\mathbb{T})$.

{\em Cusped periodic waves} (given by bounded functions with unbounded derivatives on each side of the peaks) exist in $H^1_{\rm per}(\mathbb{T})$. These cusped waves were used in \cite{B,Him} to show that local solutions to the initial-value problem for the periodic CH equation (\ref{CHconv}) are not uniformly continuous
with respect to the initial data. The same norm inflation occurs
in $H^s_{\rm per}(\mathbb{T})$ for every $s \leq \frac{3}{2}$ \cite{Molinet}.
The initial-value problem for cusped periodic waves and their perturbations in $H^1_{\rm per}(\mathbb{T})$  is ill-posed due to the lack of continuous dependence on the initial data.

Two energy quantities are well-defined for solutions to the CH equation (\ref{CHconv}) in $H^1_{\rm per}(\mathbb{T})$:
\begin{equation}
\label{conserved-quantities}
E(u) = \int_{\mathbb{T}} (u^2 + u_x^2) dx, \quad F(u) = \int_{\mathbb{T}} u (u^2 + u_x^2) dx.
\end{equation}
These quantities are constant continuously in time before the first time instance, for which the $W^{1,\infty}$-norm of the solution blows up.

By considering perturbations to the peaked periodic wave in $H^1_{\rm per}(\mathbb{T})$ and
using conservation of the mean value as well as conservation of the two energy quantities (\ref{conserved-quantities}),
it was proven in \cite{Len1,Len2} that the peaked periodic wave $u(t,x) = \varphi(x-ct)$ is orbitally stable
in the following sense.

\begin{theorem}\cite{Len1,Len2}
\label{theorem-CS}
For every $\varepsilon > 0$, there is a $\nu > 0$ such that if $u \in C([0,T),H^1_{\rm per}(\mathbb{T}))$
is a solution to the CH equation (\ref{CH}) with the initial data $u_0$ satisfying
\begin{equation}
\label{H1-initial}
\| u_0 - \varphi \|_{H^1(\mathbb{T})} < \nu,
\end{equation}
then
\begin{equation}
\label{H1-final}
\| u(t,\cdot) - \varphi(\cdot - \xi(t)) \|_{H^1(\mathbb{T})} < \varepsilon, \quad t \in [0,T),
\end{equation}
where $\xi(t) \in \mathbb{T}$ is a point where the function $u(t,\cdot)$ attains its maximum on $\mathbb{T}$
and $T > 0$ is either finite or infinite.
\end{theorem}

The main purpose of this work is to clarify the meaning of the stability result of Theorem \ref{theorem-CS}. We will
prove that the peaked periodic waves are strongly unstable with respect to perturbations in $H^1_{\rm per}(\mathbb{T}) \cap  W^{1,\infty}(\mathbb{T})$.
In this regard, Theorem \ref{theorem-CS} just ensures that the $H^1$-norm of the solution does not
drift far from the $H^1$-norm of a translated peaked periodic wave,
when the $W^{1,\infty}$ norm grows and even blows up in a finite time.

The following theorem represents the main result of this work.

\begin{theorem}
\label{theorem-nonlinear}
For every $\delta > 0$, there exist $t_0 > 0$ and $u_0 \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$
satisfying
\begin{equation}
\label{initial-bound-theorem}
\| u_0 - \varphi \|_{H^1(\mathbb{T})} + \| u_0' - \varphi' \|_{L^{\infty}(\mathbb{T})} < \delta,
\end{equation}
such that the local solution $u \in C([0,T),H^1_{\rm per}(\mathbb{T})\cap W^{1,\infty}(\mathbb{T}))$
to the CH equation (\ref{CHconv}) with the initial data $u_0$ and $T > t_0$ satisfies
\begin{equation}
\label{final-bound-theorem}
\| u_x(t_0,\cdot) - \varphi'(\cdot - \xi(t_0)) \|_{L^{\infty}(\mathbb{T})} \geq 1,
\end{equation}
where $\xi(t) \in \mathbb{T}$ is a point of peak of the function $u(t,\cdot)$ on $\mathbb{T}$.
Moreover, there exist $u_0$ such that the maximal existence time $T$ is finite.
\end{theorem}

The proof of Theorem \ref{theorem-nonlinear} is based on several recent developments.
A similar theorem for peaked solitary waves of the CH equation on an infinite line
was proven in \cite{NP}, to which the present work has many common points.
Analogous study was performed for peaked solitary waves in a different model (the Novikov equation)
in \cite{CP}, where the local well-posedness result of the initial-value problem
in $H^1(\mathbb{R}) \cap W^{1,\infty}(\mathbb{R})$ was not previously available.

In the context of the peaked periodic waves, linear instability of peaked periodic waves was obtained for
a different model (the reduced Ostrovsky equation) in \cite{GP2}.
More recently, spectral instability for perturbations in $L^2(\mathbb{T})$
was obtained in \cite{GP3} for the reduced Ostrovsky equation with either quadratic or cubic nonlinearities.
Neither local well-posedness nor the nonlinear instability was considered in the
framework of the reduced Ostrovsky equation in \cite{GP2,GP3}.

Let us explain the organization of the paper.

In Section 2, we define weak solutions to the CH equation (\ref{CHconv})
in $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$.
For solutions with a single peak at $\xi(t)$ on $\mathbb{T}$, we prove that the
single peak propagates with the local characteristic speed so that
\begin{equation}
\label{characteristic-speed}
\frac{d \xi}{dt} = u(t,\xi(t)).
\end{equation}
This allows us to define the precise form of the evolution equations for perturbations to the peaked periodic wave (see Appendix A). It follows from (\ref{characteristic-speed}) that the speed $c$ of the travelling peaked periodic wave $u(t,x) = \varphi(x-ct)$
is uniquely determined as $c = \varphi(0) = M$. The latter fact was missed in
the previous works \cite{Len1,Len2}, where $c = 1$ was suggested.

In Section 3, we study the linearized evolution equations for perturbations to the peaked periodic wave.
Similarly to \cite{NP} and \cite{CP}, we are able to simplify the linearized evolution equation
in a compact form, which requires no convolution integrals and which can be solved explicitly
by using the method of characteristics (see Appendix B). Linear instability of the peaked periodic wave
is proven both in $H^1$ and $W^{1,\infty}$. Compared to the previous work \cite{NP}, we add a new result, where we show
that the linearized instability in $H^1$ is related to the conservation of the two energy quantities
(\ref{conserved-quantities}).

In Section 4, we consider nonlinear dynamics of peaked perturbations to the peaked periodic wave.
Although the $H^1$ norm of the perturbation does not grow in the nonlinear evolution
due to to the same two conserved quantities (\ref{conserved-quantities}) as is shown in \cite{Len1,Len2},
we prove that the $W^{1,\infty}$ norm of the perturbation can grow. Moreover, we prove that the
$W^{1,\infty}$ norm of the perturbation can blow up in a finite time as follows:
\begin{equation}
\label{blow-up-criterion}
u_x(t,x) \to -\infty \quad \mbox{\rm at some} \;\; x \in \mathbb{T} \quad \mbox{\rm as} \;\; t \to T^-.
\end{equation}
The wave breakdown criterion (\ref{blow-up-criterion}) is natural for the inviscid Burgers equation
\begin{equation}
\label{Burgers}
u_t + u u_x = 0,
\end{equation}
which contributes to the local part of the CH equation (\ref{CHconv}).
The precise blow-up rate was derived for the strong solutions in $H^3_{\rm per}(\mathbb{T})$ by using the method of characteristics \cite{CE2000}.
Compared to
the previous work \cite{NP}, we do not take for granted the existence of local solutions in
$H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$
and prove local well-posedness for piecewise $C^1$ perturbations with a single peak on $\mathbb{T}$ by using the method of characteristics.
We also show that
the wave breakdown (\ref{blow-up-criterion}) does occur as a result of the nonlinear instability
of Theorem \ref{theorem-nonlinear}.

Similar to the case of peaked solitary waves in the Camassa-Holm equation \cite{NP}
and contrary to the case of peaked solitary waves in the Novikov equation \cite{CP},
we confirm that the passage from the linear to the nonlinear theory
is false in $H^1$ as was expected in \cite{CM}.
Although the linearized instability in $H^1$ is replaced by
the nonlinear stability result of Theorem \ref{theorem-CS} in $H^1$.
the linearized instability in $W^{1,\infty}$ persists
as the nonlinear instability result of Theorem \ref{theorem-nonlinear}.
This nonlinear instability arises due to the negative slopes at the right
side of the peak, which grow unboundedly from below.

Finally, we mention two open problems for the CH equation (\ref{CH}).

Peaked periodic waves (\ref{peaked-traveling-wave}) appear to be at the border between two families of smooth and cusped periodic waves (see \cite{Len3} and Appendix C). Existence of smooth periodic waves was studied in \cite{GV}, whereas cusped periodic waves were analyzed in \cite{B,Him}. It is naturally to expect that the smooth periodic waves are orbitally stable in $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ similarly to the case of the reduced
Ostrovsky equation considered in \cite{GP1}. On the other hand, it is difficult to consider stability of cusped periodic waves due to the lack of continuity with respect to initial data and the norm inflation in $H^1_{\rm per}(\mathbb{T})$.

Another interesting problem is to study how the nonlinear instability
of peaked waves complicates the interaction dynamics of multi-peaked solutions.
Asymptotic stability of multi-peaked solitary waves is proven in \cite{Molinet1} (without anti-peakons)
and in \cite{Molinet2} (in the presence of anti-peakons). Collisions of peakons and anti-peakons
lead to jumps of the energy quantities (\ref{conserved-quantities}) \cite{BC1}, see also
relevant results in \cite{Kostenko,HGH,H1,Li,Lund}. Perturbations to multi-peakons are expected to grow in the $W^{1,\infty}$ norm similarly to perturbations to single peakons considered in \cite{NP} and in this work.

\section{Peaked periodic waves as weak solutions}

Let us rewrite the initial-value problem for the CH equation (\ref{CHconv}) in the form:
\begin{equation}
\label{CHevol}
\left\{
\begin{array}{l}
u_t + u u_x + Q[u] = 0, \quad t > 0,\\
u |_{t = 0} = u_0,
\end{array} \right.
\end{equation}
where
\begin{equation}
\label{Q-def}
Q[u](x) := \frac{1}{2} \int_{\mathbb{T}} \varphi'(x-y) q[u](y) dy, \quad
q[u] := u^2 + \frac{1}{2} u_x^2, \quad x \in \mathbb{T}.
\end{equation}
and the time dependence of $Q[u]$ and $q[u]$ is dropped for convenience. The following lemma describes properties of $Q[u]$
depending on the class of functions for $u$.

\begin{lemma}
\label{lem-nonlinear-1}
If $u \in H^1_{\rm per}(\mathbb{T})$, then $Q[u] \in C^0_{\rm per}(\mathbb{T})$.
If in addition, $u \in W^{1,\infty}(\mathbb{T})$, then $Q[u]$ is Lipschitz on $\mathbb{T}$.
\end{lemma}

\begin{proof}
The integration in (\ref{Q-def}) can be split as a sum of two terms:
\begin{equation*}
Q[u](x) = \frac{m}{2} \left[ \int_{-\pi}^{x} \sinh(x-y-\pi) q[u](y) dy - \int_x^{\pi} \sinh(y-x-\pi) q[u](y) dy \right],
\end{equation*}
where $m$ is given by (\ref{minimum}).
Since $q[u]$ is absolutely integrable if $u \in H^1_{\rm per}(\mathbb{T})$, each integral is continuous on $\mathbb{T}$.
If $u \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$, then $q[u]$ is also bounded, so that
each integral is Lipschitz on $\mathbb{T}$.
\end{proof}

We say that $u \in C([0,T),H^1_{\rm per}(\mathbb{T})\cap W^{1,\infty}(\mathbb{T}))$
is a weak solution to the initial-value problem (\ref{CHevol}) for some maximal existence time $T > 0$ if
\begin{equation}
\label{CHweak}
\int_{0}^{T} \int_{\mathbb{T}} \left( u \psi_t + \frac{1}{2} u^2 \psi_x - Q[u] \psi \right) dx dt
+ \int_{\mathbb{T}} u_0(x) \psi(0,x) dx = 0
\end{equation}
is satisfied for every test function $\psi \in C^1([0,T] \times \mathbb{T})$ such that $\psi(T,\cdot) = 0$.

We consider the class of peaked periodic wave solutions with a single peak on $\mathbb{T}$
placed at the point $x = \xi(t)$ for every $t \in [0,T)$. Hence
we introduce the following notation:
\begin{equation}
\label{C-1-0}
C^1_{\xi} := \{ u \in H^1_{\rm per}(\mathbb{T})\cap W^{1,\infty}(\mathbb{T}) : \quad u_x \in C(\mathbb{T}\backslash \{\xi\}) \}.
\end{equation}
The following lemma shows that the single peak moves with its local characteristic speed
as in (\ref{characteristic-speed}).

\begin{lemma}
\label{lem-nonlinear-2}
Assume that $u \in C([0,T),H^1_{\rm per}(\mathbb{T})\cap W^{1,\infty}(\mathbb{T}))$ is a weak solution
to the CH equation in the form (\ref{CHweak}) and there exists $\xi(t) \in \mathbb{T}$ for $t \in [0,T)$
such that $u(t,\cdot) \in C^1_{\xi(t)}$ for $t \in [0,T)$. Then, $\xi \in C^1(0,T)$ satisfies
\begin{equation}
\label{peak}
\frac{d \xi}{dt} = u(t,\xi(t)), \quad t \in (0,T).
\end{equation}
\end{lemma}

\begin{proof}
Integrating (\ref{CHweak}) by parts for $x < \xi(t)$ and $x > \xi(t)$ on $\mathbb{T}$
and using the fact that $u(t,\cdot) \in C^0_{\rm per}(\mathbb{T})$ and $u(t,\cdot) \in C^1_{\xi(t)}$ for $t \in [0,T)$, we obtain
the following equations piecewise outside the peak's location:
\begin{equation}
\label{jump-1}
u_t(t,x) + u(t,x) u_x(t,x) + Q[u](t,x) = 0, \quad \pm \left[ x-\xi(t) \right] > 0,  \quad t \in (0,T).
\end{equation}
By Lemma \ref{lem-nonlinear-1}, $Q[u]$ is a continuous function of $x$ on $\mathbb{T}$ for $t \in [0,T)$,
hence it follows from (\ref{jump-1}) that
\begin{equation}
\label{jump-2}
[u_t]^+_- + u(t,\xi(t)) [u_x]^+_-  = 0, \quad t \in (0,T),
\end{equation}
where
$$
[v]^+_- := \lim_{x \to \xi(t)^+} v(t,x) - \lim_{x \to \xi(t)^-} v(t,x)
$$
is the jump of $v$ across the peak location at $x = \xi(t)$. On the other hand,
since $u(t,\cdot) \in C^1_{\xi(t)}$ for $t \in [0,T)$, we differentiate $u(t,\xi(t))$ continuously on both sides
from $x = \xi(t)$ and define
\begin{equation}
\label{jump-3}
\dot{u}^{\pm}(t) := \lim_{x \to \xi(t)^{\pm}} \left[ u_t + \frac{d \xi}{dt} u_x \right], \quad t \in (0,T).
\end{equation}
Since $u(t,\xi(t))$ is continuous for $t \in [0,T)$, we have $\dot{u}^+(t) = \dot{u}^-(t)$ almost everywhere for $t \in (0,T)$.
Therefore, it follows from (\ref{jump-3}) that
\begin{equation}
\label{jump-4}
[u_t]^+_- + \frac{d \xi}{dt} [u_x]^+_- = 0, \quad {\rm a.e.} \; t \in (0,T).
\end{equation}
Since $[u_x]^+_- \neq 0$ if $u \notin C^1_{\rm per}(\mathbb{T})$, then it follows from
(\ref{jump-2}) and (\ref{jump-4}) that $\xi(t)$ satisfies (\ref{peak}) almost everywhere for $t \in (0,T)$.
Since $u \in C([0,T) \times \mathbb{T})$ due to Sobolev embedding of $H^1_{\rm per}(\mathbb{T})$
into $C^0_{\rm per}(\mathbb{T})$, then $u(t,\xi(t)) \in C^0(0,T)$, so that equation (\ref{peak}) is satisfied
everywhere for $t \in (0,T)$ and $\xi \in C^1(0,T)$.
\end{proof}

As a corollary, it follows from Lemma \ref{lem-nonlinear-2} that if $u(t,x) = \varphi(x-ct)$ is the travelling peaked periodic wave,
then $c = \varphi(0) = M$ is uniquely defined with $M$ given by (\ref{maximum})
(compared to the incorrect value $c = 1$ used in \cite{Len1,Len2}).
The following lemma proves that $c = M$ by explicit computation.

\begin{lemma}
\label{lem-traveling}
The traveling peaked periodic wave $u(t,x) = \varphi(x-ct)$ satisfies the stationary equation
\begin{equation}
\label{statCH}
- c \varphi + \frac{1}{2} \varphi^2 + \frac{3}{4} \varphi \ast \varphi^2 = d,
\quad x \in \mathbb{T},
\end{equation}
with $c = M$ and $d = m^2$, where the nonlocal equation is piecewise $C^1$ on both sides from the peak at $x = 0$.
\end{lemma}

\begin{proof}
We shall use the relation $(\varphi')^2 = \varphi^2 - m^2$ for $x \in \mathbb{T} \backslash \{0\}$,
which follows from (\ref{Green-explicit}).
Substituting $u(t,x) = \varphi(x-ct)$ into (\ref{CHconv}) and integrating in $x$ yields
the nonlocal equation (\ref{statCH}) where $d$ is an integration constant. In order to verify the validity
of this equation and the explicit values of $c$ and $d$, we shall consider $x \in (0,2\pi)$, for which
one can use the expression $\varphi(x) = m \cosh(\pi - x)$ without the modulus sign. By using $\varphi \ast 1 = 2$
and continuing with explicit evaluation of integrals, we derive
\begin{eqnarray*}
\varphi \ast \varphi^2 & = & \frac{m^2}{4} \left[ \varphi \ast e^{2\pi - 2x} + 2 \varphi \ast 1 + \varphi \ast e^{-2\pi + 2x} \right] \\
& = & \frac{m^2}{3} \left[ 3 + 4 \cosh(\pi) \cosh(\pi - x) - \cosh(2 \pi - 2x) \right],
\end{eqnarray*}
so that
\begin{eqnarray*}
\frac{3}{4} \varphi \ast \varphi^2 + \frac{1}{2} \varphi^2 = M \varphi + m^2,
\end{eqnarray*}
which coincides with (\ref{statCH}) for $c = M$ and $d = m^2$.
\end{proof}

By using Lemmas \ref{lem-nonlinear-2} and \ref{lem-traveling}, we shall now derive
the evolution equations for perturbations near the peaked periodic wave.
We are looking for a weak solution $u \in C([0,T),H^1_{\rm per}(\mathbb{T})\cap W^{1,\infty}(\mathbb{T}))$
to the CH equation in the form (\ref{CHweak}), for which there exists $\xi(t) = c t + a(t) \in \mathbb{T}$ for $t \in [0,T)$
such that $u(t,\cdot) \in C^1_{\xi(t)}$ for $t \in [0,T)$. We present the solution
in the form:
\begin{equation}
\label{decomp}
u(t,x) = \varphi(x-ct-a(t)) + v(t,x-ct-a(t)), \quad t \in [0,T), \quad x \in \mathbb{T},
\end{equation}
where $c = M$, $a(t)$ is the deviation of the peak position from its unperturbed position moving with the speed $c$,
and $v(t,x)$ is the perturbation to the peaked periodic wave $\varphi$.
By Lemma \ref{lem-nonlinear-2}, $a \in C^1(0,T)$ satisfies the equation
\begin{equation}
\label{peak-moves}
\frac{da}{dt} = v(t,0), \quad t \in (0,T).
\end{equation}
Substituting (\ref{decomp}) and (\ref{peak-moves}) into the initial-value problem (\ref{CHevol}) yields
the following problem for the peaked perturbation $v$:
\begin{equation}
\label{CHpert}
\left\{
\begin{array}{l}
v_t = (c - \varphi) v_x + (v|_{x=0} - v) \varphi'  + (v|_{x=0} - v) v_x \\
\qquad \qquad - \varphi' \ast \left(\varphi v + \frac{1}{2} \varphi' v_x \right) - Q[v], \qquad \qquad \qquad t \in (0,T),\\
v |_{t = 0} = v_0,
\end{array} \right.
\end{equation}
where we have used the stationary equation (\ref{statCH}) piecewise on both sides from
the peak and replaced $x - c t - a(t)$ by $x$ due to the translational invariance of the system
(\ref{CHevol}) with the convolution integral (\ref{Q-def}).

\section{Linearized evolution}

Here we study the linearized equation of motion arising in the truncation
of the nonlinear equation in system (\ref{CHpert}) at the linear terms in $v$:
\begin{equation}
\label{linCH}
v_t = (c - \varphi) v_x + (v|_{x=0} - v) \varphi' - \varphi' \ast \left(\varphi v + \frac{1}{2} \varphi' v_x \right).
\end{equation}
We first simplify the linearized equation (\ref{linCH}) by using the following elementary result.

\begin{lemma}
\label{lem-0}
Assume that $v \in H^1_{\rm per}(\mathbb{T})$. Then, it is true for every $x \in \mathbb{T}$ that
\begin{equation*}
[ v(0) - v(x)] \varphi'(x) - (\varphi' \ast \varphi v)(x) - \frac{1}{2} (\varphi' \ast \varphi' v_x)(x)
= \varphi(x) \int_0^x v(y) dy - \frac{1}{2} m^2 \sinh(x) \int_{-\pi}^{\pi} v(y) dy.
\end{equation*}
\end{lemma}

\begin{proof}
Since integrals of absolutely integrable functions are continuous, the map
$$
x \mapsto \varphi' \ast \left( \varphi v + \frac{1}{2} \varphi' v_x \right)
$$
is continuous for every $x \in \mathbb{T}$. Now, $H^1_{\rm per}(\mathbb{T})$ is continuously
embedded into the space of continuous and periodic functions on $\mathbb{T}$, hence $v \in C^0_{\rm per}(\mathbb{T})$.
Integrating by parts yields the following explicit expression for every $x \in \mathbb{T}$:
\begin{eqnarray*}
\left( \varphi' \ast \varphi' v_x\right)(x)
& = & \left( \varphi'' \ast \varphi' v \right)(x) - \left( \varphi' \ast \varphi'' v \right)(x) \\
& = & \left( \varphi \ast \varphi' v \right)(x) - 2 \varphi'(x) v(x) - \left( \varphi' \ast \varphi v \right)(x) + 2\varphi'(x)  v(0),
\end{eqnarray*}
which yields
\begin{equation*}
[ v(0) - v(x)] \varphi'(x) - (\varphi' \ast \varphi v)(x) - \frac{1}{2} (\varphi' \ast \varphi' v_x)(x)
= -\frac{1}{2} (\varphi' \ast \varphi v)(x) - \frac{1}{2} (\varphi \ast \varphi' v)(x).
\end{equation*}
Furthermore, we obtain for $x \in (0,\pi]$:
\begin{eqnarray*}
& \phantom{t} & -\frac{1}{2} \left( \varphi \ast \varphi' v\right)(x) - \frac{1}{2} \left( \varphi' \ast \varphi v\right)(x) \\
& = & -\frac{m^2}{2} \left[ \int_{-\pi}^0 \sinh(x) v(y) dy
- \int_0^x \sinh(2\pi - x) v(y) dy + \int_x^{\pi} \sinh(x) v(y) dy \right] \\
& = & \varphi(x) \int_0^x v(y) dy - \frac{m^2}{2} \sinh(x) \int_{-\pi}^{\pi} v(y) dy,
\end{eqnarray*}
which completes the proof of the equality for $x \in (0,\pi]$. For $x \in [-\pi,0)$,
the computations are similar:
\begin{eqnarray*}
& \phantom{t} & -\frac{1}{2} \left( \varphi \ast \varphi' v\right)(x) - \frac{1}{2} \left( \varphi' \ast \varphi v\right)(x) \\
& = & -\frac{m^2}{2} \left[ \int_{-\pi}^x \sinh(x) v(y) dy
+ \int_x^0 \sinh(2\pi + x) v(y) dy + \int_0^{\pi} \sinh(x) v(y) dy \right] \\
& = & \varphi(x) \int_0^x v(y) dy - \frac{m^2}{2} \sinh(x) \int_{-\pi}^{\pi} v(y) dy.
\end{eqnarray*}
The zero value at $x = 0$ is recovered by taking the one-sided limits $x \to 0^{\pm}$ in the previous two expressions.
\end{proof}

By Lemma \ref{lem-0}, we can rewrite the initial-value problem for the linear equation (\ref{linCH})
in the equivalent form:
\begin{equation}
\label{linCH-equiv}
\left\{ \begin{array}{l}
v_t = (c - \varphi) v_x + \varphi w - \pi m^2 \bar{v} \sinh(x), \quad t > 0, \\
v |_{t = 0} = v_0, \end{array} \right.
\end{equation}
where we have introduced
\begin{equation}
\label{w-definition}
w(t,x) := \int_0^x v(t,y) dy, \quad \bar{v}(t) := \frac{1}{2\pi} \int_{-\pi}^{\pi} v(t,y) dy = \frac{w(t,\pi) - w(t,-\pi)}{2\pi}.
\end{equation}
Let us consider the linearized inital-value problem (\ref{linCH-equiv})
in the space $C^1_0$ defined by (\ref{C-1-0}) with $\xi \equiv 0$. The following lemma shows
that the values of $v(t,0)$ and $\bar{v}(t)$ are independent of $t$.

\begin{lemma}
\label{lem-1}
Assume that there exists a solution $v \in C(\mathbb{R}^+,C^1_0)$ to the initial-value problem (\ref{linCH-equiv}).
Then, $v(t,0) = v_0(0)$ and $\bar{v}(t) = \bar{v}_0$ for every $t \in \mathbb{R}^+$.
\end{lemma}

\begin{proof}
If $v \in C(\mathbb{R}^+,C^1_0)$, then $w \in C(\mathbb{R}^+,C^1(\mathbb{T}))$
so that $w(t,0) = 0$ follows from (\ref{w-definition}). Hence, it follows
from (\ref{linCH-equiv}) that
$$
\lim_{x \to 0^{\pm}} v_t(t,x) = 0, \quad t \in \mathbb{R}^+,
$$
due to $\varphi(0) = c$ and $v(t,\cdot) \in C^1_0$ for every $t \in \mathbb{R}^+$.
Hence, $v(t,0) = v_0(0)$ for every $t \in \mathbb{R}^+$.

Integrating the evolution equation for the solution $v \in C(\mathbb{R}^+,C^1_0)$ in $x$ on $\mathbb{T}$
and using integration by parts piecewise on $[-\pi,0]$ and $[0,\pi]$, we obtain for every $t \in \mathbb{R}^+$:
\begin{eqnarray*}
\frac{d}{dt} \bar{v}(t) & = & \frac{1}{2\pi} \int_{\mathbb{T}} (\varphi(0) - \varphi(x)) v_x(t,x) dx + \frac{1}{2\pi} \int_{\mathbb{T}} \varphi(x) w(t,x) dx\\
 & = & \frac{1}{2\pi} \left[ (\varphi(0) - \varphi(x)) v(t,x) + \varphi'(x) \int_0^{x} v(t,y) dy \right] \biggr|_{x=0^+}^{x = \pi} +
 \biggr|_{x = -\pi}^{x = 0^-}  \\
 & = & 0,
\end{eqnarray*}
due to $\varphi(\pi) = \varphi(-\pi)$, $\varphi'(\pm \pi) = 0$, and $v(t,\cdot) \in C^0_{\rm per}(\mathbb{T})$ for every $t \in \mathbb{R}^+$.
Hence, $\bar{v}(t) = \bar{v}(0)$ for every $t \in \mathbb{R}^+$.
\end{proof}

The evolution problem (\ref{linCH-equiv}) can be solved explicitly by using the method of characteristics
piecewise for $x \in \mathbb{T}$ on both sides of the peak at $x = 0$. The family of characteristic curves
$x = X(t,s)$ satisfies the initial-value problem
\begin{equation}
\label{char}
\left\{ \begin{array}{l}
\frac{dX}{dt} = \varphi(X)-\varphi(0), \\
X |_{t=0} = s, \end{array} \right.
\end{equation}
for every $s \in \mathbb{T}$.

Along each characteristic curve $x = X(t,s)$ satisfying (\ref{char}),
function $W(t,s) := w(t,X(t,s))$ satisfies the initial-value problem:
\begin{equation}
\label{w-eq}
\left\{ \begin{array}{l}
\frac{dW}{dt} = \varphi'(X(t,s)) W + \pi m^2 \bar{v} [ 1 - \cosh(X(t,s))], \\
W |_{t=0} = w_0(s), \end{array} \right.
\end{equation}
where $w_0(x) := \int_0^x v_0(y) dy$. The initial-value problem (\ref{w-eq})
is obtained by linearizing
the nonlinear equation (\ref{w-appendix}) in Appendix A and posing it at the characteristic  curve $x = X(t,s)$. It follows from (\ref{linCH-equiv}) and (\ref{char}) that
$V(t,s) := v(t,X(t,s))$ satisfies the initial-value problem:
\begin{equation}
\label{v-eq}
\left\{ \begin{array}{l}
\frac{dV}{dt} = \varphi(X(t,s)) W(t,s) - \pi m^2 \bar{v} \sinh(X(t,s)), \\
V |_{t=0} = v_0(s). \end{array} \right.
\end{equation}
The following lemma gives the solution of the initial-value problem
(\ref{linCH-equiv}) in class $C^1(\mathbb{R}^+,C^1_0)$ from
solutions to the initial-value problems (\ref{char}), (\ref{w-eq}), and (\ref{v-eq}).

\begin{lemma}
\label{lem-2}
For every $v_0 \in C^1_0$, there exists the unique solution
$v \in C^1(\mathbb{R}^+,C^1_0)$ to the initial-value problem (\ref{linCH-equiv}).
\end{lemma}

\begin{proof}
By the existence and uniqueness theory for differential equations,
there exists the unique solution $X(\cdot,s) \in C^1(\mathbb{R}^+)$ to (\ref{char}) for every $s \in \mathbb{T}$
due to the Lipschitz continuity of $\varphi$ on $\mathbb{T}$. The peak's location at $X = 0$
is a critical point which remains invariant under the time flow, hence $X(t,0) = 0$.
The ends of $\mathbb{T}$ at $x = \pm \pi$ are not invariant under the time flow.
However, the adjacent peak's location at $X = 2\pi$ is also a critical point invariant under the time flow with $X(t,2\pi) = 2\pi$, hence the interval $[0,2\pi]$ is invariant under the time flow and it makes sense to consider the initial-value problem (\ref{char}) on the interval $[0,2\pi]$.

By the continuous dependence
of solutions on the initial data, the map $(0,2\pi) \ni s \mapsto X(t,s) \in (0,2\pi)$
is $C^1$ for every $t \in \mathbb{R}^+$. Moreover, the transformation is invertible because
\begin{equation}
\label{property-1}
\frac{\partial X}{\partial s} = e^{\int_0^t \varphi'(X(t',s)) dt'} > 0, \quad t \in \mathbb{R}^+, \quad s \in (0,2\pi).
\end{equation}

By substituting the solution $X(t,s)$ to (\ref{char})
into (\ref{w-eq}) and solving the linear inhomogeneous equation for $W$,
we obtain the unique solution $W(\cdot,s) \in C^1(\mathbb{R}^+)$ to (\ref{w-eq}) for every fixed $s \in [0,2\pi]$.
Moreover, since $v_0 \in C^0_{\rm per}(\mathbb{T})$, the map $(0,2\pi) \ni s \mapsto W(t,s) \in \mathbb{R}$
is $C^1$ for every $t \in \mathbb{R}^+$. The right-hand side of (\ref{w-eq}) implies
$W(t,0) = 0 = w_0(0)$ and $W(t,2\pi) = 2\pi \bar{v} = w_0(2\pi)$, where the last equality is due to invariance of the mean value of a periodic function in $C^0_{\rm per}(\mathbb{T})$
with respect to any starting point on $\mathbb{T}$. .

Similarly, by substituting the solutions $X(t,s)$ and $W(t,s)$
to (\ref{char}) and (\ref{w-eq}) into (\ref{v-eq}) and integrating
it for $V$,
we obtain the unique solution $V(\cdot,s) \in C^1(\mathbb{R}^+)$ to (\ref{v-eq}) for every fixed $s \in [0,2\pi]$.
Moreover, since $v_0 \in C^1_0$, the map $(0,2\pi) \ni s \mapsto V(t,s) \in \mathbb{R}$
is $C^1$ for every $t \in \mathbb{R}^+$. The right-hand side of (\ref{v-eq}) implies
$V(t,0) = v_0(0)$ and $V(t,2\pi) = v_0(2\pi) = v_0(0)$. Since $v_0 \in C^1_0$,
then $V(t,\cdot) \in C^1_0$  for every $t \in \mathbb{R}^+$.

Finally, the change of coordinates $\mathbb{R}^+ \times (0,2\pi) \ni (t,s) \to (t,X) \in \mathbb{R}^+ \times (0,2\pi)$
is a diffeomorphism due to (\ref{property-1}).
Since $\mathbb{T}$ is compact, the solution $v(t,\cdot) = V(t,s = X^{-1}(t,\cdot))$ belongs to $C^1_0$ for every $t \in \mathbb{R}^+$.
\end{proof}

Although the proof of Lemma \ref{lem-2} does not rely on construction of the exact solutions
to the initial-value problems (\ref{char}), (\ref{w-eq}), and (\ref{v-eq}), it is easy to obtain exact solutions
(see Appendix B). By analyzing the exact solution (\ref{v-sol}), we show that $v(t,\cdot)$ remains bounded in the $L^{\infty}$ norm,
as in the following lemma.

\begin{lemma}
\label{lem-3}
Assume that $v_0 \in C^1_0$ in the initial-value problem (\ref{linCH-equiv}).
Then, there exists $C_0 > 0$ such that
\begin{equation}
\label{bounds-lin-1}
\| v(t,\cdot) \|_{L^{\infty}(\mathbb{T})} \leq C_0, \quad t \in \mathbb{R}^+.
\end{equation}
\end{lemma}

\begin{proof}
By Lemma \ref{lem-2}, the unique solution to the initial-value problem
(\ref{linCH-equiv}) with $v_0 \in C^1_0$ satisfies $v(t,\cdot) \in C^1_0$
for every $t \in \mathbb{R}^+$. By Sobolev's embedding, $v(t,\cdot) \in L^{\infty}(\mathbb{T})$
for every $t \in \mathbb{R}^+$. It remains to obtain the bound (\ref{bounds-lin-1}) uniformly in $t \in \mathbb{R}^+$.

It suffices to consider the exact solution (\ref{v-sol}) from Appendix B for $s \in [0,2\pi]$ and $t \in \mathbb{R}^+$
since
$$
\| v(t,\cdot) \|_{L^{\infty}(\mathbb{T})} = \max_{s \in [0,2\pi]} |V(t,s)|.
$$
Since $v_0, w_0 \in L^{\infty}(\mathbb{T})$, we only need to estimate $Y(t,s)$ in (\ref{v-sol}).
By using elementary transformations, we rewrite $Y(t,s)$ in the form:
\begin{equation*}
Y(t,s) = \frac{(1-e^{-t}) [\sinh(\pi - \frac{s}{2}) \cosh(\pi - \frac{s}{2}) + e^{-t} \sinh(\frac{s}{2}) \cosh(\frac{s}{2})]}{
[\sinh(\pi - \frac{s}{2}) + e^{-t} \sinh(\frac{s}{2})]^2 + 2 (\cosh \pi - 1) e^{-t} \sinh(\frac{s}{2}) \sinh(\pi - \frac{s}{2})}.
\end{equation*}
For $s \in [0,\pi]$, we have
$$
0 \leq Y(t,s) \leq \frac{\sinh(\pi - \frac{s}{2}) \cosh(\pi - \frac{s}{2}) + \sinh(\frac{s}{2}) \cosh(\frac{s}{2})}{\sinh^2(\pi - \frac{s}{2})},
$$
which is bounded on $[0,\pi]$ uniformly in $t \in \mathbb{R}^+$. For $s \in [\pi,2 \pi]$, it follows that
\begin{eqnarray*}
0 \leq Y(t,s) & \leq & \frac{\cosh(\frac{s}{2}) [\sinh(\pi - \frac{s}{2}) + e^{-t} \sinh(\frac{s}{2})] - \sinh(\pi - \frac{s}{2})
[\cosh(\frac{s}{2}) - \cosh(\pi - \frac{s}{2})]}{
	[\sinh(\pi - \frac{s}{2}) + e^{-t} \sinh(\frac{s}{2})]^2} \\
& \leq & \frac{\cosh(\frac{s}{2})}{\sinh(\pi - \frac{s}{2}) + e^{-t} \sinh(\frac{s}{2})} =: Z(t,s).
\end{eqnarray*}
Since
$$
\frac{\partial Z}{\partial s} = \frac{\cosh(\pi) - e^{-t}}{2 [\sinh(\pi - \frac{s}{2}) + e^{-t} \sinh(\frac{s}{2})]^2} > 0,
$$
the map $[\pi,2\pi] \ni s \mapsto Z(t,s) \in \mathbb{R}^+$ is monotonically increasing for $t \in \mathbb{R}^+$ so that
$$
\max_{s \in [\pi,2\pi]} Z(t,s) = Z(t,2\pi) = M e^{t},
$$
which is exponentially growing in $t \in \mathbb{R}^+$. We also have
\begin{eqnarray}
\nonumber
w_0(s) - \pi m^2 \bar{v} (\cosh s - 1) (1 - e^{-t}) \\
= \int_{2\pi}^s v_0(s') ds'
+ \pi m^2 \bar{v} ( \cosh 2\pi - \cosh s) + \pi m^2 \bar{v} e^{-t} (\cosh s - 1).
\label{tech-eq}
\end{eqnarray}
The third term in (\ref{tech-eq}) multiplied by $Y(t,s)$ is bounded on $[\pi,2\pi]$ by
$$
\pi m^2 |\bar{v}| e^{-t} (\cosh s - 1) Y(t,s) \leq 2 \pi |\bar{v}| e^{-t} Z(t,2\pi) \leq 2 \pi M |\bar{v}|.
$$
The second term in (\ref{tech-eq}) multiplied by $Y(t,s)$ is bounded on $[\pi,2\pi]$ by
\begin{eqnarray*}
\pi m^2 |\bar{v}| ( \cosh 2\pi - \cosh s) Y(t,s) & \leq &
2 \pi m^2 |\bar{v}|
\sinh\left(\pi + \frac{s}{2}\right) \sinh\left(\pi - \frac{s}{2}\right) Z(t,s) \\
& \leq & 4 \pi M |\bar{v}| \frac{\sinh(\pi - \frac{s}{2}) \cosh(\frac{s}{2})}{\sinh(\pi - \frac{s}{2}) + e^{-t} \sinh(\frac{s}{2})} \\
& \leq & 4 \pi M |\bar{v}| \cosh(\pi).
\end{eqnarray*}
Finally, since $v_0 \in L^{\infty}(\mathbb{T})$, it follows for every $s \in [\pi,2\pi]$ that
$$
\left| \int_{2\pi}^s v_0(s') ds' \right| \leq 2 \| v_0 \|_{L^{\infty}(\mathbb{T})} \left( \pi - \frac{s}{2} \right)
\leq 2 \| v_0 \|_{L^{\infty}(\mathbb{T})} \sinh\left( \pi - \frac{s}{2} \right).
$$
Hence, the first term in (\ref{tech-eq}) multiplied by $Y(t,s)$ is bounded by the same estimate as the second term is.
All estimates together yield the bound (\ref{bounds-lin-1}).
\end{proof}

The result of Lemma \ref{lem-3} suggests the linear stability of peaked perturbations in the $L^{\infty}$ norm.
The following lemma shows that the perturbations grow in the $W^{1,\infty}$ norm and this growth is generic
at the right side of the peak.

\begin{lemma}
\label{lem-4}
Assume that $v_0 \in C^1_0$ in the initial-value problem (\ref{linCH-equiv}).
Then, it follows that
\begin{equation}
\label{bounds-lin-2}
\lim_{x \to 0^+} v_x(t,x) = e^t v_0'(0^+) + \left(M v_0(0) - \pi m^2 \bar{v}\right) (e^t - 1)
\end{equation}
and
\begin{equation}
\label{bounds-lin-3}
\lim_{x \to 0^-} v_x(t,x) = e^{-t} v_0'(0^-) + \left(M v_0(0) - \pi m^2 \bar{v}\right) (1 - e^{-t}).
\end{equation}
Consequently, if $v_0 \in C^1_{\rm per}(\mathbb{T})$ with either $v_0(0) \neq \frac{m^2}{2M} \bar{v}$ or
$v_0'(0) \neq 0$ or both, then $v(t,\cdot) \notin C^1_{\rm per}(\mathbb{T})$ for almost every $t \in \mathbb{R}^+$.
\end{lemma}

\begin{proof}
Along each characteristic curve $x = X(t,s)$ satisfying (\ref{char}), function $U(t,s) := v_x(t,X(t,s))$ satisfies the initial-value problem:
\begin{equation}
\label{u-eq}
\left\{ \begin{array}{l}
\frac{dU}{dt} = \varphi'(X(t,s)) \left[ W(t,s) - U(t,s) \right] + \varphi(X(t,s)) V(t,s) - \pi m^2 \bar{v} \cosh(X(t,s)), \\
U |_{t=0} = v_0'(s). \end{array} \right.
\end{equation}
The initial-value problem (\ref{u-eq})
is obtained by linearizing the nonlinear equation (\ref{u-appendix}) in Appendix A
and posing it at the characteristic curve $x = X(t,s)$.
Coefficients of the differential equation for $U$ are all $C^1(0,2\pi)$. The limits $s \to 0^+$ and $s \to 2\pi^-$ yield
the differential equations
\begin{equation}
\label{dif-eq-1}
\frac{d}{dt} U(t,0^+) = U(t,0^+) + M v_0(0) - \pi m^2 \bar{v}
\end{equation}
and
\begin{equation}
\label{dif-eq-2}
\frac{d}{dt} U(t,2\pi^-) = -U(t,2\pi^-) + M v_0(0) - \pi m^2 \bar{v},
\end{equation}
where we have used  $V(t,0) = V(t,2\pi) = v_0(0)$, $w_0(2\pi) = 2 \pi \bar{v}$, and
$$
w_0(2\pi) - \pi m^2 \bar{v} \cosh(2\pi) = -\pi m^2 \bar{v}.
$$
Since $U(t,0^-) = U(t,2\pi^-)$ due to periodic continuation of all coefficients of the differential equation (\ref{u-eq})
and its solution as $C^1(-2\pi,0)$, we obtain (\ref{bounds-lin-2}) and (\ref{bounds-lin-3}) by solving the linear differential equations
(\ref{dif-eq-1}) and (\ref{dif-eq-2}) with $U(0,0^+) = v_0'(0^+)$ and $U(0,2\pi^-) = U(0,0^-) = v_0'(0^-)$.
\end{proof}

The exponential growth of the $W^{1,\infty}$ norm in Lemma \ref{lem-4}
due to the exact solution (\ref{bounds-lin-2}) at $x = 0^+$
is equivalent to the linear instability
of the peaked perturbations in $W^{1,\infty}$. The following lemma
shows that the peaked perturbations are also linearly unstable in $H^1$.

\begin{lemma}
\label{lem-linear}
Assume that $v_0 \in C^1_0$ in the initial-value problem (\ref{linCH-equiv}).
Then, it follows that
\begin{equation}
\label{bound-1}
\| v(t,\cdot) \|_{H^1(\mathbb{T})}^2 = C_+ e^{t} + C_0 + C_- e^{-t},
\end{equation}
for some uniquely defined constants $C_+, C_0, C_-$.
\end{lemma}

\begin{proof}
Since $v(t,\cdot) \in C^1_0$ for every $t \in \mathbb{R}^+$ is the unique solution to the initial-value
problem (\ref{linCH-equiv}) by Lemma \ref{lem-2}, the $H^1$ norm is independent on the choice of the fundamental interval.
Hence we proceed with apriori energy estimates on $[0,2\pi]$ instead of $\mathbb{T}$.
We write the evolution equation for $v$ and $v_x$ explicitly:
\begin{equation}
\label{v-evol}
v_t = (c - \varphi) v_x + \varphi w - \pi m^2 \bar{v} \sinh(x)
\end{equation}
and
\begin{equation}
\label{u-evol}
v_{tx} = (c - \varphi) v_{xx} + \varphi' (w - v_x) + \varphi v - \pi m^2 \bar{v} \cosh(x).
\end{equation}
By multiplying (\ref{v-evol}) by $v$ and (\ref{u-evol}) by $v_x$,
integrating on $[0,2\pi]$, and using integration by parts for $v(t,\cdot) \in C^1_0$, we obtain
\begin{eqnarray*}
\frac{1}{2} \frac{d}{dt} \| v(t,\cdot) \|^2_{L^2(0,2\pi)} & = &
\int_0^{2\pi} \left[ (c - \varphi) v v_x + \varphi v \int_0^x v dy - \pi m^2 \bar{v} \sinh(x) v \right] dx \\
& = & \int_0^{2\pi} \left[ \frac{1}{2} \varphi' v^2 + \varphi v \int_0^x v dy - \pi m^2 \bar{v} \sinh(x) v \right] dx
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{1}{2} \frac{d}{dt} \| v_x(t,\cdot) \|^2_{L^2(0,2\pi)} & = &
\int_0^{2\pi} \left[ (c - \varphi) v_x v_{xx} + \varphi' v_x \int_0^x v dy - \varphi' v_x^2 + \varphi v v_x
- \pi m^2 \bar{v} \cosh(x) v_x \right] dx \\
& = & \int_0^{2\pi} \left[ - \frac{1}{2} \varphi' v_x^2 - \frac{3}{2} \varphi' v^2
- \varphi v \int_0^x v dy  + \pi m^2 \bar{v} \sinh(x) v \right] dx,
\end{eqnarray*}
where we have used $c = \varphi(0) = \varphi(2\pi)$, $\varphi'(2\pi^-) = 1$, and $v(t,0) = v(t,2\pi)$.
Adding these two lines together yields
\begin{equation}
\label{eq-1}
\frac{d}{dt} E(v) = -2 \int_0^{2\pi} \varphi' \left[ v^2 + \frac{1}{2} v_x^2 \right] dx,
\end{equation}
where $E(v) := \| v(t,\cdot) \|^2_{L^2(0,2\pi)} + \| v_x(t,\cdot) \|^2_{L^2(0,2\pi)}$ in agreement with (\ref{conserved-quantities}).

Next, we multiply (\ref{v-evol}) by $\varphi v$ and (\ref{u-evol}) by $\varphi v_x$,
integrate on $[0,2\pi]$, and use integration by parts for $v(t,\cdot) \in C^1_0$.
After straightforward computations, we obtain
\begin{eqnarray*}
\frac{1}{2} \frac{d}{dt} \int_0^{2\pi} \varphi v^2 dx & = &
\int_0^{2\pi} \left[ -\frac{1}{2} (c - 2 \varphi) \varphi' v^2 + \varphi^2 v \int_0^x v dy - \pi m^2 \bar{v} \varphi \sinh(x) v \right] dx
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{1}{2} \frac{d}{dt} \int_0^{2\pi} \varphi v_x^2 dx
& = & \int_0^{2\pi} \left[ - \frac{1}{2} c \varphi' v_x^2 - 2 \varphi \varphi' v^2
- (\varphi^2 + (\varphi')^2) v \int_0^x v dy  \right] dx \\
& \phantom{t} & + \pi m^2 \bar{v} \int_0^{2\pi} \left[ \varphi \sinh(x)
+ \varphi' \cosh(x) \right] v dx,
\end{eqnarray*}
where we can use $(\varphi')^2 = \varphi^2 - m^2$ and
$\varphi'(x) \cosh(x) - \varphi(x) \sinh(x)  = -1$.
Adding a linear combination of these two lines multiplied by $2$ and $1$ respectively yields
\begin{eqnarray}
\frac{d}{dt} \int_0^{2\pi} \varphi \left[ v^2 + \frac{1}{2} v_x^2 \right] dx =
- c \int_0^{2\pi} \varphi' \left[ v^2 + \frac{1}{2} v_x^2 \right] dx,
\label{eq-2}
\end{eqnarray}
where we have used that
$$
\int_0^{2\pi} \left[ v \int_0^x v dy - \frac{1}{2} v \int_0^{2\pi} v dy \right] dx = 0.
$$

Let us define
\begin{equation}
\label{P-S-def}
P(t) := \int_0^{2\pi} \varphi \left[ v^2 + \frac{1}{2} v_x^2 \right] dx, \quad S(t) := \int_0^{2\pi} \varphi' \left[ v^2 + \frac{1}{2} v_x^2 \right] dx.
\end{equation}
It follows from (\ref{eq-1}) and (\ref{eq-2}) that
\begin{equation}
\label{eq-conservation}
c E(v) = 2 P(t) + C_1,
\end{equation}
where $C_1$ is arbitrary constant.

Finally, we multiply (\ref{v-evol}) by $\varphi' v$ and (\ref{u-evol}) by $\varphi' v_x$,
integrate on $[0,2\pi]$, and use integration by parts for $v(t,\cdot) \in C^1_0$.
After straightforward computations, we obtain
\begin{eqnarray*}
\frac{1}{2} \frac{d}{dt} \int_0^{2\pi} \varphi' v^2 dx & = &
\int_0^{2\pi} \left[ -\frac{1}{2} (c \varphi - \varphi^2 - (\varphi')^2) v^2 +
\varphi \varphi' v \int_0^x v dy - \pi m^2 \bar{v} \varphi' \sinh(x) v \right] dx
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{1}{2} \frac{d}{dt} \int_0^{2\pi} \varphi' v_x^2 dx
& = & \int_0^{2\pi} \left[ - \frac{1}{2} (c \varphi - \varphi^2 + (\varphi')^2) v_x^2
- \frac{1}{2} (\varphi^2 + 3 (\varphi')^2) v^2
- 2 \varphi \varphi' v \int_0^x v dy  \right] dx \\
& \phantom{t} & +\pi m^2 \bar{v} \int_0^{2\pi} \left[ \varphi' \sinh(x)
+ \varphi  \cosh(x) \right] v dx + M (v|_{x = 0})^2 - 2 \pi m^2 \bar{v} v|_{x = 0},
\end{eqnarray*}
where we can use $\varphi(x) \cosh(x) -  \varphi'(x) \sinh(x)  = M$.
Adding a linear combination of these two lines multiplied by $2$ and $1$ respectively yields
\begin{eqnarray}
\label{eq-3}
\frac{d}{dt} S(t) = - c P(t) + \frac{1}{2} m^2 E(v) + C_2,
\end{eqnarray}
where $C_2$ is a constant defined by
$$
C_2 := \pi m^2 M \bar{v} + M (v|_{x = 0})^2 - 2 \pi m^2 \bar{v} v|_{x = 0},
$$
since $v|_{x=0}$ and $\bar{v}$ are independent of $t$.
Equations (\ref{eq-2}) and (\ref{eq-3}) with conservation (\ref{eq-conservation}) yields
the system of differential equations
\begin{equation}
\label{eq-4}
P'(t) = - M S(t), \quad S'(t) = -M^{-1} P(t) + C_3,
\end{equation}
where $C_3 := \frac{m^2 C_1}{2 M} + C_2$ and we have used that $m^2  - M^2 = -1$.
Thus, $S''(t) = S(t)$, so that the most general solution
of the system (\ref{eq-4}) is given by
\begin{equation}
\label{eq-solution}
P(t) = -M S_+ e^t + M S_- e^{-t} + M C_3, \quad S(t) = S_+ e^t + S_- e^{-t},
\end{equation}
where $S_+$ and $S_-$ are arbitrary constants. Substituting (\ref{eq-solution}) into (\ref{eq-conservation})
yields (\ref{bound-1}) for some constants $C_+$, $C_0$, and $C_-$.
\end{proof}

We end this section with two remarks.
First we show that the conserved quantity (\ref{eq-conservation}) for the linearized evolution
is obtained from the conserved quantities (\ref{conserved-quantities}) after substituing the solution in the form (\ref{decomp})
and expanding in powers of the perturbation. By using (\ref{Green}) and integrating by parts, we obtain
\begin{eqnarray*}
E(u) = E(\varphi) + 2 \int_{\mathbb{T}} (\varphi v + \varphi' v_x) dx + E(v) = E(\varphi) + 4 v|_{x=0} + E(v)
\end{eqnarray*}
and
\begin{eqnarray*}
F(u) & = & F(\varphi) + 2 \int_{\mathbb{T}} \varphi (\varphi v + \varphi' v_x) dx
+ \int_{\mathbb{T}} v [\varphi^2 + (\varphi')^2] dx + 2 \int_{\mathbb{T}} v (\varphi v + \varphi' v_x) dx \\
&& \qquad \qquad \qquad \qquad  + \int_{\mathbb{T}} \varphi [v^2 + (v_x)^2] dx + F(v) \\
& = & F(\varphi) + 4 M (v|_{x=0})
+ 2 \pi m^2 \bar{v}+ 2 P(t) + 2 (v|_{x=0})^2 + F(v),
\end{eqnarray*}
where $P(t)$ is defined by (\ref{P-S-def}). By eliminating $v|_{x=0}$ from the first equality and substituting it to the second equality, we obtain
\begin{equation}
\label{P-E-conservation}
2 P(t) - M E(v) - \frac{1}{4} [E(u) - E(\varphi)] E(v) + \frac{1}{8} [E(v)]^2 + F(v) = C,
\end{equation}
where
$$
C := F(u) - F(\varphi) - 2 \pi m^2 \bar{v} - M [E(u) - E(\varphi)] - \frac{1}{8} [E(u) - E(\varphi)]^2
$$
is constant in the time evolution due to the energy conservation. Neglecting the cubic and quartic terms of $v$ in (\ref{P-E-conservation}) recovers the conserved quantity  (\ref{eq-conservation}) of the linearized evolution since $c = M$.

Finally, we illustrate the exact solutions to the initial-value problems (\ref{char}), (\ref{w-eq}), and (\ref{v-eq}) obtained in Appendix B.
The solution $v = v(t,x)$ is available in the parametric form
(\ref{char-sol}) and (\ref{v-sol})
due to the method of characteristics with parameter $s$ on $[0,2\pi]$. The solution is extended
periodically to $[-2\pi,0]$.
Figure \ref{wave-cos} shows the plots of $v(t,x)$ versus $x$ on $[-2\pi,2\pi]$  for different values of $t$ for two initial conditions: $v_0(x) = \sin(x)$ (top) and $v_0(x) = \cos(x)$ (bottom), in both cases, $\bar{v} = 0$.
These panels give clear illustrations that $v(t,\cdot)$ remains bounded in the $L^{\infty}$ norm (Lemma \ref{lem-3}) and that the slope of the perturbation grows at the right side of the peak (Lemma \ref{lem-4}).

\begin{figure}[htb!]
	\includegraphics[width=0.7\textwidth]{sin1}
	\includegraphics[width=0.7\textwidth]{cos1}
	\caption{The plots of $v(t,x)$ versus $x$ on $[-2\pi,2\pi]$ for different values of $t$ in the case $v_0(x) = \sin(x)$ (top) and
		$v_0(x) = \cos(x)$ (bottom). } \label{wave-cos}
\end{figure}

\section{Nonlinear evolution}

Here we analyze the initial-value problem (\ref{CHpert}) with $v_0 \in C^1_0$ and prove Theorem \ref{theorem-nonlinear}. The challenge is that the local well-posedness in $C^1_0$ has not been established in the periodic domain $\mathbb{T}$. Neverthless, we will get the local well-posedness result by using the method of characteristics and the ODE theory.

By simplifying the linear terms with Lemma \ref{lem-0},
we rewrite the initial-value problem (\ref{CHpert}) in the equivalent form:
\begin{equation}
\label{v-nonlinear}
\left\{
\begin{array}{l}
v_t = (c - \varphi) v_x + \varphi w - \pi m^2 \bar{v} \sinh(x)
+ (v|_{x=0} - v) v_x - Q[v],\\
v |_{t = 0} = v_0,
\end{array} \right.
\end{equation}
where $w$ and $\bar{v}$ are defined by the same expressions as in (\ref{w-definition}) and
\begin{equation}
\label{Q-def-again}
Q[v](x) := \frac{1}{2} \int_{\mathbb{T}} \varphi'(x-y) q[v](y) dy, \quad
q[v] := v^2 + \frac{1}{2} v_x^2, \quad x \in \mathbb{T}.
\end{equation}
As is derived in Appendix A, $w$ satisfies the initial-value problem:
\begin{equation}
\label{w-nonlinear}
\left\{
\begin{array}{l}
w_t = (c - \varphi) w_x + \varphi' w - \pi m^2 \bar{v} [\cosh(x) - 1]
- \frac{1}{2} (v|_{x=0} - v)^2 - P[v] + P[v] |_{x=0},\\
w |_{t = 0} = w_0,
\end{array} \right.
\end{equation}
where $w_0(x) := \int_0^x v_0(y) dy$ and
\begin{equation}
\label{P-def}
P[v](x) := \frac{1}{2} \int_{\mathbb{T}} \varphi(x-y) q[v](y) dy, \quad
q[v] := v^2 + \frac{1}{2} v_x^2, \quad x \in \mathbb{T}.
\end{equation}

Similarly to Lemma \ref{lem-1}, we prove that $\bar{v}$ is independent of $t$.
At the same time, $v|_{x=0}$ does depend on $t$ in the nonlinear evolution.

\begin{lemma}
	\label{lem-nonlinear-0}
	Assume that there exists a solution $v \in C([0,T),C^1_0)$ to the initial-value problem (\ref{v-nonlinear})
	Then, $\bar{v}(t) = \bar{v}_0$ for every $t \in [0,T)$.
\end{lemma}

\begin{proof}
	We integrate the evolution equation in system (\ref{v-nonlinear}) for the solution $v \in C([0,T),C^1_0)$ in $x$ on $\mathbb{T}$ and use cancelation of the linear terms in $v$ as in the proof of Lemma \ref{lem-1}. Then, it is true
	that $\frac{d}{dt} \bar{v}(t) = 0$ if and only if
\begin{equation}
\label{zero-mean-Q}
	\int_{\mathbb{T}} Q[v](t,x) dx = \frac{1}{2}
	\int_{\mathbb{T}} \int_{\mathbb{T}} \varphi'(x-y) q[v](t,y) dy dx = 0,
\end{equation}
	where both $\varphi'$ and $q[v]$ are absolutely integrable. Interchanging the integrations by Fubini's theorem and integrating $\varphi'$ piecewise on both sides of the peak yields for every $t \in [0,T)$:
	\begin{eqnarray*}
\int_{\mathbb{T}} \int_{\mathbb{T}} \varphi'(x-y) q[v](t,y) dy dx
& = & \int_{\mathbb{T}} q[v](t,y) \left(\int_{\mathbb{T}} \varphi'(x-y) dx \right) dy \\
& = & \int_{\mathbb{T}} q[v](t,y) \left( \varphi(\pi - y) - \varphi(-\pi - y) \right) dy \\
& = & 0,
	\end{eqnarray*}
	due to periodicity of $\varphi \in C^0_{\rm per}(\mathbb{T})$.
	Hence, $\bar{v}(t) = \bar{v_0}$ for every $t \in [0,T)$.
\end{proof}

We can now develop local well-posedness theory of the initial-value
problem (\ref{v-nonlinear}) by means of the
method of characteristics. The family of characteristic curves $x = X(t,s)$
satisfies the initial-value problem:
\begin{equation}
\label{char-X}
\left\{
\begin{array}{l}
\frac{dX}{dt} = \varphi(X) - \varphi(0) + v(t,X) - v(t,0), \\
X |_{t = 0} = s,
\end{array} \right.
\end{equation}
for every $s \in \mathbb{T}$. Assuming that $v(t,\cdot) \in C^1_0$ for every $t \in [0,T)$, we can differentiate (\ref{char-X}) piecewise in $s \in \mathbb{T} \backslash \{0\}$ and obtain
\begin{equation}
\label{char-X-der}
\left\{
\begin{array}{l}
\frac{d}{dt} \frac{\partial X}{\partial s} =
\left[ \varphi'(X) + v_x(t,X) \right]  \frac{\partial X}{\partial s}, \\
 \frac{\partial X}{\partial s} |_{t = 0} = 1.
\end{array} \right.
\end{equation}
with the exact solution
\begin{equation}
\label{char-X-der-solution}
 \frac{\partial X}{\partial s}(t,s) = \exp\left(\int_0^t \left[ \varphi'(X(t',s)) + v_x(t,X(t',s)) \right] dt' \right).
\end{equation}
The peak's locations at $X(t,0) = 0$ and $X(t,2\pi) = 2\pi$ are invariant
in the time evolution if $v(t,\cdot) \in C^1_0$ for every $t \in [0,T)$.

Along each characteristic curve $x = X(t,s)$ satisfying (\ref{char-X}), functions
$V(t,s) := v(t,X(t,s))$, $W(t,s) = w(t,X(t,s))$, and $U(t,s) = v_x(t,X(t,s))$ satisfy the following initial-value problems:
\begin{equation}
\label{char-V}
\left\{
\begin{array}{l}
\frac{dV}{dt} = \varphi(X) W - \pi m^2 \bar{v} \sinh(X) - Q[v](X), \\
V |_{t = 0} = v_0(s),
\end{array} \right.
\end{equation}
\begin{equation}
\label{char-W}
\left\{
\begin{array}{l}
\frac{dW}{dt} = \varphi'(X) W
- \pi m^2 \bar{v} [ \cosh(X) - 1]
+ \frac{1}{2} [ V^2 - (v|_{X=0})^2 ] - P[v](X) + P[v](0), \\
W |_{t = 0} = w_0(s),
\end{array} \right.
\end{equation}
and
\begin{equation}
\label{char-U}
\left\{
\begin{array}{l}
\frac{dU}{dt} = \varphi'(X) [W - U] + \varphi(X) V - \pi m^2 \bar{v} \cosh(X)
- \frac{1}{2} U^2 + V^2 - P[v](X), \\
U |_{t = 0} = v_0'(s).
\end{array} \right.
\end{equation}
Derivation of (\ref{char-V}), (\ref{char-W}), and (\ref{char-U})  follow
from (\ref{v-appendix}), (\ref{w-appendix}), and (\ref{u-appendix}) in Appendix A
after using (\ref{char-X}). The following lemma transfers the local well-posedness theory for differential equations to the initial-value problem (\ref{v-nonlinear}).

\begin{lemma}
\label{lem-nonlinear-3}
For every $v_0 \in C^1_0$, there exists the maximal existence time $T > 0$ (finite or infinite) and the unique solution $v \in C^1([0,T),C^1_0)$ to the initial-value problem (\ref{v-nonlinear}) that depends continuously on $v_0 \in C^1_0$.
\end{lemma}

\begin{proof}
If $v \in C^1_0$, then $Q[v] \in C^0_{\rm per}(\mathbb{T}) \cap {\rm Lip}(\mathbb{T})$ by Lemma \ref{lem-nonlinear-1}. It follows from Lemma
\ref{lem-nonlinear-0} due to (\ref{zero-mean-Q}) that if $v \in C^1_0$,
then $P[v] \in C^1_{\rm per}(\mathbb{T})$. Therefore, the nonlocal parts of the initial-value problems (\ref{char-V}), (\ref{char-W}), and (\ref{char-U}) are well-defined
and can be considered for $X \in [0,2\pi]$ that corresponds to $s \in [0,2\pi]$.

For $s \in [0,2\pi]$, we rewrite the evolution equations in (\ref{char-X}), (\ref{char-V}), (\ref{char-W}), and (\ref{char-U}) as the dynamical system
\begin{equation*}

\frac{d}{dt} \begin{bmatrix} X \\ V \\ W  \\ U \end{bmatrix} =
\begin{bmatrix} \varphi(X) - \varphi(0) + V - V|_{s=0} \\
\varphi(X) W - \pi m^2 \bar{v} \sinh(X) - Q[v](X) \\
\varphi'(X) W - \pi m^2 \bar{v} [ \cosh(X) - 1 ]
+ \frac{1}{2} [ V^2 - (V|_{s=0})^2 ] - P[v](X) + P[v] |_{s=0} \\
 \varphi'(X) [W - U] + \varphi(X) V - \pi m^2 \bar{v} \cosh(X)
 - \frac{1}{2} U^2 + V^2 - P[v](X) \end{bmatrix}
\end{equation*}
subject to the initial condition
\begin{equation*}

\begin{bmatrix} X \\ V \\ W \\ U\end{bmatrix} \biggr|_{t = 0} =
\begin{bmatrix} s \\ v_0(s) \\ w_0(s) \\ v_0'(s) \end{bmatrix}
\end{equation*}
and the boundary conditions
\begin{equation*}

\left\{ \begin{array}{ll}
X(t,0) = 0, & \quad X(t,2\pi) = 2\pi, \\
V(t,0) = V|_{s=0}, & \quad V(t,2\pi) = V|_{s=0}, \\
W(t,0) = 0, & \quad  W(t,2\pi) = 2 \pi \bar{v}, \end{array} \right.
\end{equation*}
where $V|_{s=0}$ satisfies
$$
\frac{d}{dt} V \biggr|_{s = 0} = -Q[v](0).
$$

Since the vector field of the dynamical system is
$C^1$ in $(X,V,W,U)$ on $[0,2\pi] \times \mathbb{R}\times \mathbb{R} \times \mathbb{R}$, there exists a unique local solution
$X(\cdot,s), V(\cdot,s), W(\cdot,s), U(\cdot,s) \in C^1([0,T))$ to the initial-value problem for some maximal existence time $T > 0$. The solution depends continuously
on the initial data for every $s \in [0,2\pi]$. Moreover,
since the initial data is $C^1(0,2\pi)$, then
$X(t,\cdot), V(t,\cdot), W(t,\cdot), U(t,\cdot) \in C^1(0,2\pi)$ for every $t \in [0,T)$.
The invertibility of the transformation $[0,2\pi] \ni s \mapsto X(t,s) \in [0,2\pi]$ is guaranteed along the solution by
\begin{equation}
\label{invertibility}
\frac{\partial X}{\partial s}(t,s) = \exp\left(\int_0^t \left[ \varphi'(X(t',s)) + U(t,s) \right] dt' \right) > 0.
\end{equation}
which follows from (\ref{char-X-der-solution}). Since $0$ and $2 \pi$ are
equilibrium points of (\ref{char-X}), we have $X(t,0) = 0$ and $X(t,2\pi) = 2\pi$.
Boundary conditions are preserved along the solution due to consistence
of (\ref{char-W}) and (\ref{char-U}) with the main equation (\ref{char-V}).
Due to the boundary conditions, the solution $V(t,\cdot) \in C^1(0,2\pi)$
is extended to $V(t,\cdot) \in C^1_0$ on $\mathbb{T}$.
Due to invertibility of the transformation $[0,2\pi] \ni s \mapsto X(t,s) \in [0,2\pi]$,
we have $v(t,\cdot) \in C^1_0$ for $t \in [0,T)$ and moreover, $v \in C^1([0,T), C^1_0)$.
\end{proof}

The proof of Theorem \ref{theorem-nonlinear} relies on the study of evolution of
$v_x \in C^1([0,T),C^0(\mathbb{T}\backslash \{0\}))$ at the right side of the peak.
By Lemma \ref{lem-nonlinear-3}, we are allowed to define the one-sided limits
$U^{\pm}(t) := \lim_{s \to 0^{\pm}} U(t,s)$ for $t \in [0,T)$,
where $U(t,0^-) = U(t,2\pi^-)$. The functions $U^{\pm} \in C^1(0,T)$
satisfy for $t \in (0,T)$:
\begin{equation}
\label{U-0}
\frac{d U^{\pm}}{dt} = \pm U^{\pm} + M V|_{s=0}
- \pi m^2 \bar{v} - \frac{1}{2} (U^{\pm})^2 + (V|_{s=0})^2 - P[v](0),
\end{equation}
which follow from taking the limits $s \to 0^{\pm}$ in (\ref{char-U}).
We will now prove Theorem \ref{theorem-nonlinear}.

Due to the decomposition (\ref{decomp}), we can rewrite the initial bound (\ref{initial-bound-theorem}) in the form:
\begin{equation}
\label{initial-bound}
\| v_0 \|_{H^1(\mathbb{T})} + \| v_0' \|_{L^{\infty}(\mathbb{T})} < \delta,
\end{equation}
where $v_0 \in C^1_0 \subset H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$
and $\delta > 0$ is arbitrary small parameter. We first show that there exists $t_0 \in (0,T)$ and $v_0 \in C^1_0$ such that the unique local solution $v \in C^1([0,T),C^1_0)$ to the initial-value problem (\ref{v-nonlinear})  constructed by
Lemma \ref{lem-nonlinear-3}  satisfies
\begin{equation}
\label{final-bound}
\| v_x(t_0,\cdot) \|_{L^{\infty}(\mathbb{T})} \geq 1.
\end{equation}

It follows from $2 P[v](0) = \int_{\mathbb{T}} \varphi(-y)q[v](y) dy > 0$
that equation (\ref{U-0}) for the upper sign can be estimated by
\begin{equation}
\label{diff-ineq}
\frac{d U^+}{dt} \leq U^+ + M V|_{s=0}
- \pi m^2 \bar{v} + (V|_{s=0})^2,
\end{equation}
By Theorem \ref{theorem-CS}, for every small $\varepsilon > 0$,
there exists $\nu(\varepsilon) > 0$ such that
if $\| v_0 \|_{H^1(\mathbb{T})} < \nu(\varepsilon)$, then
$\| v(t,\cdot) \|_{H^1(\mathbb{T})} < \varepsilon$ for every $t \in [0,T)$.
By Sobolev's embedding, there is a positive constant $C$ such that
\begin{equation}
\label{bound-on-V0}
\| v(t,\cdot) \|_{L^{\infty}(\mathbb{T})} \leq C \| v(t,\cdot) \|_{H^1} < C \varepsilon.
\end{equation}
By using (\ref{bound-on-V0}), we can simplify (\ref{diff-ineq})
for every $\varepsilon \in (0,1]$ to the form:
\begin{equation}
\label{diff-ineq-2}
\frac{d U^+}{dt} \leq U^+ + M C \varepsilon
+ \pi m^2 C \varepsilon + C^2 \varepsilon^2 \leq U^+ + C_1 \varepsilon,
\end{equation}
for some $\varepsilon$-independent constant $C_1 > 0$.
Let us assume that the initial data $v_0 \in C^1_0$ satisfies
\begin{equation}
\label{eq-alpha}
\lim_{x \to 0^+} v_0'(x) = -\| v_0' \|_{L^{\infty}(\mathbb{T})} = -2 C_1 \varepsilon,
\end{equation}
The initial bound (\ref{initial-bound}) is consistent with
(\ref{eq-alpha}) if for every small $\delta > 0$, the small value
of $\varepsilon$ satisfies
$$
\nu(\varepsilon) + 2 C_1 \varepsilon < \delta,
$$
which just specifies small $\varepsilon$ in terms of small $\delta$. By integrating
(\ref{diff-ineq-2}) and using (\ref{eq-alpha}), we obtain
\begin{equation*}
U^+(t) \leq e^{t} \left[  U^+(0) + C_1 \varepsilon \right] = - C_1 \varepsilon e^{t}.
\end{equation*}
Hence, for every small $\varepsilon > 0$ there exists a sufficiently large
$t_1 = -\log(C_1 \varepsilon)$ such that $U^+(t_1) \leq -1$.
If $t_1 < T$, the bound (\ref{final-bound}) is true with some $t_0 \in (0,t_1]$ since
\begin{equation}
\label{lower-bound}
\| v_x(t,\cdot) \|_{L^{\infty}(\mathbb{T})} =
\| U(t,\cdot) \|_{L^{\infty}(0,2\pi)} \geq |U^+(t)|, \quad t \in [0,T).
\end{equation}
If $t_1 \geq T$, then $T$ is the finite maximal existence time in Lemma \ref{lem-nonlinear-3} for the solution
$$
X(\cdot,s), V(\cdot,s), W(\cdot,s), U(\cdot,s) \in C^1([0,T)), \quad
s \in [0,2\pi].
$$
By (\ref{invertibility}), we have $X \in C^1([0,T),C^1(0,2\pi))$
if and only if $U \in C^1([0,T),C^0(0,2\pi))$. By the bound
(\ref{bound-on-V0}), we have $W,V \in C^1([0,T),C^0(0,2\pi))$
with bounded limits of $\| W(t,\cdot) \|_{L^{\infty}(0,2\pi)}$
and $\| V(t,\cdot) \|_{L^{\infty}(0,2\pi)}$ as $t \to T^-$.
Then, necessarily, if $T < \infty$, we have
\begin{equation}
\label{blow-up-ode}
\| U(t,\cdot) \|_{L^{\infty}(0,2\pi)} \to \infty \quad
\mbox{\rm as} \quad t \to T^-,
\end{equation}
so that there exists $t_0 \in (0,T)$ such that  $\| U(t_0,\cdot) \|_{L^{\infty}(0,2\pi)} \geq 1$ and the bound (\ref{final-bound}) is true
due to (\ref{lower-bound}).

Finally, we show that there exists $v_0 \in C^1_0$ such that the maximal existence time $T$ is finite.

Due to the bound (\ref{bound-on-V0}), we have for every $\varepsilon \in (0,1)$:
$$
\left| M V|_{s=0}
- \pi m^2 \bar{v} + (V|_{s=0})^2 - P[v](0) \right|
\leq M C \varepsilon + \pi m^2 C \varepsilon + C^2 \varepsilon^2
+ \frac{1}{2} M \varepsilon^2 \leq C_2 \varepsilon,
$$
for some $\varepsilon$-independent constant $C_2 > 0$.
Let $U_0(\varepsilon)$ be the negative root of the quadratic equation
$$
U - \frac{1}{2} U^2 + C_2 \varepsilon = 0.
$$
It is clear that $U_0(\varepsilon) \in (-C_2 \varepsilon,0)$. Assume that the initial data $v_0 \in C^1_0$ satisfies
\begin{equation}
\label{eq-alpha-new}
\lim_{x \to 0^+} v_0'(x) = -\| v_0' \|_{L^{\infty}(\mathbb{T})} = 2 U_0(\varepsilon).
\end{equation}
The initial bound (\ref{initial-bound}) is consistent with
(\ref{eq-alpha-new}) if for every small $\delta > 0$, the small value
of $\varepsilon$ satisfies
$$
\nu(\varepsilon) + 2 |U_0(\varepsilon)| < \delta,
$$
which again specifies small $\varepsilon$ in terms of small $\delta$. If $U^+(0) = 2 U_0(\varepsilon)$, then differential equation (\ref{U-0}) for $U^+$ implies that $\frac{d}{dt} U^+ < 0$, hence $U^+(t) < U^+(0)$ for small positive $t$ and the map $t \mapsto U(t)$ is monotonically decreasing. Let $\overline{U}$ be the supersolution which satisfies:
\begin{equation*}
\label{Ricatti-inequality}
\frac{d \overline{U}}{dt} = \overline{U} - \frac{1}{2} \overline{U}^2 + C_2 \varepsilon.
\end{equation*}
with $\overline{U}(0) = U^+(0) = 2 U_0(\varepsilon)$. It follows by the comparison theory
for differential equations that $U^+(t) \leq \overline{U}(t)$ for every $t > 0$ for which $U^+(t)$ exists.
Since there exists a finite $\overline{T} > 0$ such that
$\overline{U}(t) \to -\infty$ as $t \to \overline{T}^-$,
then there exists $T \in (0,\overline{T}]$ such that $U^+(t) \to -\infty$ as $t
\to T^-$, which implies that the blow-up criterion (\ref{blow-up-ode}) is satisfied for finite $T > 0$.

Both parts of Theorem \ref{theorem-nonlinear} have been proven.

\appendix

\section{Derivation of the evolution equations for $w$ and $v_x$}

We consider the evolution equation for $v$ given by
\begin{equation}
\label{v-appendix}
v_t = (c - \varphi) v_x + \varphi w - \pi m^2 \bar{v} \sinh(x)
 + (v|_{x=0} - v) v_x - Q[v],
\end{equation}
where
$$
Q[v](x) := \frac{1}{2} \int_{\mathbb{T}} \varphi'(x-y) \left[
v^2 +  \frac{1}{2}(v_y)^2 \right](y) dy.
$$
Substituting $v = w_x$ into (\ref{v-appendix}) yields
$$
w_{tx} = (c - \varphi) w_{xx}
+ \varphi w - \pi m^2 \bar{v} \sinh(x)
 +  (v|_{x=0} - w_x) w_{xx} - Q[v].
$$
By integrating this evolution equation in $x$
and picking the constant of integration from the boundary conditions $w(t,0) = 0$, we obtain
\begin{equation}
\label{w-appendix}
w_t = (c - \varphi) w_x + \varphi' w - \pi m^2 \bar{v} [\cosh(x) - 1]
- \frac{1}{2} (v|_{x=0} - v)^2 - P[v] + P[v] |_{x=0},
\end{equation}
where
$$
P[v](x) := \frac{1}{2} \int_{\mathbb{T}} \varphi(x-y) \left[
v^2 +  \frac{1}{2}(v_y)^2 \right](y) dy.
$$

Differentiating (\ref{v-appendix}) in $x$ yields
\begin{equation}
\label{u-appendix}
v_{tx} = (c - \varphi) v_{xx}
+ \varphi' (w - v_x) + \varphi v - \pi m^2 \bar{v} \cosh(x)
+ (v|_{x=0} - v) v_{xx} + v^2 - \frac{1}{2} (v_x)^2
- P[v],
\end{equation}
where we have used the relation
$$
\frac{1}{2} \int_{\mathbb{T}} \varphi''(x-y) \left[
v^2 +  \frac{1}{2}(v_y)^2 \right] dy =
\frac{1}{2} \int_{\mathbb{T}} \varphi(x-y) \left[
v^2 + \frac{1}{2} (v_y)^2 \right] dy - v^2 - \frac{1}{2} v_x^2
$$

\section{Explicit solutions to the linearized equations}

Separation of variables in (\ref{char}) gives
$$
m t = \int_s^X \frac{dx}{\cosh(\pi - x) - \cosh(\pi)} = m \log \frac{(e^{2\pi} - e^X) (1 - e^s)}{(1 - e^X) (e^{2\pi} - e^s)},
$$
from which we derive
\begin{equation}
\label{char-sol}
X(t,s) = \log \frac{(e^{2\pi} - e^s) + e^{2\pi - t} (e^s - 1)}{(e^{2\pi} - e^s) + e^{-t} (e^s - 1)}, \quad t \in \mathbb{R}^+, \quad s \in [0,2\pi],
\end{equation}
such that
$$
\lim_{s \to 0^+} X(t,s) = 0 \quad \mbox{\rm and} \quad \lim_{s \to 2 \pi^-} X(t,s) = 2\pi.
$$
After lengthy manipulations,
we can find that
\begin{equation*}
\frac{\partial X}{\partial s}(t,s) = \frac{(e^{2\pi}-1)^2 e^{s-t}}{
[(e^{2\pi}-e^s) + e^{-t} (e^s - 1)][(e^{2\pi}-e^s) + e^{2\pi - t} (e^s - 1)]}, \quad t \in \mathbb{R}^+, \quad s \in [0,2\pi],
\end{equation*}
such that
$$
\lim_{s \to 0^+} \frac{\partial X}{\partial s}(t,s) = e^{-t} \quad \mbox{\rm and} \quad
\lim_{s \to 2 \pi^-} \frac{\partial X}{\partial s}(t,s) = e^t.
$$

Integrating (\ref{w-eq}) with an integrating factor yields
$$
W(t,s) = \frac{\partial X}{\partial s} \left[ w_0(s) + \pi m^2 \bar{v}
\int_0^t \frac{1 - \cosh X(t',s)}{\frac{\partial X}{\partial s}(t',s)} dt' \right].
$$
By using (\ref{char-sol}), we derive the following simple expression:
\begin{equation}
\label{w-sol}
W(t,s) = \frac{\partial X}{\partial s} \left[ w_0(s) - \pi m^2 \bar{v}
 (\cosh s - 1) (1 - e^{-t}) \right], \quad t \in \mathbb{R}^+, \quad s \in [0,2\pi],
\end{equation}
such that
$$
\lim_{s \to 0^+} W(t,s) = 0 \quad \mbox{\rm and} \quad
\lim_{s \to 2 \pi^-} W(t,s) = w_0(2 \pi) = 2 \pi \bar{v}.
$$

Instead of integrating (\ref{v-eq}), we can use the chain rule
$$
\frac{\partial W}{\partial s}(t,s) = V(t,s) \frac{\partial X}{\partial s}(t,s),
$$
from which we derive
\begin{equation}
\label{v-sol}
V(t,s) = v_0(s) - \pi m^2 \bar{v} \sinh s (1 - e^{-t}) +
\left[ w_0(s) - \pi m^2 \bar{v} (\cosh s - 1) (1 - e^{-t})\right] Y(t,s)
\end{equation}
with
\begin{eqnarray*}
Y(t,s) & := & \frac{\partial}{\partial s} \log \frac{\partial X}{\partial s}(t,s) \\
& = & \frac{(1-e^{-t}) [\sinh(2\pi - s) + e^{-t} \sinh(s)]}{\cosh(2 \pi - s)
-1 + [\cosh(2\pi) + 1 - \cosh(2\pi - s) - \cosh(s)] e^{-t} + (\cosh s - 1) e^{-2t}},
\end{eqnarray*}
such that
$$
\lim_{s \to 0^+} V(t,s) = v_0(0) \quad \mbox{\rm and} \quad
\lim_{s \to 2 \pi^-} V(t,s) = v_0(2 \pi) = v_0(0).
$$

\section{Smooth, peaked, and cusped periodic waves}

Traveling wave solutions to the CH equation (\ref{CH}) are of the form
$u(t,x) = \phi(x-ct)$, where $c$ is the wave speed and $\phi(x) : \mathbb{T}\mapsto \mathbb{R}$ satisfies the third-order differential equation
\begin{equation}
\label{third-order}
(c-\phi) (\phi''' - \phi') + 2 \phi \phi' - 2 \phi' \phi'' = 0.
\end{equation}
Multiplying (\ref{third-order}) by $(c - \phi)$ allows us to integrate it once:
\begin{equation}
\label{second-order}
\frac{d}{dx} \left[ (c - \phi)^2 (\phi'' - \phi) \right] = 0 \quad \Rightarrow \quad (c - \phi)^2 (\phi'' - \phi) = a,
\end{equation}
where $a \in \mathbb{R}$ is the integration constant. The level $\phi = c$ is singular and in what follows, we consider $c > 0$. For every $\phi \neq c$,
the second-order differential equation (\ref{second-order}) can be integrated once:
\begin{equation}
\label{first-order}
\phi'' - \phi - \frac{a}{(c - \phi)^2} = 0 \quad \Rightarrow \quad
(\phi')^2 - \phi^2 - \frac{2a}{c-\phi} = b,
\end{equation}
where $b \in \mathbb{R}$ is another integration constant.

Let $W(\phi) := -\phi^2 - 2a/(c-\phi)$. If $a = 0$, there is only one critical point of $W$ at $\phi = 0$ so that $(0,0)$ is a saddle point of the dynamical system
on the phase plane $(\phi,\phi')$. Peaked periodic waves are constructed from orbits of the linear equation $\phi'' - \phi = 0$ intersecting the singularity line $\phi = c$ over the length of $2 \pi$. If
$$
M_{\phi} := \max_{x \in \mathbb{T}} \phi(x), \quad
m_{\phi} := \min_{x \in \mathbb{T}} \phi(x),
$$
then $b = -m_{\phi}^2$ and $c = M_{\phi}$. The exact $2\pi$-periodic solution to $\phi'' - \phi = 0$ is given by
\begin{equation}
\label{wave-appendix}
\phi(x) = m_{\phi} \cosh(\pi - |x|), \quad M_{\phi} = m_{\phi} \cosh(\pi).
\end{equation}
The uniquely selected peaked wave (\ref{unique-wave}) corresponds to $\phi = \varphi$ with $M_{\phi} = M$ and $m_{\phi} = m$ given by (\ref{maximum}) and (\ref{minimum}). The one-parameter
family (\ref{wave-appendix}) corresponds to the one-parameter family of peaked $2\pi$-periodic waves (\ref{peaked-traveling-wave}), this family is given by
$\phi = \gamma \varphi$ with $\gamma := \frac{m_{\phi}}{m}$ being a free parameter and $c = M_{\phi} = \gamma M$ in agreeement between (\ref{peaked-traveling-wave}) and (\ref{wave-appendix}).

If $a > 0$, there is only one critical point of $W$ at some $\phi_0 < 0$
so that $(\phi_0,0)$ is a saddle point on the phase plane $(\phi,\phi')$.
The periodic waves are constructed from orbits squeezed between the stable and unstable curves from $(\phi_0,0)$ and the singularity line $\phi = c$. These orbits give the cusped periodic waves with the following local behavior near the singularity:
$$
\phi(x) \sim c - \alpha x^{2/3} \quad \mbox{\rm as} \quad x \to 0,
$$
where $\alpha > 0$ is related to parameter $a > 0$.

If $a < 0$, there are three critical points of $W$ given by $\{ \phi_1,\phi_2,\phi_3\}$ ordered as $\phi_1 < \phi_2 < c < \phi_3$ so that
$(\phi_1,0)$ and $(\phi_3,0)$ are saddle points, whereas
$(\phi_2,0)$ is a center point.
Smooth periodic waves are located inside the homoclinic loop connected to $(\phi_1,0)$ which surrounds $(\phi_2,0)$.

Cusped waves also exist for $\phi > c$ but not for $\phi < c$.

config: arxiv


TITLE: How to prove the Taylor series identity
QUESTION [1 upvotes]: How to prove the following identity:
$(1-t)^{-L}=\sum_{j\geq 0}
\binom{j+L-1}{L-1}t^j$.
I will appreciate for your helpful suggestion.

REPLY [0 votes]: HINT
Refer to the Binomial Series
$$(1 + x)^\alpha = \sum_{k=0}^{\infty} \; {\alpha \choose k} \; x^k   $$
for the special case with negative integer exponent.


TITLE: How to position a rotated rectangle to the perimeter of a circle at specific radius
QUESTION [0 upvotes]: Let's say we have a circle (position of its center and radius are known) and a rotated rectangle (width, height and rotation are known).
The rectangle has 8 "anchor points": 4 at its corners, 4 at the mid-point of its edges.
Now imagine there's a given radius at a specific angle, and we want to move the rectangle along with that line starting from the center of the circle to the edge until one of its eight "anchor points" reaches the perimeter of the circle. How do I compute the center point of the rectangle at this position?
Moreover, if we keep moving the rectangle until the last of its anchor point located on the perimeter, what is the center point of the rectangle?
Here's the image demo

REPLY [0 votes]: Referring to the picture provided, let us assume $\alpha = 0, \beta = 0$. Now we will assume the rectangle moves with the radius as if it is fixed to the radius, so $\alpha = \beta$.

Suppose you rotated the radius by $\alpha$ anticlockwise and so now $\beta = \alpha$. Now  in this position we rotate back the rectangle by $\gamma$ clockwise, such that $\beta = \alpha - \gamma$ is the desired angle you wanted to rotate the rectangle.
Now going back to the picture above we see that in this position if it moves outwards then both 3 and 5 will touch the perimeter together, 4 will never touch first; for this matter no points on the mid of the sides will touch the perimeter first as it is concave inside a circle.
In this position if we start rotating the rectangle say clockwise then we see immediately 3 will become the point which will touch the perimeter first.This will continue till we rotate full 90 degree. At that point both 3 and 1 will touch the perimeter first.
Now to summarize, the points will touch;  $\gamma = 0\rightarrow (3,5), 0 <\gamma < 90 \rightarrow (3), \gamma = 90\rightarrow (3,1), 90 <\gamma < 180 \rightarrow (1), \gamma = 180\rightarrow (1,7), 180 <\gamma < 270 \rightarrow (7), \gamma = 270\rightarrow (7,5), 270 <\gamma < 360 \rightarrow (5)$ ,
Note that we calculated $\gamma$ in earlier equation ($\gamma = \alpha - \beta$).
Now to find out the last point while going out we observe circle is convex outside so (referring to same picture above) 8 will be the last point when $\gamma = 0$, to summarize
$\gamma = 0\rightarrow (8),0 <\gamma < 90 \rightarrow (7), \gamma = 90\rightarrow (6), 90 <\gamma < 180 \rightarrow (5), \gamma = 180\rightarrow (4), 180 <\gamma < 270 \rightarrow (3), \gamma = 270\rightarrow (2), 270 <\gamma < 360 \rightarrow (1)$
Now to determine the center of the rectangle, let us assume length of its longer side is a and shorter side is b. So for the picture where $\gamma = 0$ the distance from center will be $\sqrt{r^2 - (b/2)^2} - (a/2)$, Note that here you only need to know this distance as the angle subtended by the radius ($\beta$) is already given (and the center of the rectangle lies on the line of the radius).
Now in the following picture

The rectangle is rotated by $\gamma$ clockwise. Let $g= \sqrt{(a/2)^2+(b/2)^2}$ and $\sin\delta = \dfrac{b/2}{g}$, we can calculate $\delta$ from here. So above length of line segment drawn from 3 on O4, which is perpendicular to O4 is $h = g\sin(\delta - \gamma)$, so distance from center O is $\sqrt{r^2-h^2} - g\cos{(\delta - \gamma)}$. Note that when $\gamma = 0$, this falls back to earlier equation.
Now this is valid for $0 \leq\gamma \leq 90$
For $90 \leq\gamma \leq 180$, we note that g will remain same, $\sin\delta = \dfrac{a/2}{g}$, $\gamma = \gamma - 90$.
So $h = g\sin(\delta - (\gamma - 90) ) = g\sin(\delta + 90 -\gamma )$; diatance = $\sqrt{r^2-h^2} - g\cos{(\delta +90 - \gamma)}$
For $180 \leq\gamma \leq 270$, g will remain same, $\sin\delta = \dfrac{b/2}{g}$, $\gamma = \gamma - 180$.
For $270 \leq\gamma \leq 360$, g will remain same, $\sin\delta = \dfrac{a/2}{g}$, $\gamma = \gamma - 270$.


TITLE: Galois group of $\mathbb{Q}(\sqrt[5]{3},\zeta):\mathbb{Q}$
QUESTION [1 upvotes]: I'm trying to determine the Galois group of $\mathbb{Q}(\sqrt[5]{3},\zeta):\mathbb{Q}$, where $\zeta=e^{2\pi i/5}$. I have the following information:

$\mathbb{Q}(\sqrt[5]{3},\zeta)$ is the splitting field for $x^5-3$, whose roots are $\sqrt[5]{3}, \zeta \sqrt[5]{3}, \zeta^2 \sqrt[5]{3}, \zeta^3 \sqrt[5]{3},$ and $\zeta^4 \sqrt[5]{3}$.
$\mathbb{Q}(\sqrt[5]{3},\zeta):\mathbb{Q}$ has degree 20.
If $\sigma$ is an element of the Galois group, then $\sigma$ must send a root of $x^5-3$ to a root of $x^5-3$. So the Galois group is isomorphic to a subgroup of $S_5$ with order 20.
$\sigma$ is completely determined by where it sends $\sqrt[5]{3}$ and $\zeta$ (the basis elements for $\mathbb{Q}(\sqrt[5]{3},\zeta):\mathbb{Q}$).

There is a unique subgroup of $S_5$ of order 20, which is $\langle (1\,2\,3\,4\,5),(1\,2\,4\,3)\rangle$, but without using this, how can I determine the Galois group structure?
Also, how can I show it's not abelian?

REPLY [2 votes]: This is the Galois group of $x^5-3$.
In general, the Galois group of $x^n-a$ is explicitly known, see here:
Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$
Proposition [Jacobson, Velez]: One has $G\cong \mathbb{Z}/n \rtimes (\mathbb{Z}/n)^{\times}$ if and only if $n$ is odd, or $n$ is even and $\sqrt{a}\not\in \mathbb{Q}(\zeta_n)$.
Here $n=5$ is odd, so we know that $G\cong \Bbb Z/5\rtimes \Bbb Z/4$ is a solvable, non-abelian group of order $20$.


TITLE: Remainder when $2009^{2009}-1982^{2009}-1972^{2009}+1945^{2009}$ is divided by $1998$
QUESTION [3 upvotes]: Find Remainder when $N=2009^{2009}-1982^{2009}-1972^{2009}+1945^{2009}$ is divided by 1998
I have written the given number $N$ as
$$N=(1998+11)^{2009}-(1998-16)^{2009}-(1998-26)^{2009}+(1998-53)^{2009}$$ and by binomial theorem the remainder when $N$ divided by $1998$ is
$$R=11^{2009}+16^{2009}+26^{2009}-53^{2009}$$
but how to find remainder when $R$ is divided by $1998$

REPLY [3 votes]: The problem has innate $\rm\color{#c00}{symmetry}$ that greatly simplifies matters once brought to the fore.
$$\begin{align} 2009-1982 = 27,\quad 2009-1972 = 37\\
1972-1945 = 27,\quad 1982-1945 = 37\end{align}$$
Thus $\phantom{\Rightarrow}\ \  \{ 2009,\ \ \  1945\}\ \  \equiv\, \{1982,\ \ \ 1972\}\ \ \ \ {\rm mod}\,\ 27\ \&\ 37,\ $
$\qquad\Rightarrow\ \{2009^n,\ \ 1945^n\} \equiv \{1982^n,\ \,1972^n\}\,\ {\rm mod}\,\ 27 \ \&\ 27,\ $ by the Congruence Power Rule
$\qquad\Rightarrow\ \ \ 2009^n\!+\! 1945^n\ \ \equiv \ \, 1982^n\!+1972^n\ \ \ {\rm mod}\,\ 27\ \&\ 37,\  $ so also $\,{\rm mod}\ 999 = {\rm lcm}(27,37)$
since addition $\,f(x,y)\, =\, x + y\ $ is $\rm\color{#c00}{symmetric}$ $\,f(x,y)= f(y,x),\, $  therefore its value depends only upon the (multi-)set $\,\{x,\ y\}.\, $   Parity $\,\Rightarrow\,$ congruent mod $2,\,$ so also mod  $\,2\cdot 999 = 1998.$
Remark $ $ Generally if a polynomial $\,f\in\Bbb Z[x,y]\,$ is $\rm\color{#c00}{symmetric}$ then as above we deduce
$\qquad\qquad\quad \{A, B\}\, \equiv\, \{a,b\}\,\ {\rm mod}\,\ m\ \&\ n\ \Rightarrow\  f(A,B)\equiv f(a,b)\, \bmod{\,{\rm lcm}(m,n)}$
a generalization of  CCRT =constant-case optimization  of CRT = Chinese Remainder,  plus a generalization of the Polynomial Congruence Rule to (symmetric) bivariate polynomials.


text:
\section{Statements used in the proofs}
\label{sect:cited}
Theorem \ref{theorem:BE16} and Lemma \ref{lemma:AC_GSA} are used in the proofs of the main results; these statements had been derived in our earlier paper \citep{Zhilova2020}. Here we provide improved lower bounds for constant $M$ (in Theorem \ref{theorem:BE16}), and we also describe the constants in the error term which appear in the main results. In Remark \ref{ref:AC_opt} we show optimality (with respect to $d$) of the Gaussian anti-concentration bound over set $\BCl$. Lemma \ref{lemma:moment_conditions} is a concise version of Lemmas 3.1 and 3.2 in \citep{Zhilova2020}, which provide conditions on distribution $X_{i}$ that are sufficient for fulfilling condition \eqref{eq:moments_2} of Theorem \ref{theorem:BE16} (as well as condition \eqref{condit:L_moments} of Theorem \ref{theorem:symm}).

Let random vectors $\{X_{i}\}_{i=1}^{n}$ in $\R^{d}$ be i.i.d. and such that $\E |X_{i}^{\otimes K}|<\infty$ for some integer $K\geq 3$,  and   $\Var(X_{i}) = \Sigma$ is p.d. Suppose that there exist i.i.d. approximating random vectors $\{Y_{i}\}_{i=1}^{n}$ such that $\E |Y_{i}^{\otimes K}|<\infty$,
\begin{equation}
\label{eq:moments_2}
\begin{gathered}
\E (X_{i}^{\otimes j})=\E (Y_{i}^{\otimes j})\ \forall j=1,\dots,K-1,\text{ and}\\
 Y_{i}=\Zy_{i}+\Uy_{i} \text{ for independent r.v. $\Zy_{i},\Uy_{i}\in\R^{d}$ s.t.,}\\
\text{$\Zy_{i}\sim \mathcal{N}(0,\Sigmaz)$ for a p.d. $\Sigmaz$.}
\end{gathered}
\end{equation}
Denote $S_{n}\define n^{-1/2}{ \sum\nolimits_{i=1}^{n}} X_{i}$, $\St_{n}\define n^{-1/2}{{\textstyle \sum\nolimits_{i=1}^{n}}} Y_{i},$  and let $\lminz^{2}>0$ be equal to the smallest eigenvalue of the covariance matrix $\Sigmaz$.

\begin{theorem}
\label{theorem:BE16}
Let $\{X_{i}\}_{i=1}^{n}$ and $\{Y_{i}\}_{i=1}^{n}$ meet the conditions above. Suppose, without loss of generality, that $\E X_{i}=\E Y_{i}=0$, then it holds
\begin{align*}
\Delta_{\BCl}(S_{n},\tilde{S}_{n})
&\leq  C_{B,K}{ \left\{\lminz^{-K}\E\left(\|X_{1}\|^{K}+\|Y_{1}\|^{K}\right)\right\}^{{1}/{(K-2)}}}{n^{-1/2}},
\end{align*}
where constant $C_{B,K}=M(K) C_{\ell_{2}}C_{\phi,K}$ depends only on $K$. For the quantity $M(K)$, one can take $M(3)=54.1$, $M(4)=9.5$, $M(6)=2.9$.
\end{theorem}
\noindent
 $C_{\ell_{2}}\define\max\{1,\aClbt\}$ for a numeric constant $\aClbt$ given in Lemma \ref{lemma:AC_GSA} further in this section.
 $C_{\phi,K}\define\max\{1,{C}_{\phi,1,K-2}, {C}_{\phi,1,K-1}\}$ is specified as follows: fix $\epsc>0$ and a Euclidean ball $B= B(x_{0},r)\define \left\{x \in \R^{\dimp}: \|x-x_{0}\|\leq r\right\}$ in $\R^{d}$, let function $\psi: \R^{d}\mapsto \R$ be defined as $\psi(x;B)\define \phi(\tilde{\rho}(x;B)/\tilde{\epsc})$, where $\tilde{\rho}(x;B)= \{\|x-x_{0}\|^{2}-r^{2}\}\Ind\{x\neq B\}$, $\tilde{\epsc}=\epsc^{2}+2r\epsc$, and $\phi(x)$ is a sufficiently smooth approximation of a step function
 \begin{gather*}
0\leq \phi(x)\leq 1, \quad
\phi(x)=
\begin{cases}
1,& x\leq 0;\\
0,& x\geq 1.
\end{cases}
 \end{gather*}
 Constants ${C}_{\phi,1,j}>0$ for $j=K-2,K-1$ are such that $\forall B\in \BCl$, $\forall\,x,h\in \R^{d}$
$$\bigl|\psi^{(j)}(x;B)h^{j}\bigr|\leq {{C}_{\phi,1,j} \|h\|^{j}}{(\epsc/2)^{-j}}\Ind\{x\in B(x,r+\epsc)\setminus B\}.$$
Further details about these terms are provided in Lemma A.3 in  \citep{Zhilova2020supp}.

The smaller values of $M=M(K)$ (compared to $M\geq 72.5$ in  \citep{Zhilova2020}) are obtained by optimizing the following expression w.r.t.  $(\adj,\cmKK,M)$ (this expression is contained in the last inequality in the end of the proof of Theorem 2.1 in \citep{Zhilova2020supp}):
\begin{align*}
 \frac{\adj}{\cM}
 +
 \frac{1.5 (\adj/2)^{-(K-2)} }{(K-2)!}\frac{(2\cM+\adj)}{\cM}
 \nonumber
+
 \frac{4\sqrt{2}\cmKK }{(\adj/2)^{(K-1)}}
\frac{(2\cM+\adj)}{\cM K! }&
\\+\frac{2\sqrt{2} }{\sqrt{K!}\cmKK^{K-2}}
+  2^{(K-3)/(K-2)}\frac{ 2.6}{M^{K-2}}
& \leq 1,
\end{align*}
we take $(K,M,\adj,\cmKK)\in\{(3, 54.1, 27.46, 14), (4,9.5, 6.33, 8.5), (6,2.9, 2.07, 8.5)\}$.

In the following lemma we study the anti-concentration properties of $Z\sim \mathcal{N}(0,\Sigma)$ over the class $\BCl$. Similar bounds for the standard normal distribution were considered by \cite{Sazonov1972,Ball1993Reverse,Klivans2008lGaussSurface}).

Let $B=B(x_{0},r)$, denote for  $\epsc>0$ $B^{\epsc}\define B(x_{0},r+\epsc).$
\begin{lemma}[Anti-concentration inequality for $\ell_{2}$-balls in $\R^{d}$]
\label{lemma:AC_GSA}
Let $Z\sim \mathcal{N}(\mu,\Sigma)$ for arbitrary $\mu\in\R^{d}$ and p.d. $\Sigma$ with the smallest eigenvalue $\lmin^{2}>0$. It holds for any $\epsc>0$ and for a numeric constant $\aClbt>0$
\begin{align}
\nonumber
\sup\nolimits_{B\in\BCl}
\P\bigl(Z\in B^{\epsc}\setminus B\bigl)
&\leq   \epsc\aClbt/\lmin.
\end{align}
\end{lemma}

\begin{remark}
\label{ref:AC_opt}
This inequality is sharp w.r.t. dimension $d$. Indeed, consider $Z\sim \mathcal{N}(0,\Id_{d})$, $B=B(0,r)$, and $\epsc>0$, Let $f_{\chi_{d}}(t)$ denote the p.d.f. of $\|Z\|\sim \chi_{d}$, then
$$\sup\nolimits_{r\geq 0} P(Z\in B^{\epsc}\setminus B)/\epsc = \sup\nolimits_{t\geq 0} f_{\chi_{d}}(t) +O(1)= O(1), \quad d\to \infty.$$
\end{remark}

\begin{lemma}
\label{lemma:moment_conditions}
\begin{enumerate}[wide,
labelwidth=!,
labelindent=0pt,
label={\Roman*.}]
\item Let $K=4$ and $X_{i}$ be a random vector in $\R^{d}$ with $\E |X_{i}^{\otimes 4}|<\infty$ and p.d. $\Var X_{i}$. Then there exists  an approximating distribution $Y_{i}$ satisfying \eqref{eq:moments_2} such that the smallest eigenvalue of $\Sigmaz$ corresponding to the normal part of the convolution $Y_{i}$ equals to an arbitrary predetermined number between $0$ and the smallest eigenvalue of $\Var X_{i}$.
\item Now let $K$ be an arbitrary integer number $\geq 3$.
Suppose that random vector $X_{i}$ is supported in a closed set $A\subseteq \R^{d}$. Let also $\E |X_{i}^{\otimes K+1}|<\infty$ and $\Var X_{i}$ be p.d. Then the existence of the an approximating distribution $Y_{i}$ satisfying \eqref{eq:moments_2} is guaranteed either by continuity of $X_{i}$ or by a sufficiently large cardinality of $X_{i}$'s support, namely, if $X_{i}$ has a discrete probability distribution supported on $M$ points in $\R^{d}$ such that each coordinate of $X_{i}$ is supported on at least $m$ points in $\R$, it is sufficient require $M\geq 1+(K+1)m^{d-1}$.
\end{enumerate}
\end{lemma}


TITLE: Find all positive integers $x, y, z$ so that $(x+2y)(y+2z)(z+2x)$ is equal to prime power
QUESTION [6 upvotes]: Find all positive integers $x, y, z$ so that $(x+2y)(y+2z)(z+2x)$ is equal to prime power.
My try
Since $x, y, z$ are positive integers it follows that $x+2y, y+2z, z+2x \geq 3$. We have following system of equations:
$$\begin{align}
x+2y &= p^r \tag{1}\label{eq1} \\
y+2z &= p^s \tag{2}\label{eq2} \\
z+2x &= p^t \tag{3}\label{eq3}
\end{align}$$
Where $p$ is prime number and $r, s, t$ are positive integers.
Solving this system of equations for $x, y, z$ gives us $x=\frac{4p^t-2p^s+p^r}{9}$, $y=\frac{4p^r-2p^t+p^s}{9}$, $z=\frac{4p^s-2p^r+p^t}{9}$. Since $x, y, z$ are integers following equations must hold:
$$\begin{align}
4p^t-2p^s+p^r &\equiv 0 \pmod 9 \\
4p^r-2p^t+p^s &\equiv 0 \pmod 9 \\
4p^s-2p^r+p^t &\equiv 0 \pmod 9
\end{align}$$
I don't know how to continue from there. Can somebody help?

REPLY [5 votes]: Suppose we have a solution $(x, y, z)$. Note that if $d = \gcd(x, y, z) > 1$, then $d$ must also be a power of $p$, and we get a smaller solution $(x', y', z') = (x/d, y/d, z/d)$ with $\gcd(x', y', z') = 1$.
Thus without loss of generality we can assume our solution has $\gcd(x, y, z) = 1$. Now, if $p \neq 3$, then by the formulas $x = (4p^t - 2p^s + p^r)/9$, etc., we must have that $x, y, z$ are divisible by $p$, contradicting $\gcd(x, y, z) = 1$. It follows that there can only be solutions in the case $p = 3$.
Now suppose without loss of generality that $z$ is the largest of $x, y, z$. Again write
$$x + 2y = 3^r \qquad y + 2z = 3^s \qquad z + 2x = 3^t.$$
Since $y \leq z$, we have $3^s \leq 3z$, and similarly since $x \leq z$, we have $3^t \leq 3z$. But then $3^{s-1} \leq z < z + 2x = 3^t$, so $s \leq t$, and conversely, $2(3^{t-1}) \leq 2z < y + 2z = 3^s$, so $t \leq s$, and thus we must have $s = t$. This gives $y + 2z = z + 2x$, hence $z = 2x - y \leq 2x$, so $3^t \leq 4x$, and $x \geq 3^t/4$. It follows that $3^r = x + 2y > 3^t/4 > 3^{t-2}$, so $r \geq t-1$. In the other direction, since $z-x=x-y$ and $z \geq x$, we have $x \geq y$, hence $3^r = x + 2y \leq z + 2x = 3^t$, so $r \leq t$.
Thus our only solutions for $(r, s, t)$ are of the forms $(t, t, t)$ and $(t-1, t, t)$. In the case $(t, t, t)$, it immediately follows that $x = y = z$, hence $(x, y, z) = (1, 1, 1)$. Solving the second case for $x, y, z$ gives
$$(x, y, z) = (3^{t-3}(12-6+1), 3^{t-3}(4-6+3), 3^{t-3}(12-2+3)) = (3^{t-3} \cdot 7, 3^{t-3}, 3^{t-3} \cdot 13)$$
and since we assumed $\gcd(x, y, z) = 1$, our solution is $(7, 1, 13)$.
Now relaxing the assumption that $z$ was the largest, so we allow for cyclic shifts, the solutions with $\gcd(x, y, z) = 1$ are exactly $(1, 1, 1), (7, 1, 13), (1, 13, 7), (13, 7, 1)$. Relaxing the other assumption that $\gcd(x, y, z) = 1$, the general solutions are exactly all scalings of these primitive solutions by a power of $3$.


TITLE: On the sine of a two by two matrix with integer entries
QUESTION [3 upvotes]: Let $A \in M(2,\mathbb Z)$ ; Let
$\sin A :=\sum _{n=0}^\infty \dfrac {(-1)^n}{(2n+1)!}A^{2n+1}$ (This series is known to converge absolutely in the Banach space $\mathcal L(\mathbb R^2)$ under Operator norm ) .
If $\sin A \in M(2,\mathbb Z)$ , then is it true that $A^2=O$ ?

REPLY [5 votes]: Note - Haven't received that many downvotes for a single answer for a while.
I suspect people downvote this because it uses something too advanced for
a problem that looks trivial. In fact, if the eigenvalues $\lambda_1, \lambda_2$
of $A$ are both real (say $A$ is real symmetric), then this problem is indeed trivial. This is because when $\lambda_1, \lambda_2$ are real,
$$|\det\sin(A)| = |\sin(\lambda_1)\sin(\lambda_2)| \le 1 \quad\text{ and }\quad |{\rm tr} \sin(A)| = |\sin(\lambda_1) + \sin(\lambda_2)| \le 2$$
and the fact $\pi$ is irrational is enough to force $\lambda_1 = \lambda_2 = 0$.
The problem is $\lambda_1, \lambda_2$ need not be real....

Original answer
This answer uses a little bit of transcendental number theory. It might be an overkill for this particular problem. However, it confirm whenever both $A$ and $\sin A$ are integer matrices, then $A^2 = 0$.
Let $\lambda_1, \lambda_2$ be eigenvalues of $A$.
When $A \in M(2,\mathbb{Z})$, its characteristic polynomial
$\chi_A(\lambda) \stackrel{def}{=}\det(\lambda I - A)$ will be a polynomial with integer coefficients. Since $\lambda_1, \lambda_2$ are roots of this polynomial, they will be algebraic numbers.
Notice $\sin A$ has $\sin(\lambda_1)$ and $\sin(\lambda_2)$ as its eigenvalues.
If $\sin A \in M(2,\mathbb{Z})$, then by same argument, $\sin\lambda_1$ and $\sin\lambda_2$ will be algebraic numbers.
By Lindermann Weierstrass theorem.
If $\alpha$ is any non-zero algebraic number, then $\sin\alpha$ will be transcendental. In order for all four numbers $\lambda_1, \lambda_2, \sin\lambda_1, \sin\lambda_2$ to be algebraic, we need $\lambda_1 = \lambda_2 = 0$. This leaves us with two possibilities:

$A$ is diagonalizable and hence is equals to the zero matrix.
$A$ is similar to a Jordan block of order $2$ with zero diagonal.
In this case, $A$ will be nilpotent and satisfies $A^2 = 0$.

In both cases, if both $A, \sin A \in M(2,\mathbb{Z})$, then $A^2 = 0$.


TITLE: Is a 1-form locally expressible as $dx$?
QUESTION [1 upvotes]: Given a 1-form $\alpha$ which is non-zero at every point of a manifold $M^n$, is it true that locally I can express the form as $dx$? (that is around each point there is a coordinate neighborhood such that $\alpha$ has that form).
I think this is false because then the form will be closed in every neighborhood and hence closed.
The problem that I have is the following reasoning:
at a point $p$ in the manifold, extend $\alpha_p$ to a basis of the cotangent space at $p$, as being linearly independent is an open condition, that implies that I can extend it to $n$ (I guess smooth) sections of the cotangent bundle linearly independent at each point of a neighborhood of $p$. Now on each point of this neighborhood take the dual basis of tangent vectors, this will produce (I guess smooth) $n$ vector fields on this neighborhood, where the dual of $\alpha$ can be written in another coordinate system as $\partial/\partial x$ and therefore $\alpha$ as $dx$.
I think the first guess is correct but not sure about the second one, can someone explain it to me? thanks

REPLY [1 votes]: It's clearly necessary for $\alpha$ to be closed. Since the property in question is local, it's sufficient to prove the result for $M = \mathbb{R}^n$. Choose an arbitrary point $p\in M$. Since $M$ is contractible, we have $\alpha = df$ for some $f\in C^\infty(M)$. Since $\alpha(p)\not = 0$, there exists by the constant rank theorem some coordinates $(x_1, \dots, x_n)$ on a neighborhood $U$ centered at $p$ with $f(x_1, \dots, x_n) = x_1$. Thus $\alpha = dx_1$ on $U$.


TITLE: Commutation classes of reduced decompositions of the longest element of the Weyl group with one element
QUESTION [8 upvotes]: For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the number of such decompositions is discussed.
Following Stembridge, we define a relation $∼$ on the set of reduced expressions for $w_0$. Let $\mathbf{w}$ and $\mathbf{w}'$ be two reduced expressions for $w_0$ and define $\mathbf{w} \sim \mathbf{w}'$ if we can obtain $\mathbf{w}'$ from $\mathbf{w}$ by applying a single commutation. Now, define the equivalence relation $\simeq$ by taking the reflexive transitive
closure of $\sim$. Each equivalence class under $\simeq$ is called a commutation class.
The question of the number of commutation classes (of reduced expressions for $w_0 \in S_n$) is still an open problem. However, is there an formula for the number of equivalence classes containing just one element?

REPLY [10 votes]: The reduced words that are in their own commutation classes are:

$s = [123\cdots(n-2)(n-1)(n-2)\cdots321][23\cdots(n-3)(n-2)(n-3)\cdots32][3\cdots(n-4)(n-3)(n-4)\cdots3]\cdots$
the reversal of $s$ (writing the word in reverse order)
the complement of $s$ (mapping each letter $i$ to $n-i$)
the reverse complement of $s$

This is just one word in $S_2$ and two words in $S_3$ (no big surprises there!). Otherwise it is four words. For example, in $S_6$, they are

123454321234323
323432123454321
543212345432343
343234543212345

On a separate note, to add to the list of related topics, you might want to look at Fishel et al. (https://doi.org/10.1016/j.ejc.2018.07.002) about how commutation classes and braid classes interact.

REPLY [8 votes]: I believe that for $n \geq 4$ there will be exactly $4$ such reduced words. One such word, call it $R_n$ can be constructed by starting with $s_{n-1}s_{n-2} \cdots s_2s_1s_2 \cdots s_{n-2}s_{n-1}$ and appending $R_{n-1}$ with all indices shifted up by 1. For example, $$R_6=s_5s_4s_3s_2s_1s_2s_3s_4s_5s_4s_3s_2s_3s_4s_3.$$
Three more words can be obtained by replacing each $s_i$ with $s_{n-i}$, reversing the word, or doing both.
For $n=4,5,6$ these are the only possibilities, and this probably shouldn't be too hard to argue in general.


TITLE: Suppose $f(z) \in H(D)$ is it possible to have...
QUESTION [2 upvotes]: (a) Suppose $f(z) \in H(D)$ is it possible to have $$|f^{(n)}(0)| \ge e^n *n!$$
for infinitely many n?
(b) Suppose there is a constant M such that $$|f^{(n)}(0)| \le M^n$$ for all n sufficiently large. Show that $f(z)$ can be extended to all of $C$ as an entire function.
My thinking:
I am having trouble grasping what is going on here for example how can $|f^{(n)}(0)| \le e^n$ essentially from part (b) but the be greater than $e^n *n!$ something doesn't seem to add up to me.

REPLY [1 votes]: The idea here is that on any disk centered at the origin where the function is analytic you have
$$f(z) = \sum_{n = 0}^{\infty} {f^{(n)}(0) \over n!} z^n$$
So for a), you could pick any $z$ for which $|ez| > 1$ and you can see the above series must diverge.
For b), the above series converges uniformly on any disk centered at the origin by the estimate you are given, and the resulting function is analytic. It must equal $f(z)$ on $D$ since $f(z)$ is equal to its power series there.


TITLE: How to solve this equation: $e^x=(\sqrt{y}+1)^2$ for y?
QUESTION [0 upvotes]: We have $e^x=(\sqrt{y}+1)^2$. I tried to solve this equation for y, but I'm stuck
when I get the expression: $e^x= y + 2\sqrt{y}+1$ because there is still an root of y. Can someone give me a step-by-step solution for this problem?

REPLY [2 votes]: taking the square root on both sides we get
$$\sqrt {e^x}=\sqrt{y}+1$$
squaring again we have
$$y=\left(\sqrt{e^x}-1\right)^2$$


TITLE: Classify all groups with normal subgroups of order 3 and 4
QUESTION [3 upvotes]: Suppose a group $G$ has normal subgroups of size 3 and 4. I want to classify all such groups.
The subgroups are clearly $\mathbf Z_3$ and $\mathbf Z_4$ or $K_4=\mathbf Z_2\times\mathbf Z_2$. If $H_4$ denotes one of $\mathbf Z_4$ and $K_4$, then any group of the form
$$\mathbf Z_3\times H_4\times L$$
where $L$ is any group, satisfies the required condition.
I suspect that these are the only groups which satisfy this this condition, but I don't know how to prove it. I am familiar with the following result for obtaining equations of direct products like what I'm after:

If $H,K\leqslant G$, $H\cap K = \{1\}$ and the elements of $H$ and $K$ commute, then $HK\simeq H\times K$.

But I don't believe I can apply it to my situation since I don't know if elements commute.
I appreciate any help with this.

REPLY [7 votes]: I think this question is too broad/vague. It is equivalent to asking for all (finite) groups that have an abelian normal subgroup of order $12$.
Indeed, if $N_1$ is normal of order $3$ and $N_2$ is normal of order $4$, then $N_1N_2$ is normal of order $12$, and has a normal Sylow $3$-subgroup and a normal Sylow $2$-subgroup. But the nonabelian groups of order $12$ are $A_4$ (which does not have a normal Sylow $3$-subgroup), the dihedral group of order $12$, $D_6$ (which does not have a normal Sylow $2$-subgroup), and the dicyclic group $\langle a,s\mid a^6=e, s^2=a^3, sas^{-1}=a^{-1}\rangle$ (which does not have a normal Sylow $2$-subgroup). Thus, $N_1N_2$ is abelian so $G$ has a normal abelian subgroup of order $12$. Conversely, if $G$ has a normal abelian subgroup $N$ of order $12$, then the Sylow subgroups of $N$ are characteristic in $N$, hence normal in $G$, so $G$ has a normal subgroup of order $4$ and one of order $3$.
However, classifying all groups with a normal abelian subgroup of order $12$ (or any given order, for that matter) seems rather broad. The group need not split into $A\times K$ with $A$ the normal abelian group of order $12$; it need not even split into $A\rtimes K$ with $A$ the normal abelian group of order $12$:

$C_{24}$ does not split into $A\times C_2$ with $A$ abelian of order $12$.

$Q_8\times C_3$ has a normal abelian subgroup of order $12$, namely $\{1,-1,i,-i\}\times C_3$, which is cyclic of order $12$ (being the direct product of two cyclic groups of relatively prime order), but it is not a semidirect product $C_{12}\rtimes C_2$, because $Q_8\times C_3$ has three subgroups of order $12$ (depending which subgroup of order $4$ of $Q_8$ you take), but $C_{12}\rtimes C_2$ only has two: if $N\triangleleft C_{12}\rtimes C_2 = \langle x\rangle\rtimes C_2$ has order $12$, the intersection with $\langle x\rangle$ has index at most $2$, hence is either $\langle x\rangle$, so $N=\langle x\rangle$, or else $N\cap\langle x\rangle = \langle x^2\rangle$ and then $N=\langle x^2\rangle\rtimes C_2$.


TITLE: Example of a Manifold which has One Non-zero Component of Ric which equals the Scalar Curvature
QUESTION [1 upvotes]: I am wondering if there is a simple example of a manifold such that, given a value for the scalar curvature $R$, I can find a manifold such that the Ricci tensor has all zero components except for one component which takes the value $R$.
I know this could be achieved using a warped product of two metrics to separate out one coordinate and then solve the differential equations so that the first coefficient vanishes, but obviously the coefficient of the non-zero component needs to be $R$.

REPLY [1 votes]: Since you asked for a simple example, I was curious about what happens for a 2D manifold. Let's try to fix the Ricci scalar $R$ to be some value $S$. (Note that it is twice the Gaussian curvature in this case $K$, i.e. $R = 2K$).
In 2D, we have (see: [1], [2], [3]):
\begin{align}
 R_{abcd} &= \frac{R}{2}(g_{ac}g_{bd} - g_{ad}g_{bc}) \\
R_{\alpha\gamma} &= g^{mn} R_{n\alpha m\gamma} \\
       &= g^{mn}\frac{R}{2}(g_{ac}g_{bd} - g_{ad}g_{bc}) \\
       &= \frac{R}{2}(2g_{\alpha\gamma} - \delta^m_\gamma g_{\alpha m}) \\
       &= Kg_{\alpha\gamma} \\
R &= \text{tr}_g (\text{Ric})=g^{ij}R_{ij} = 2K
\end{align}
You are requiring two constraints:

$S = R$
Every component of $R_{ij}$ is zero except one.

Notice that in this case you are requiring every entry of the metric tensor to be zero except one entry (which must be on the diagonal, since $g$ is symmetric) that must equal $1$. Such a metric is not invertible (nor positive definite)!
I guess in 2D, asking for this is too harsh of a constraint on the metric tensor.