and by Lemma \ref{lem:lamk}, each of these intersections is both
nowhere dense and measure zero with respect to $2n$-dimensional
Hausdorff measure in $\B R^{2n+1}$, and the first part of the
theorem follows.

For the second part, we note that the equation $\d_k(J,\Theta)=0$
can be regarded as an equation for the unknown $\t_m$ (or
$k\piot\t_m$), which is $2\pi$-periodic, namely
\begin{equation}
\label{tmeq}
\d_k = \ul z\,R(k\piot\t_m)\,\ol z = 0,
\end{equation}
similar to \eqref{dkz}, for some nonzero vectors $\ul z$ and
$\ol z$.  Note that as $k$ increases, solutions of \eqref{tmeq} get
closer together.  Given an arbitrary measurable entropy
profile $S(x)$ and $\epsilon>0$, we can approximate the entropy
profile by a piecewise constant profile $s(J,\T)$, with
\[
\big\|S-s(J,\T)\big\|_{L^1} < \epsilon/3.
\]
Next, by choosing $k$ large enough, we find a small perturbation
$\wh\Theta$ of $\Theta$ with $\theta_m$ a solution of \eqref{tmeq},
so that
\[
(J,\wh\T)\in\Lambda_k \com{with}
\big\|s(J,\wh\T)-s(J,\T)\big\|_{L^1} < \epsilon/3.
\]
Finally, by genericity, we find a small perturbation
$(\wt J,\wt\T)\in\Lambda_k$ of $(J,\wh\T)$ which is nonresonant, and
for which
\[
\big\|s(\wt J,\wt \T)-s(J,\wh\T)\big\|_{L^1} < \epsilon/3;
\]
combining these completes the proof.
\end{proof}

\subsection{Dependence on Reference Period}

As in Section~\ref{sec:Tvar}, we again allow the piecewise constant
entropy field, parameterized by $(J,)$, to be arbitrary, and
consider the dependence of the divisors $_̣k(J,;T)$ on the
reference period $T$.  The total width of the evolution (in the
material frame) is $∑_m_m = ℓ$, and by scale invariance of our
system \eqref{lagr}, we can without loss of generality take $ℓ=1$.
This means that our parameter space which fully describes any
piecewise constant entropy profile on $[0,ℓ]=[0,1]$ is
\[
J \in\B R^n_+, \qquad
\T \in \Delta^n \subset \B R^{n+1}_+,
\]
where $Δ^n$ is the $n$-simplex,
\[
\Delta^n := \Big\{ \Theta\in\B R^{n+1}_+\;\Big|\;
\sum_{m=0}^n\t_m = 1 \Big\},
\]
which is the convex hull of the standard coordinate vectors.  It will
be convenient to parameterize $Δ^n$ by the open solid simplex in
\[
\Delta_\circ^n := \Big\{ \Theta^\circ\in\B R^n_+\;\Big|\;
\sum_{m=0}^{n-1}\t_m < 1 \Big\}, \com{with}
\t_n := 1 - \sum_{m=0}^{n-1}\t_m > 0;
\]
here we write
\[
\T^\circ := (\t_0,\dots,\t_{n-1}) \com{and}
\T := (\t_0,\dots,\t_n) = (\T^\circ,\t_n).
\]
Note that whenever any $J_m=1$, this description becomes degenerate,
and in fact the actual number of entropy jumps is $#{J_m1}$.

For a given $T$, the basis vectors $T_k(T)$ and divisors
$_̣k(J,;T)$ are given by \eqref{Tk} and \eqref{dkpw}, respectively,
and so again satisfy the degeneracy \eqref{d2j}.  As in our earlier
development, we write $= k$, recalling we have assumed
$ℓ=1$, and we express each $_̣k(J,;T)$ in terms of $$, as
\[
\begin{aligned}
\d_k(J,\T;T) = \(0&1\)&P^{-k}\,
\end{aligned}
\]
Since we are interested in those values of $T$ for which $_̣k$
vanishes, and since the matrices $R( _m)$ and $M(J_m)$ are
invertible, we again use the function $h$ defined in \eqref{hdef} to
describe the angles at each jump.

Proceeding inductively, we thus define
\begin{equation}
\label{gamdef}
\begin{aligned}
\g_1(\w) &:= h(J_1,\w\t_0), \com{and}\\
\g_{m+1}(\w) &:= h\big(J_{m+1},\w\t_m+\g_m), \quad m = 1,\dots,n-1.
\end{aligned}
\end{equation}
Thus $_n()$ is the angle obtained after the final jump $J_n$, and
our condition that $_̣k=0$ is again that $_n+_n$ lie exactly on
a coordinate axis.  Thus, in analogy with \eqref{weq1}, we
again define the $k$-th \emph{base frequency}
$^(k)=^(k)(J,)$ by the implicit equation
\begin{equation}
\label{weq}
\w^{(k)}\t_n + \g_n\big(\w^{(k)}\big) = k\,\frac\pi2.
\end{equation}
Here we have suppressed dependence on the entropy profile parameter
$(J,)$.  Note, however, that $_m$ depends only on the
initial part of the parameter,
\[
\g_m =
\g_m\big(\w;J_1,\dots,J_m,\t_0,\dots,\t_{m-1}\big) =: \g_m(\w).
\]
To illustrate, in Figure~\ref{fig:freqs}, we show the corresponding
rotations and jumps (scaled for visibility) for the first four nonzero
eigenfrequencies.  Here the entropy field consists of four constant
states separated by three jumps.  The circular arcs represent linear
evolution of the $k$-mode through the entropy level $_m$ so are
rotations by $^(k)_m$, and the vertical segments represent the
action of the jumps, in which the second component is scaled but the
first remains constant.  The four curves are color coded, and some
arcs are labeled: for example, $^(1)_1$ (blue) is the evolution
of the 1-mode between jumps $J_1$ and $J_2$.

\begin{figure}[thb]
\begin{tikzpicture}[>=stealth]
\draw[->] (-4.5,0) -- (4.5,0);
\draw[->] (0,-4.5) -- (0,5);

\def\th0{22}

\foreach \rad/\mult/\C in
\fill[color=\C] (\rad, 0) circle (2 pt);
\draw[very thick,color=\C,>->,shorten <=2.4pt,shorten >=2.5pt]
(\rad0, 0) arc (0:\mult*\th0:\rad) coordinate (ee);
\foreach \J/\th in {1.3/20,1.5/15,0.8/20}
\draw[very thick,color=C7]
let \p2 = (ee) in (ee) -- (\x2,\J*\y2) coordinate (ff);
\fill[color=\C] (ee) circle (2 pt);
\fill[color=\C] (ff) circle (2 pt);
\draw[very thick,color=\C,>->,shorten <=2.4pt,shorten >=2.5pt]
let \p1 = (ff), \n1 = {veclen(\x1,\y1)},
\n2 = {atan(\y1/\x1)+180*(\x1<0)}
in (ff) arc [radius=\n1, start angle=\n2, delta angle=\mult*\th]
coordinate (ee);
\fill[color=\C] (ee) circle (2 pt);

\node[C0] at (3.6,2.3) {$\w^{(1)}\theta_1$};
\node[C2] at (-1.5,3) {$\w^{(2)}\theta_2$};
\node[C1] at (-3.55,2.8) {$\w^{(4)}\theta_1$};
\node[C3] at (-1.15,-1.5) {$\w^{(3)}\theta_3$};
\node[C1] at (2.3,-2.85) {$\w^{(4)}\theta_3$};
\end{tikzpicture}
\caption{Rotations and jumps generating frequencies}
\label{fig:freqs}
\end{figure}

\begin{lemma}
\label{lem:wsol}
For any given $J\in\B R^n_+$, $\T\in\Delta^n$, there are infinitely
many bounded time periods
\[
T^{(k)} := k\,\frac{2\pi}{w^{(k)}}\,\ell, \com{or}
T^{(k)} := k\,\frac{2\pi}{w^{(k)}}, \quad \ell=1,
\]
such that the $k$-mode with corresponding reference period
$T^{(k)}$, that is $\mc T_k\big(T^{(k)}\big)$, has a zero divisor,
\end{lemma}

\begin{proof}
We have seen that $\d_k(J,\T;T)=0$ if and only if \eqref{weq} holds
for appropriate $k$.  To show that this uniquely defines $\w^{(k)}$,
we differentiate \eqref{gamdef}: inductively, we have
\[
\begin{aligned}
\frac{\del\g_1}{\del\w}
&= \frac{\del h}{\del x}\Big|_{(J_1,\w\t_0)}\,\t_0 \com{and}\\
\frac{\del\g_{m+1}}{\del\w}
&= \frac{\del h}{\del x}\Big|_{(J_{m+1},\w\t_m+\g_m)}\,
\Big(\t_m + \frac{\del\g_m}{\del\w}\Big).
\end{aligned}
\]
Now by \eqref{hder}, each of these terms is positive, and so
differentiating the left hand side of \eqref{weq}, we get
\[
\frac{\del}{\del\w}\Big(\g_n(\w)+\w\,\t_n\Big) > \t_n > 0,
\]
so by the implicit function theorem there exists a unique positive
solution $\w^{(k)}$ for each $k\ge1$.

We now show that $\w^{(k)}$ grows like $k$ for $k$ large.  Note from
\eqref{hdef} that $h(J,x)$ is always in the same quadrant as $x$,
and $|h(J,x)-x\big|<\frac\pi2$.  Applying this to \eqref{gamdef}, we
\[
\big|\g_1 - \w\t_0\big| < \frac\pi2, \com{and}
\big|\g_{m+1} - (\w\t_m+\g_n)\big| < \frac\pi2,
\]
for $m=1,\dots,n-1$.  Telescoping and using the triangle inequality
we get
\[
\Big|\g_n - \w\sum_{m=0}^{n-1}\t_m\Big| < n\,\frac\pi2,
\]
and finally using \eqref{weq} to eliminate $\g_n$ yields
\[
\Big|k\,\frac\pi2 - \w^{(k)}\sum_{m=0}^n\t_m\Big|
< n\,\frac\pi2,
\]
and since $\sum_m\t_m=1$, this yields
\[
(k-n)\,\frac\pi2 < \w^{(k)} < (k+n)\,\frac\pi2,
\]
and the boundedness of reference periods $T^{(k)}$ follows.
\end{proof}

Having described the frequencies $^(k)$ and corresponding periods
$T^(k)$, we again look for the conditions that determine resonance
of distinct frequencies.  We have shown above that each $^(k)$ can
be regarded as a single-valued, non-degenerate smooth function
\[
\w^{(k)}:\B R^n_+\times\Delta^n\to\B R, \qquad
\w^{(k)} = \w^{(k)}\big(J,\T\big),
\]
in which the simplex $Δ^n⊂R^n+1_+$ is independently
parameterized by the $^∘∈Δ^n_∘⊂R^n$.
Because the addition of extra entropy intervals introduces more
degrees of freedom, the following generalization of
Theorem~\ref{thm:T} to a general step entropy profile is unsurprising.

\begin{theorem}
\label{thm:T2}
There is a meagre subset $\mc Z\subset\Delta^n\times\B R^n$ of
($2n$-dimensional) measure zero, such that, for any
$(J,\T)\notin\mc Z$, \emph{every} frequency $\w^{(k)}$ is
non-resonant.  For each of the corresponding piecewise constant
entropy profiles, the $k$-mode with reference period $T^{(k)}$,
$k\ge 1$, is non-resonant, and so perturbs to a finite amplitude
solution of the nonlinear equation \eqref{F=0pw}.  This in turn
generates a space and time periodic solution of the nonlinear
Euler equations.
\end{theorem}

\begin{proof}
We proceed as in the proof of Theorem~\ref{thm:T}.  Thus we fix a
$k$-mode, and assume that the $j$ mode resonates with this $k$ mode.
The $k$-mode determines the reference period
$T^{(k)}=k\,2\pi/\w^{(k)}$, and resonance of the $j$-mode means that
also $\d_j\big(J,\T;T^{(k)}\big)=0$.  This in turn corresponds to
$\w = j\,2\pi/T^{(k)}$ being a solution of \eqref{weq}, say
\begin{equation}
\label{res}
\w = j\,\frac{2\pi}{T^{(k)}} = \w^{(p)}, \com{so that}
\w^{(p)} = q\,\w^{(k)}, \quad q :=\frac jk\in\B Q_+.
\end{equation}
Thus, fixing positive integers $k$, $j\ne k$ and $p$, we define
\[
\mc Z_{k,p,j} := \Big\{(J,\T)\in\B R^n_+\times\Delta^n\;\Big|\;
\w^{(p)} = q\,\w^{(k)}\Big\},
\]
where $\w^{(k)}$ and $\w^{(p)}$ are functions of $(J,\T)$, regarded
as known.  We wish to show that this is a non-degenerate constraint
on the parameters, which will imply that $\mc Z_{k,p,j}$ is a
codimension one submanifold, and so is small, in both the measure
and Baire senses.

We again get a more explicit version of the angles, and so also
$\w^{(k)}$, by using the fact that $\t_n = 1 - \sum_{m<n}\t_m$ in
\eqref{weq}, which gives, with $\w=\w^{(k)}$,
\begin{equation}
\label{woft}
\w = \sum_{m=0}^{n-1} \w\t_m - \g_n(\w) + k\,\frac\pi2.
\end{equation}
Now whenever $\t_m$ appears in \eqref{gamdef}, it is scaled by $\w$,
and this is the only context in which it appears.  We thus make the
substitution $\w\t_m\to x_m$ for $m<n$.  That is, we use $\T^\circ$
to parameterize the simplex, and scale this up to a vector
\[
x = (x_0,\dots,x_{n-1}) \in\B R^n, \com{by}
x_m = \w\,\t_m, \com{or} x = \w\,\T^\circ.
\]
We now use \eqref{gamdef} to define $\g_m$ as a function of $x$,
\begin{equation}
\label{gx}
\begin{aligned}
\g_1(x) &:= h(J_1,x_0), \com{and}\\
\g_{m+1}(x) &:= h\big(J_{m+1},x_m+\g_m\big), \quad m = 1,\dots,n-1,
\end{aligned}
\end{equation}
which now yields the explicit function $\g_n = \g_n(J,x)$.

We can combine this function and the scaling $x=\w\,\T^\circ$ to get
a geometric description of $w^{(k)}$, as follows.  Referring to
\eqref{woft}, we see that \eqref{weq} is equivalent to the coupled
\begin{equation}
\label{wxk}
\w^{(k)} = \sum_{m=0}^{n-1}x^{(k)}_m - \g_n(x^{(k)}) + k\,\frac\pi2, \qquad
x^{(k)}_m = \w^{(k)}\,\t_m.
\end{equation}
We thus interpret $\w^{(k)}$ as being determined by the intersection
of a ray $\w\,\T^\circ$ and the graph of the explicit function
$\sum x_m+\g_n(x)+k\frac\pi2$.

We now use \eqref{wxk} to re-express the resonance condition
\eqref{res} as a single equation: that is, we have
\[
\w^{(p)} = q\,\w^{(k)} \com{if and only if}
g(J,x) = (p-j)\,\frac\pi2,
\]
where we have defined the function
\begin{equation}
\label{gJx}
g:\B R^{2n}_+\to\B R \com{by}
g(J,x) := \g_n(q\,x) - q\,\g_n(x),
\end{equation}
and written
\[
x := x^{(k)}, \quad \w := \w^{(k)}, \quad
\w^{(p)} = q\,\w, \com{and} x^{(p)} = q\,x,
\]
with $q=j/k\in\B Q_+$ fixed and $\g_n$ is given by \eqref{gx}.  Here
the vector identity $x^{(p)} = q\,x^{(k)}$ follows from the second
equation in \eqref{wxk}.  Thus we have effectively described the
resonant set as a level surface of the explicit function $g$, namely
\[
\mc Z_{j,k,p} = \Big\{(J,x)\in\B R^{2n}\;\Big|\;
g(J,x) = (p-j)\,\frac\pi2\Big\},
\]
and we recover the original coordinates $(J,\T)\in\B
R^n_+\times\Delta^n$ by setting
\[
\T^\circ := \frac x{\w^{(k)}}, \com{and}
\t_n = 1 - \sum_{m=0}^{n-1}\t_m,
\]
in which $\w^{(k)}$ is given explicitly by \eqref{wxk}.

We will show that $g$ is non-degenerate, that is
\[
\nabla_{(J,x)}g \ne 0 \com{whenever} g(J,x) = b,
\]
for any nonzero constant $b\ne0$.  From the definition \eqref{gJx}, and
using \eqref{hder}, we calculate
\[
\begin{aligned}
\frac{\del g}{\del x_{n-1}}
&= \frac{\del h}{\del x}
\bigg|_{\big(J_n,q\,x_{n-1}+\g_{n-1}(qx)\big)}\,q
- q\,\frac{\del h}{\del x}
\bigg|_{\big(J_n,x_{n-1}+\g_{n-1}(x)\big)}\\[3pt]
&= \frac{q\,J_n\,(1+\G_{n-1}^2(qx))}{1 + J^2\,\G_{n-1}^2(qx)}
- \frac{q\,J\,(1+\G_{n-1}^2(x))}{1 + J^2\,\G_{n-1}^2(x)}\\[3pt]
&= \frac{q\,J\,(J^2-1)\,\big(\G_{n-1}^2(x)-\G_{n-1}^2(qx)\big)}
{\big(1 + J^2\,\G_{n-1}^2(qx)\big)\,\big(1 + J^2\,\G_{n-1}^2(x)\big)},
\end{aligned}
\]
where we have set
\[
\G_{n-1}(x) := \tf\big(x_{n-1}+\g_{n-1}(x)\big).
\]
It follows that
\[
\frac{\del g}{\del x_{n-1}} = 0
\com{iff}
\G_{n-1}(qx) = \pm\G_{n-1}(x),
\]
unless $J_n=1$.  Similarly,
\[
\begin{aligned}
\frac{\del g}{\del J_n}
&= \frac{\del h}{\del J}
\bigg|_{\big(J_n,q\,x_{n-1}+\g_{n-1}(qx)\big)}
- q\,\frac{\del h}{\del J}
\bigg|_{\big(J_n,x_{n-1}+\g_{n-1}(x)\big)}\\
&= \frac{\G_{n-1}(qx)}{1 + J^2\,\G_{n-1}^2(qx)}
- q\,\frac{\G_{n-1}(x)}{1 + J^2\,\G_{n-1}^2(x)}.
\end{aligned}
\]
Thus, if we assume $J_n\ne 1$, and assume both
\[
\frac{\del g}{\del x_{n-1}}\bigg|_{(J,x)} = 0 \com{and}
\frac{\del g}{\del x_{n-1}}\bigg|_{(J,x)} = 0,
\]
then we must have both
\[
\G_{n-1}(qx) = \pm\G_{n-1}(x) \com{and}
\G_{n-1}(qx) - q\,\G_{n-1}(x) = 0,
\]
which in turn implies $\G_{n-1}(qx) = \G_{n-1}(x) = 0$, so that, by
\eqref{gJx}, \eqref{gx} and \eqref{hdef}, we get
\[
g(J,x) = \arctan\big(J_n\,\G_{n-1}(qx)\big)
- q\,\arctan\big(J_n\,\G_{n-1}(x)\big) = 0.
\]

On the other hand, if $J_n=1$, which is the degenerate case, we
carry out the same calculation for $\frac{\del g}{\del x_{n-2}}$ and
$\frac{\del g}{\del J_{n-1}}$, and continue by backward induction as
necessary.  If all $J_m=1$, which is the most degenerate isentropic
case, then since $h(1,x)=x$, we get $g = 0$ identically.

We have thus shown that, as long as at least one $J_m\ne 1$,
\[
\nabla_{(J,x)} g = 0 \com{implies} g(J,x) = 0.
\]
The implicit function theorem now tells us that non-zero level sets
of $g$, and in particular $\mc Z_{k,p,j}$, are codimension one
submanifolds of $\B R^{2n}$, and so these are again both measure
zero and nowhere dense.

As before, we now set
\[
\mc Z := \bigcup_{k,p,j}\mc Z_{k,p,j},
\]
and since this is a countable union, $\mc Z$ has measure zero and is
meagre.  Since $\mc Z$ contains all possible resonances, the proof
is complete.
\end{proof}

\section{Generalizations}
\label{sec:SL}

We return now to the compressible Euler equations \eqref{lagr} with a
general constitutive law, $p = p(v,s)$, in which we solve for $v$, so
that $v=v(p,s)$.  For classical solutions, in a Lagrangian frame, we
write the system as
\begin{equation}
\label{genls}
p_x + u_t = 0, \qquad
u_x - v(p,s)_t = 0,
\end{equation}
or, in quasilinear form,
\[
p_x + u_t = 0, \qquad
u_x - v_p(p,s)\,p_t = 0,
\]
where we are again evolving in $x$, and since $s_t=0$, we regard the
entropy as a given fixed function $s(x)$.  As above we make the
observation that the nonlinear equations respect the symmetry $p$
even, $u$ odd, as functions of $t$.  That is, if we specify
$p(x_0,·)$ even, $u(x_0,·)$ odd at one point $x_0$, then this
is satisfied throughout the interval of classical existence, that is
as long as gradients remain finite.

\subsection{Minimal Tile for Periodicity and Boundary Conditions}

We again build a tile which generates a periodic solution by assuming
the data is time periodic and imposing extra symmetry conditions at
the ends $x=0$ and $x=ℓ$ of the evolution.  As in \eqref{olsymm},
\eqref{ols}, we impose a regular reflective boundary condition at
$x=0$: that is, we impose the \emph{acoustic reflection boundary
condition} $u(0,·)=0$, with $p(0,·)$ even and periodic.
Physically, this acoustic boundary condition describes pure (lossless)
reflection of a sound wave off a wall.  On the other hand, in
\eqref{ulsymm}, \eqref{uls}, the boundary condition which is a shifted
reflection, provides a mechanism for the generation of periodic tiles,
but is not realizable as a simple physical boundary condition.  This
is because when reflecting the profile around $x=ℓ$ to generate a
(half) periodic tile, the velocity need not actually vanish at
$x=ℓ$.  This can be explained by noting the effect of the
quarter-period shift $S^T/4$ on a $k$-mode: we have
\[
\mc S^{T/4}\s\big(k\piot t\big) =
\s\big(k\piot t-k{\TS{\frac\pi2}}\big) =
\begin{cases}
\pm\c\big(k\piot t\big), & k \text{ odd},\\
\pm\s\big(k\piot t\big), & k \text{ even},
\end{cases}
\]
and similarly for $S^T/4(kt)$.  It follows that,
for $k$-modes with $k$ even,
\[
\IR-\,\mc S^{T/4}\,\mc T_{k} = 0 \com{iff}
\IR-\,\mc T_{k} = 0,
\]
while this fails for odd $k$-modes.  As a consequence, at least at the
linear level, if we double the time frequency of tones, then we can
drop the $1/4$-period shift in posing the right boundary condition,
and this then becomes an acoustic reflection boundary condition at
$x=ℓ$, exactly as at $x=0$.

It follows that we can pose two problems that effectively have the
same method of solution:  as in previous sections, we assume $p$ even
and $u$ odd as functions of $t$, and set
\begin{equation}
\label{ypu}
y(x,t) := p(x,t) + u(x,t),
\end{equation}
and define a nonlocal scalar flux by
\[
g(y) := u + v(p,s) = \IR-y + v\big(\IR+y,s\big),
\]
so that $y$ solves the nonlocal scalar conservation law
\begin{equation}
\label{ycl}
y_x + g(y)_t = 0.
\end{equation}
Note that this is distinct from our earlier treatment of piecewise
constant entropy, in which we used the isentropic equations at each
entropy level together with the rescaling of $x$, whereas here we use
the full equations and unscaled $x$ directly.

The two problems we then pose are:
\begin{itemize}
\item the \emph{periodic tile problem}, namely
\begin{equation}
\label{yper}
\mc F_P(y^0) := \IR-\,\mc S^{-T/4}\,\mc E^\ell\,y^0 = 0;
\end{equation}
\item the \emph{acoustic boundary value problem}, namely
\begin{equation}
\label{yab}
\mc F_A(y^0) := \IR-\,\mc E^\ell\,y^0 = 0.
\end{equation}
\end{itemize}
In both of these problems, as above, $E^ℓ$ denotes nonlinear
evolution through the varying entropy profile from $x=0$ to $x=ℓ$,
and the data $y^0=y^0(t)$ is again assumed to be even and $T$-periodic.

For a given entropy profile $s(x)$, each solution of \eqref{yper}
generates a space and time periodic solution, generated by a
reflection at the left boundary and a shifted reflection on the right,
so the space period of the solution is $4ℓ$.  On the other hand, a
solution of \eqref{yab} generates a periodic solution by just one
reflection in $x$, so has space period $2ℓ$.  The two problems are
related as follows: any $k$-mode solution of the linearization of
\eqref{yper} with $k$ even is also a solution of that of \eqref{yab},
and conversely an even $2j$-mode linearized solution of \eqref{yab}
can also be realized as a solution of the linearization of
\eqref{yper}.  Moreover, if these linearized solutions perturb, then
by uniqueness they will coincide as solutions of the corresponding
nonlinear problems.

It appears that \eqref{yper} generates more solutions than
\eqref{yab}, but we regard \eqref{yab} as a more physically relevant
problem.  This is because the acoustic boundary condition \eqref{yab}
is simply a reflection off a wall, so the system models sound waves
bouncing between two walls which bound an unrestricted varying entropy
profile on $[0,ℓ]$.  On the other hand, the velocity $u$ need not
vanish at $x=ℓ$ with \eqref{yper}, and it is hard to ascribe
\eqref{yper} to a physical condition, because we expect that
controlling an entropy profile to be perfectly symmetric between two
walls (at $x=0$ and $x=2ℓ$) is practically infeasible.  However, we
regard periodic solutions in which compression and rarefaction are in
perfect balance as being of fundamental importance.

\subsection{Linearization and Sturm-Liouville systems}

We now analyze \eqref{genls} following the same steps as before.  We
regard the entropy field $s(x)$ as a given piecewise continuous
function on the interval $x∈[0,ℓ]$, continuous at the endpoints,
and assume a constitutive law
\[
v = v(p,s), \com{with} v_p<0,
\]
and we will make further assumptions as necessary.

We begin by noting that the quiet state $(p,0)$ is a solution of the
Euler equations satisfying the boundary conditions; although it is a
constant solution of \eqref{genls}, it is part of a non-constant
standing wave solution of \eqref{lagr}.  Linearizing \eqref{genls}
around $(p,0)$ gives the system
\begin{equation}
\label{UP}
\begin{gathered}
P_x + U_t = 0, \qquad
U_x + \sg^2\,P_t = 0,\\ \com{where}
\sg = \sg(x) := \sqrt{-v_p(\ol p,s)},
\end{gathered}
\end{equation}
in which we use the convention that $(P,U)$ solve linearized
equations, while $(p,u)$ solve the nonlinear system.

We use separation of variables to solve \eqref{UP}, in which we look
for $T$-periodic solutions of \eqref{UP} in which $P$ and $U$ are even
and odd, respectively, so we set
\begin{equation}
\label{UPsub}
\begin{aligned}
P(x,t) &:= \sum_{n\ge 0} \vp_n(x)\,\c\big(n\piot t\big), \\
U(x,t) &:= \sum_{n> 0} \psi_n(x)\,\s\big(n\piot t\big).
\end{aligned}
\end{equation}
Plugging in to \eqref{UP} and simplifying, we get the ODE system
\begin{equation}
\label{SL1}
\dot\vp_n + \w\,\psi_n = 0, \qquad
\dot\psi_n - \sg^2\,\w\,\vp_n = 0,  \qquad
\w := n\piot,
\end{equation}
where $□̇$ denotes $d/dx□$.

We denote nonlinear evolution through $x$ according to \eqref{ycl} by
$E^x$, and we denote the linearization of this around the constant
quiet state $(p,0)$ by $L^x := DE^x(p,0)$.  That is,
$L^x$ denotes evolution by \eqref{UP}, or more precisely by the
linear nonlocal scalar equation for $Y := P+U$,
\begin{equation}
\label{Ylin}
Y_x + \big(\IR-Y+\sg^2\,\IR+Y\big)_t = 0.
\end{equation}
We again use the notation \eqref{Tk}, so that \eqref{UPsub} can be
\begin{equation}
\label{YasT}
Y(x,t) = P + U = \sum_{n\ge 0} \mc T_n\,\(\vp_n(x)\\\psi_n(x)\),
\end{equation}
and we obtain an analogue of Lemma~\ref{lem:Tk}, whose proof is

\begin{lemma}
\label{lem:Lx}
The linearized evolution operator $\mc L^x = D\mc E^x(\ol p,0)$ acts
on the $k$-mode row vector $\mc T_k = \mc T_k(T)$ given by
\eqref{Tk}, by
\[
\mc L^x\,\mc T_k = \mc T_k\,\Psi\big(x;k\piot\big), \com{so}
\mc L^x\,\mc T_k\(a\\b\) = \mc T_k\,\Psi\(a\\b\),
\]
where $\Psi = \Psi(x;\w)$ is the fundamental solution of
\eqref{SL1}, so satisfies
\begin{equation}
\label{Psi}
\dot\Psi(x;\w) = \w\,
\(0&-1\\\sg^2(x)&0\)\,\Psi(x;\w),
\qquad \Psi(0;\w) = I.
\end{equation}
\end{lemma}

Alternatively, we can express the linearized system \eqref{UP} as a
wave equation with varying speed, namely
\begin{equation}
\label{Peq}
P_{xx} - \sg^2\,P_{tt} = 0,
\end{equation}
and after use of \eqref{UPsub}, by separation of variables, we get the
second-order linear system
\begin{equation}
\label{SL2}
\ddot \vp_n + \big(n\piot\big)^2\,\sg^2\,\vp_n = 0,
\end{equation}
which is equivalent to \eqref{SL1}.

We now consider boundary values for these ODE systems.  In both
\eqref{yper} and \eqref{yab}, the data $y^0$ (and so also $Y^0$) posed
at $x=0$ is even, so using \eqref{ypu} and \eqref{UPsub}, this becomes
the condition
\begin{equation}
\label{bc0}
U(0,\cdot) = 0, \com{equivalently} \psi_n(0) = \dot\vp_n(0) = 0.
\end{equation}
If we are solving \eqref{yab}, we get the same condition at $x=ℓ$,
\begin{equation}
\label{bcab}
U(\ell,\cdot) = 0, \com{equivalently} \psi_n(\ell) = \dot\vp_n(\ell) = 0.
\end{equation}
On the other hand, if we are solving \eqref{yper}, the boundary
condition is
\[
\begin{aligned}
0 &= \IR-\,\mc S^{-T/4}\big(P(\ell,\cdot)+U(\ell,\cdot)\big)\\
&= \IR-\sum\Big(\vp_n(\ell)\,\c\big(n\piot t-n{\TS\frac\pi2}\big)
+\psi_n(\ell)\,\s\big(n\piot t-n{\TS\frac\pi2}\big)\Big),
\end{aligned}
\]
which yields the conditions
\begin{equation}
\label{bcper}
\begin{gathered}
\dot\vp_n(\ell) = \psi_n(\ell) = 0, \quad n \text{ even},\\
\vp_n(\ell) = \dot\psi_n(\ell) = 0, \quad n \text{ odd}.
\end{gathered}
\end{equation}

Our boundary conditions can now be expressed succinctly using the
fundamental solution \eqref{Psi}, in analogy with Lemma~\ref{lem:pws},
as follows.

\begin{lemma}
\label{lem:fund}
Suppose that constant ambient pressure $\ol p$, entropy profile
$s(x)$ and reference period $T$ are given.  Then the constant quiet
state $y^0=\ol p$ solves \eqref{yper} and \eqref{yab}, and the
respective linearizations around $\ol p$ are
\[
D\mc F_P(\ol p) = \IR-\,\mc S^{-T/4}\,\mc L^\ell \com{and}
D\mc F_A(\ol p) = \IR-\,\mc L^\ell,
\]
and moreover the nonlinear functionals $\mc F_P$ and $\mc F_A$
factor as
\begin{equation}
\label{Ngenl}
\mc F_P = \IR-\,\mc S^{-T/4}\,\mc L^\ell\,\mc N^\ell \com{and}
\mc F_A = \IR-\,\mc L^\ell\,\mc N^\ell,
\end{equation}
respectively, where
$\mc N^\ell := \big(\mc L^\ell\big)^{-1}\mc E^\ell$ in both cases.
The linearized operators respect $k$-modes, and we get
\[
\begin{aligned}
D\mc F_P(\ol p)\Big[\c\big(k\piot t\big)\Big]
&= \d_{P,k}(T)\,\s\big(k\piot t\big) \com{and}\\
D\mc F_A(\ol p)\Big[\c\big(k\piot t\big)\Big]
&= \d_{A,k}(T)\,\s\big(k\piot t\big),
\end{aligned}
\]
where the $k$-th divisor is given by
\begin{equation}
\label{dkcts}
\begin{aligned}
\d_{P,k}(T) &:= \(0&1\)\,P^{-k}\,
\Psi\big(\ell;k\piot\big)\(1\\0\) \com{or}\\
\d_{A,k}(T) &:= \(0&1\)\,
\Psi\big(\ell;k\piot\big)\(1\\0\),
\end{aligned}
\end{equation}
for the periodic or acoustic boundary conditions, respectively.
\end{lemma}

It follows that the single mode $(kt)$ provides a
solution given by $T_k Ψ(x;k)$ of the linearized
problem satisfying the boundary conditions if and only if
\[
\d_{P,k}(T)=0 \com{or} \d_{A,k}(T)=0,
\]
respectively.  In this case,
solves \eqref{SL1}, and since it satisfies the boundary conditions, it
is an eigenfunction of the Sturm-Liouville problem \eqref{SL2},
corresponding to eigenvalue $λ_k := (k)^2$.  As in
previous sections, we expect that if the corresponding mode is
nonresonant, then this should perturb to a nonlinear pure tone
solution of \eqref{genls}.

We are thus led to introduce the Sturm-Liouville (SL) operator
\begin{equation}
\label{SLop}
\mc L := - \frac1{\sg^2}\,\frac{d^2}{dx^2}, \com{so that}
\mc L \phi = - \frac1{\sg^2}\,\ddot\phi,
\end{equation}
and to introduce the associated SL eigenvalue problem,
\begin{equation}
\label{SLev}
\mc L\,\vp = \lambda\,\vp, \com{which is}
- \ddot\vp = \lambda\,\sg^2\,\vp,\com{on} 0<x<\ell,
\end{equation}
and subject to the boundary conditions \eqref{bc0} at $x=0$ and either
\eqref{bcper} or \eqref{bcab} at $x=ℓ$.  We note that the boundary
conditions imply self-adjointness, implying that this is a
\emph{regular SL problem} as long as the weight
$^2(x) = -v_p(p,s(x))$ is a positive, bounded piecewise
continuous function~[28, 1, 5].  In particular, the
setup we have here is a direct generalization of the cases studied
earlier in this paper, and the results here apply unchanged in that
context.  Moreover, because the corresponding ODEs are linear, the
solutions can be expressed as integrals, and our methods extend to BV
entropy profiles with only minor technical changes.

We collect some well-known classical results for regular SL eigenvalue
problems in the following lemma; see [28, 5] for details.
This is the direct generalization of the first statement of
Lemma~\ref{lem:Tj}; indeed, the existence of infinitely many reference
periods for a piecewise constant entropy profile is a special case of
the classical Sturm-Liouville theory.  Many of the statements in this
lemma will be explicitly verified below.

\begin{lemma}
\label{lem:SLprops}
Assume that the entropy profile is piecewise continuous.  The
Sturm-Liouville system \eqref{SLev}, with boundary conditions
\eqref{bc0} and \eqref{bcper} or \eqref{bcab}, has infinitely many
eigenvalues $\lambda_k$.  These are positive, simple, monotone
increasing and isolated, and satisfy the growth condition
$\lambda_k=O(k^2)$.  The corresponding eigenfunctions form an
orthogonal $L^2$ basis in the Hilbert space with weight function
\end{lemma}

We label the eigenvalues $λ_k$ and corresponding eigenfunctions
$_k$, scaled so that $_k(0)=1$, so that \eqref{SLev} and
\eqref{SL1} hold for each $k≥1$, and the $λ_k$'s are
increasing with $k$.  Because we prefer to work with the frequencies,
we define the $k$-th \emph{eigenfrequency} $_k$ and corresponding
\emph{reference period} $T_k$ by
\begin{equation}
\label{Tn}
\w_k := \sqrt{\lambda_k}, \com{and}
T_k := k\,\frac{2\pi}{\w_k} = k\,\frac{2\pi}{\sqrt{\lambda_k}},
\end{equation}
respectively.  It follows that the $k$-modes
\[
\begin{aligned}
Y_k(x,t) &:= \mc T_k(T_k)\,\Psi\big(x;k{\TS\frac{2\pi}{T_k}}\big)
\,{\TS\begin{pmatrix}1\\0\end{pmatrix}}
\com{or}\\
P_k(x,t) &:= \vp_k(x)\,\c\big(k{\TS\frac{2\pi}{T_k}}t\big) = \vp_k(x)\,\c(\w_kt),
\end{aligned}
\]
solve \eqref{Ylin} or \eqref{Peq}, respectively, while also satisfying
the appropriate boundary conditions.  These $k$-modes lie in the
kernel of the linearized operator $DF(p)$, and we will show
that these perturb to pure
tone solutions of the nonlinear problem.  We note that for given $k$,
the frequency $_k$ is determined by the entropy profile, and this in
turn yields the reference period.  Consideration of other resonant and
nonresonant modes then refers to this fixed period $T_k$.
\subsection{Nonresonant modes}

As in our above development, we wish to perturb $Y_k$ (or $P_k$) to a
time periodic solution of the corresponding nonlinear equation.
Following \eqref{y0}, and having identified the reference period,
we consider perturbations of the initial data of the form
\begin{equation}
\label{pert}
y^0(0) = p^0(t) := \ol p + \a\,\c\big(k{\TS\frac{2\pi}{T_k}}t\big)
+ z + \sum_{j\ge 1,j\ne k}a_j\,\c\big(j{\TS\frac{2\pi}{T_k}}t\big),
\end{equation}
in which the reference period $T_k$ is determined by the choice of
SL eigenvalue $λ_k$ in \eqref{Tn}, and we now regard this as

We again declare the $k$-mode to be \emph{nonresonant} if no other
$j$-modes satisfy the boundary condition, that is
\[
\d_{P,j}(T_k)\ne0 \com{or} \d_{A,j}(T_k)\ne0,
\com{for all} j\ne k,
\]
respectively for the appropriate boundary conditions \eqref{yper} or
\eqref{yab}.  Here we note that the reference period $T_k$ is
fixed, while $j$ varies, and the collection of $j$-mode basis elements
\[
\big\{\mc T_j(T_k)\;|\;j\ge 1\big\} =
\Big\{\(\c(j\piotk t)&\s(j\piotk t)\)\;|\;j\ge 1\Big\}
\]
span the even and odd functions of period $T_k$, respectively.

\begin{lemma}
\label{lem:nonres}
The $j$-mode is resonant with the fixed $k$-mode if and only if two
distinct SL frequencies are rationally related: that is,
\[
\d_j(T_k) = 0 \com{iff}
k\,\w_p = j\,\w_k,
\]
for some index $p = p(j)\ne k$.
\end{lemma}

\begin{proof}
By construction, we have
\[
\d_k(T_k) = 0, \com{with}
T_k := k\,\frac{2\pi}{\w_k}.
\]
If $\d_j(T_k) = 0$, then $j\frac{2\pi}{T_k}$ also corresponds to some
SL frequency, and since $j\ne k$, we must have
\[
\w_p = j\frac{2\pi}{T_k} \com{for some}
p\ne k,
\]
and substituting in $T_k$ gives the result.
\end{proof}

The resonance or nonresonance of modes corresponding to SL eigenvalues
depends on the entropy profile and constitutive law.  Moreover,
because the SL eigenvalues, and hence also frequencies, vary
continuously with the entropy profile, we expect that in a strong
sense, as in the previous cases, the set of profiles with resonant
modes should be small, so that generically, all modes should be

\subsection{Angle Variable}

Once again we treat the general entropy profile as given, and solve
for the base frequencies $_k$ that yield $k$-mode solutions of the
linearized equation.  Starting with the linearization \eqref{UP}, we
separate variables by setting
\[
P(x,t) := \vp(x)\,\c(\w\,t), \qquad
U(x,t) := \psi(x)\,\s(\w\,t),
\]
which yields the SL system \eqref{SL1}, namely
\begin{equation}
\label{SL3}
\dot\vp + \w\,\psi = 0, \qquad
\dot\psi - \w\,\sg^2\,\vp = 0,
\end{equation}
with $^2 = - v_p(p,s) = ^2(x)$.  According to \eqref{bc0},
we first consider initial values
\[
\vp(0) = c_0 \com{and} \psi(0) = 0,
\]
for appropriate $c_00$.  Finding the values of $_k$ that meet
the boundary conditions \eqref{bcab} or \eqref{bcper} will then
determine the appropriate reference period
$T_k:=k 2π/_k$.

As we have seen earlier, the most important part of the evolution is
the angle $=(x)$ of the vector $(,ψ)$.  This can be
effectively captured with the use of \emph{modified Pr\"ufer
variables}, which are
\begin{equation}
\label{pruf}
\vp(x) := r(x)\,\frac1{\rho(x)}\,\c\big(\t(x)\big), \qquad
\psi(x) := r(x)\,\rho(x)\,\s\big(\t(x)\big),
\end{equation}
see [28, 5, 1].  Here we interpret $r(x)$ as the
radial length or amplitude, $ρ(x)$ as the eccentricity or aspect,
and $(x)$ as the angle variable.  This is a degenerate description
in which we are free to choose the aspect $ρ(x)>0$, and having done
so, both $r(x)$ and $(x)$ will be determined by the equations.
Plugging in \eqref{pruf} into \eqref{SL3} and simplifying, we get the
\[
\begin{aligned}
\frac{\dot r}r\,\c(\t) - \frac{\dot\rho}{\rho}\,\c(\t)
- \s(\t)\,\dot\t + \w\,\rho^2\,\s(\t) &= 0, \\
\frac{\dot r}r\,\s(\t) + \frac{\dot\rho}{\rho}\,\s(\t)
+ \c(\t)\,\dot\t - \w\,\frac{\sg^2}{\rho^2}\,\c(\t) &= 0,
\end{aligned}
\]
with initial conditions
\[
\t(0) = 0 \com{and} r(0) = 1,
\com{so} c_0 = \frac1{\rho(0)}.
\]
After use of elementary trig identities, this in turn becomes
\[
\begin{aligned}
\frac{\dot r}r - \frac{\dot\rho}{\rho}\,\c(2\t)
+ \w\,\Big(\rho^2-\frac{\sg^2}{\rho^2}\Big)\s(\t)\,\c(\t) &=0, \\
\dot\t + \frac{\dot\rho}{\rho}\,\s(2\t)
- \w\,\Big(\frac{\sg^2}{\rho^2}\,\c^2(\t) + \rho^2\,\s^2(\t)\Big) &=0.
\end{aligned}
\]
Thus the system is reduced to a nonlinear scalar ODE for the angle
$(x)$, coupled with an integration for the amplitude $r(x)$.
Moreover, it is now clear that we should choose $ρ$ such that
\begin{equation}
\label{rho}
\rho^2 = \frac{\sg^2}{\rho^2}, \com{that is}
\rho (x) := \sqrt{\sg(x)}.
\end{equation}
With this choice the equations simplify further, and we get the scalar
equation for $(x)$,
\begin{equation}
\label{theta}
\dot\t = \w\,\sg - \frac{\dot\sg}{2\sg}\,\s(2\t),
\qquad \t(0) = 0,
\end{equation}
coupled with a linear homogeneous equation for $r(x)$, namely
\[
\frac{\dot r}r = \frac{\dot\sg}{2\sg}\,\c(2\t),
\qquad r(0) = 1,
\]
which immediately yields the quadrature
\begin{equation}
\label{rint}
r(x) = \exp\Big\{ \int_0^x \c\big(2\t(y)\big)\;d\log\sqrt\sg(y)\Big\}.
\end{equation}

We note that \eqref{theta} is consistent with our treatment of
piecewise constant entropy; in that case, we have $=const$ on
intervals, while each jump contributes a $$̣-function.  Thus the
angle changes linearly on each entropy interval, with a finite jump in
angle (independent of $\w$) at each entropy jump.  In particular, our
framework extends without change to bounded measurable entropy
profiles for which $\log\sg$ has bounded variation.  Moreover, in the
prototypical case of a $\g$-law gas, $\log\sg$ is a multiple of the

As above, we are interested in characterizing the base $k$-mode
frequencies and periods, that yield periodic solutions of the
linearized equations.  That is, we want to characterize periods $T_k$,
such that the $k$-mode $\mc T_k(T_k)$ satisfies the boundary
conditions (<ref>) or (<ref>), respectively, or

\[
\d_{A,k}(T_k)=0 \com{or} \d_{P,k}(T_k)=0,
\]

respectively, these being given by (<ref>).  Using (<ref>),
we express the boundary conditions in terms of the angle variable
$\t(x)$, as follows.  For the periodic boundary condition
(<ref>), we require

\begin{equation}
\label{kodd}
\vp_k(\ell) = 0, \com{so} \t(\ell) = \Big(n+\frac12\Big)\pi,
\end{equation}

for $k$ odd, and for $k$ even or the acoustic boundary condition, we

\begin{equation}
\label{keven}
\psi_k(\ell) = 0, \com{so} \t(\ell) = n\,\pi,
\end{equation}

for some $n$.

We can methodically enumerate all such conditions in terms of the
frequency $\w$ and angle $\t(x) = \t(x,\w)$, using the angle
boundary condition

\begin{equation}
\label{abc}
\t(\ell,\w) = k\,\frac\pi2,
\end{equation}

which we interpret as an implicit condition for the coefficient
$\w=\w_k$ of (<ref>).  Integrating, this becomes the condition

\begin{equation}
\label{wkexp}
k\,\frac\pi2 = \w_k\,\int_0^\ell\sg\;dy -
\int_0^\ell \s\big(2\t(y,\w_k)\big)\;d\log\sqrt\sg(y),
\end{equation}

which is an implicit equation for $\w_k$.

Thus, given an entropy profile, we find the base reference period
$T_k$ of the $k$-mode by solving (<ref>) for $\w_k$, where
$\t(x,\w)$ solves (<ref>), and then using (<ref>).  Each such
$k$-mode with reference period $T_k$ then determines a periodic
problem of the linearized equation.

Note that (<ref>) is equivalent to the periodic boundary condition
(<ref>), but permits more frequencies than the acoustic
reflection boundary condition (<ref>) allows, these latter two
being equivalent only for even values of $k$.  However, the even modes
form a closed subspace, so if we start from $a_m=0$ for all odd $m$,
this persists when we perturb to the nonlinear problem.  Thus for
notational convenience, we use (<ref>) to identify all
frequencies, with the understanding that if we are using (<ref>),
all linearizations and perturbations are restricted to even modes.

For each integer $k\ge 1$, and any entropy profile $s(x)$ such that
$\log\sg\in BV[0,\ell]$, there is a unique $\w_k$ such that the
solution $\t(x)$ of (<ref>) satisfies (<ref>).  Moreover,
$\{\w_k\}$ is a monotone increasing sequence which grows like $k$,
that is $\w_k/k\in[1/C,C]$ for some constant $C>0$.

We will show that for any fixed $x>0$, the function $\t(x)$ is
strictly monotone increasing as a function of $\w$.  In particular,
taking $x=\ell$ in (<ref>) implies that $\w_k$ exists for each
$k\ge1$ and is increasing.  To get the growth rate for $\w_k$, we
use (<ref>) to write

\[
\bigg|\,\w_k\,\int_0^\ell\sg\;dy - k\,\frac\pi2\bigg|
\le \int_0^\ell d\,|\!\log\sqrt\sg(y)|.
\]

To show monotonicity of $\t(\cdot,\w)$, we note that $\sg$ is
determined by the entropy $s(x)$ and differentiate the ODE
(<ref>) with respect to $\w$.  Denoting
$\frac{\del\t}{\del \w}$ by $\zeta$, this yields

\begin{equation}
\label{zeta}
\dot\zeta = \sg - \frac{\dot\sg}{\sg}\,\c(2\t)\,\zeta,
\qquad \zeta(0) = 0,
\end{equation}

which by (<ref>) can also be written

\[
\dot\zeta = \sg - 2\,\frac{\dot r}{r}\,\zeta,
\qquad \zeta(0) = 0.
\]

It follows that $r^2$ is an integrating factor, and we integrate to

\[
r^2(x)\,\zeta(x) = \int_0^xr(y)^2\,\sg(y)\;dy.
\]

Thus $\zeta(x)>0$ for $x>0$, and the proof is complete.

§.§ Perturbation and Auxiliary equation

Assuming now that the $k$-th mode is nonresonant, we show that the
$k$-mode bifurcates to a time periodic solution of the compressible
Euler equations with periodic or acoustic reflective boundary
conditions (<ref>) or (<ref>), respectively.  As noted
above, it is enough to consider the periodic condition (<ref>)
only, and restrict our attention to even modes when using the
reflective condition (<ref>).

Our strategy is unchanged: we build Hilbert spaces as in
(<ref>) and (<ref>), but separating the $k$-mode rather
than the 1-mode.  The analogues of Lemmas <ref>
and <ref> then follow in exactly the same way.  For clarity and
brevity, we define the spaces and state the lemmas without proof.

For $k$ fixed, the angle boundary condition (<ref>), or
equivalently (<ref>), determines the base frequency $\w_k$, and
by (<ref>) this gives both the SL eigenvalue $\lambda_k$ and
reference period $T=T_k$, namely

\[
\lambda_k := \w_k^2 \com{and} T_k := k\,\frac{2\pi}{\w_k},
\]

respectively, and while considering other modes, this period $T=T_k$
remains fixed.

Following (<ref>), we define

\begin{equation}
\label{Hsplit2}
\begin{aligned}
\mc H_1 &:= \big\{z + \a\,\c(k\piotk t)\;\big|\;z, \a\in\B R\big\}
\com{and}\\
\mc H_2 &:= \Big\{ \sum_{j\ne k}a_j\,\c(j\piotk t)\;\Big|
\;\sum_{j\ne k} a_j^2\,j^{2s}<\infty\Big\},
\end{aligned}
\end{equation}

so the domain is $H^s = \mc H_1\oplus\mc H_2$.  Similarly, from
(<ref>), we describe the range $\mc H$ by

\begin{equation}
\begin{aligned}
\mc H_+ &:= \Big\{ y = \sum_{j\ne k}a_j\,\s(j\piotk t)\;\Big|
\;\|y\|<\infty\Big\}, \com{and}\\
\mc H &:= \big\{ \beta\,\s(k\piotk t) \big\} \oplus \mc H_+, \com{with norm}\\
\|y\|^2 &:= \beta^2 + \sum_{j>1} a_j^2\,\d_j^{-2}\,j^{2s},
\end{aligned}
\label{Hplus2}
\end{equation}

and where we have set

\[
\d_j := \d_{P,j}(T_k) \com{or} \d_j := \d_{A,j}(T_k),
\]

respectively for the periodic or acoustic boundary condition.  We
similarly define the orthogonal projection,

\[
\Pi:\mc H\to \mc H_+ \com{by}
\Pi\Big[\beta\,\s(k\piotk t)+\sum_{j\ne k}a_j\,\s(jt)\Big] :=
\sum_{j\ne k}a_j\,\s(jt),
\]

which projects onto all but the $k$-mode.

Now let $\mc F$ denote $\mc F_P$ or $\mc F_A$, given by (<ref>),
depending on whether periodic or acoustic boundary conditions
(<ref>) or (<ref>) are used, respectively.  Note that the
data consists only of even functions and $\mc F$ incorporates the
projection onto odd modes, and the Hilbert spaces respect this
structure, being spaces of even and odd functions for the domain and
range, respectively.

As above, the nonlinear problem is treated as a
bifurcation problem, which consists of an infinite dimensional
auxiliary equation, namely

\begin{equation}
\label{auxil}
\Pi\,\mc F\big(y^0\big) = 0, \com{with}
y^0 = \ol p + z + \a\,\c(k\piotk t) + W,
\end{equation}

together with a scalar bifurcation equation, namely

\begin{equation}
\label{bifurc}
\Big\langle \s(k\piotk t), \mc F\big(y^0\big)\Big\rangle = 0.
\end{equation}

Here the amplitude $\a$ parameterizes the $k$-mode linear solution
that we are perturbing, $z$ is the free parameter of the 0-mode, which
is always in the kernel, and $W$ is the nonlinear correction off the
kernel.  As usual, we first solve (<ref>) to get a solution
$W(\a,z)$ uniform in $\a$ and $z$, and then we solve for the 0-mode
correction $z$ as a function of $\a$.  Here $z$ is the 0-mode
correction that gives a nonzero derivative in the bifurcation equation
as a consequence of genuine nonlinearity.  Physically, this is a
correction to the ambient pressure that balances rarefaction and
compression, and ultimately this is the physical mechanism for
avoiding shock formation and generating time periodic solutions.

The following summarizes the conclusions of Lemmas <ref>
and <ref>, and follows from the implicit function theorem
exactly as those lemmas do.

If the $k$-mode is nonresonant, there is a neighborhood
$\mc U\subset\mc H_1$ of the origin and a unique $C^1$ map

\[
W:\mc U\to\mc H_2, \com{written}
W\big(\ol p + z+\a\,\c(k\piotk t)\big) =: W(\a,z) \in\mc H_2,
\]

such that, for all $z+\a\,\c(k\piotk t)\in\mc U$, we have
a solution of the auxiliary equation (<ref>), given by

\[
\Pi\,\mc F\Big(\ol p + z+\a\,\c(k\piotk t) + W(\a,z)\Big) = 0.
\]

Moreover, the map $W(\a,z)$ satisfies the estimate

\[
W(\a,z) = o(|\a|),
\]

uniformly for $z$ in a neighborhood of 0.

§.§ Bifurcation equation

It remains to solve the bifurcation equation (<ref>), which is
scalar.  Proceeding as before, we define scalar functions $f$ and $g$
in analogy with (<ref>), (<ref>).  That is, we set

\begin{equation}
\label{fdef}
f(\a,z) := \Big\langle \s(\w\, t),
\mc F\big(\ol p + z+\a\,\c(\w\, t) + W(\a,z)\big)\Big\rangle,
\end{equation}

where $\w := \w_k = k\,2\pi/T_k$, and we set

\[
\begin{aligned}
g(\a,z) &:= \frac1\a\, f(\a,z), \quad \a\ne0,\\[2pt]
g(0,z) &:=  \frac{\del f}{\del\a}(0,z).
\end{aligned}
\]

Our goal is to show that there is some $z=z(\a)$ such that

\begin{equation}
\label{fgbif}
f\big(\a,z(\a)\big) = 0, \com{or equivalently}
g\big(\a,z(\a)\big) = 0.
\end{equation}

As above, we cannot apply the implicit function theorem to $f$
directly, because $\frac{\del f}{\del z}\big|_{\a=0}=0$, but we can
apply it to $g$: to do so, we must show that

\begin{equation}
\label{faz}
\frac{\del g}{\del z}\Big|_{(0,0)} \ne 0, \com{which is}
\frac{\del^2f}{\del z\,\del\a}\Big|_{(0,0)} \ne 0.
\end{equation}

We can rewrite $f(\a,z)$ using (<ref>), to get

\[
f(\a,z) =
\Big\langle\s\big(\w\,t),\IR-\,\mc S^{-T/4}\,\mc E^\ell y^0\Big\rangle
= \Big\langle\s\big(\w\,t-k\,\frac\pi2\big),\mc E^\ell y^0\Big\rangle,
\]

where we have used self-adjointness of $\IR-$ and the
fact that $\s(k\piotk t)$ is odd as a function of $t$, together

\[
\big(\mc S^{-T/4}\big)^\dag\big[\s(\w\,t)\big] =
\mc S^{T/4}\big[\s(\w\,t)\big] =
\s\Big(\w\,\big(t - \frac T4\big)\Big)
= \s\big(\w\,t-k\,\frac\pi2\big),
\]

where $\square^\dag$ denotes the adjoint (in $t$).  It follows that

\begin{equation}
\label{d2fdaz}
\frac{\del^2f}{\del z\,\del\a}\Big|_{(0,0)} =
\Big\langle \s\big(\w\,t -k\,{\TS\frac\pi2}\big),
\frac{\del^2}{\del z\,\del\a}\mc E^\ell y^0
\Big|_{(0,0)}\Big\rangle.
\end{equation}

\begin{equation}
\label{bifdat}
y^0 := \ol p + z+\a\,\c(\w\,t) + W(\a,z).
\end{equation}

Recall that the acoustic boundary problem (<ref>) is obtained by
limiting ourselves to $k$ even.

To proceed we must thus calculate the second derivative of the evolution,
evaluated at the constant state, namely

\begin{equation}
\label{d2Edadz}
\frac{\del^2}{\del z\,\del\a}
\mc E^\ell y^0\Big|_{(0,0)}.
\end{equation}

We postpone this calculation to
the next section so that we can complete the bifurcation argument.
The proof of the following lemma is the topic of Section <ref>

For a genuinely nonlinear constitutive equation, the second
derivative given in (<ref>) is not equal to zero, that is

\[
\frac{\del^2f}{\del z\,\del\a}\Big|_{(0,0)} =
\Big\langle \s\big(\w\,t -k\,{\TS\frac\pi2}\big),
\frac{\del^2}{\del z\,\del\a}\mc E^\ell y^0
\Big|_{(0,0)}\Big\rangle \ne 0.
\]

Applying the lemma to (<ref>) now reduces the solvability of
the bifurcation equation to a single non-degeneracy condition, namely
that the $k$-mode be nonresonant.

Assume that the $k$-mode is nonresonant.  Then there exists a
one-parameter family of solutions of the form (<ref>) of
equation (<ref>) or (<ref>), respectively, parameterized by
the amplitude $\a$ in a neighborhood of $0$.  This in turn generates
a periodic pure tone solution of the compressible Euler equations.

Lemma <ref> shows that the auxiliary equation has a unique
solution in a neighborhood of the origin.  It remains only to show that
the bifurcation equation (<ref>), namely
$f\big(\a,z\big) = 0$, for the $k$-mode can always be solved
uniquely in a neighborhood of the origin.  Using (<ref>),
Lemma <ref> and the implicit function theorem imply the
existence of a unique $z(\a)$ such that (<ref>) gives the
data $y^0$ which solves the equation (<ref>) for $\a$ in a
neighborhood of the origin.

§.§ Nonresonance conditions

As in our earlier development, we now consider conditions for
resonance or non-resonance of linear modes.  As in
Section <ref>, we regard the resonance conditions as
conditions on the infinite dimensional entropy field, and show that
the set of entropy fields for which any resonance condition holds is

According to Lemma <ref>, the $j$-mode is resonant with the
$k$-mode if and only if there is some $p$ such that

\begin{equation}
\label{resw}
\w_p = q\,\w_k, \com{with} q := \frac jk.
\end{equation}

The frequencies $\w_k$ and $\w_p$ are in turn chosen by the condition
(<ref>) or (<ref>), which we rewrite as the implicit

\[
\begin{aligned}
k\,\frac\pi2 &= \w_k\,\int_0^\ell\sg\;dy -
\int_0^\ell \s\big(2\t(y,\w_k)\big)\;d\log\sqrt\sg(y),\\
p\,\frac\pi2 &= \w_p\,\int_0^\ell\sg\;dy -
\int_0^\ell \s\big(2\t(y,\w_p)\big)\;d\log\sqrt\sg(y).
\end{aligned}
\]

Assuming (<ref>) and eliminating the linear term, we get the

\begin{equation}
\label{wq}
\int_0^\ell\Big(q\,\s\big(2\t(y,\w_k)\big)
- \s\big(2\t(y,q\,\w_k)\big)\Big)\;d\log\sqrt\sg(y)
= (p-j)\,\frac\pi2,
\end{equation}

which is necessary and sufficient for (<ref>) to hold.

We view (<ref>) as a restriction on the entropy profile $s(x)$, and
as before we set

\[
\mc Z_{k,j,p} := \Big\{s(x)\;\Big|\;
\w_p = \frac jk\,\w_k \Big\},
\]

so that $s(x) \in \mc Z_{k,j,p}$ if and only if (<ref>) holds.
As before, we then form the union

\[
\mc Z := \bigcup_{k,p,j}\mc Z_{k,p,j},
\]

which is the set of all entropy profiles having some resonant

In order to show that the resonant set is small, we need to introduce
a convenient topology on the set of entropy profiles.  We thus define
the subset

\begin{equation}
\label{BVsp}
\mc B := \Big\{s\in L^1[0,\ell]\;\Big|\;
\sg \in L^1,\ \log\sg\in BV\Big\},
\end{equation}

together with the $L^1$ topology, where $\sg$ is given by (<ref>).

Each of the sets $\mc Z_{k,j,p}$ is nowhere dense in $\mc B$, and
the resonant set $\mc Z$ is meagre in $\mc B$.

Since the map $\mc B\to\B R$ given by (<ref>),

\[
s(\cdot) \mapsto
\int_0^\ell\Big(q\,\s\big(2\t(y,\w_k)\big)
- \s\big(2\t(y,q\,\w_k)\big)\Big)\;d\log\sqrt\sg(y)
\]

is evidently continuous, each of the sets $\mc Z_{k,j,p}$ is closed
in $\mc B$.  Let $\epsilon>0$ and $s=s(x)\in\mc Z_{k,j,p}$ be given.
Since piecewise constant functions are dense, we use
Theorem <ref> to find a piecewise constant entropy profile
$\ol s(J,\T)$ which approximates $s(x)$ but is fully nonresonant, so
in particular,

\[
\big\|s(x) - \ol s(J,\T)\big\|_{L^1} < \epsilon, \com{with}
\ol s(J,\T) \notin \mc Z_{k,j,p}.
\]

It follows that the interior of the closure of $\mc Z_{k,j,p}$ is
empty, that is $\mc Z_{k,j,p}$ is nowhere dense, and, being a
countable union of nowhere dense sets, $\mc Z$ is meagre.

We cannot use this proof to show directly that $\mc Z$ is nowhere
dense, because $\mc Z$ is not itself closed in $\mc B$.  We expect
that similar to our earlier results, if one were to regard $\mc B$ as
a measure or probability space, then the resonant set $\mc Z$ would
also have zero measure.  However we will not pursue this here.  Our
next theorem is a summary the foregoing lemmas.

Given any entropy profile $s(x)$ with $s\in\mc B$, the linearization
(<ref>) around the constant state solution $(\ol p,0)$
determines a Sturm-Liouville operator (<ref>), (<ref>).
Imposing the periodic tiling boundary conditions (<ref>),
respectively the acoustic reflection boundary conditions
(<ref>), determines an increasing sequence $\w_k$, respectively
$\w_{2k}$, of linearized SL frequencies.  To each of these
frequencies corresponds a time periodic solution of the linearized
equations (<ref>), (<ref>).  For any such frequency which is
nonresonant, there is a one-parameter family of perturbations of the
linearized $k$-mode (resp. $2k$-mode) to a pure tone solution of the
nonlinear system (<ref>), parameterized by the amplitude $\a$
of the $k$-mode component of the linearized data.  The set of fully
nonresonant profiles, for which every $k$-mode (resp. $2k$-mode)
perturbs to a solution of the nonlinear problem, is generic in
$\mc B$, in that it is residual, or the complement of a countable
union of nowhere dense sets.

§ DIFFERENTIATION OF THE EVOLUTION OPERATOR

To complete the proof of the existence of periodic solutions of the
compressible Euler equations with generic entropy profile, we must
prove Lemma <ref> stated and used above, which requires
calculation of the second derivative of the evolution operator.

In differentiating the solution twice, we cannot apply
Lemma <ref> or Corollary <ref> directly, because we no
longer have exact expressions for the nonlinear solutions, although
this can be carried out: see [49].  However, because our
gradients remain finite and our solutions to the PDE are classical, we
can expand the solution and evaluate the derivative at the origin.

We briefly describe our strategy for evaluating (<ref>).  Here
we view $\mc E^\ell$ as a Banach space valued function of the two real
variables $\a$ and $z$, evaluated on data given by (<ref>).
Formally we assume that $(p,u)$ are general functions of variables
$(\a,z)$, satisfying (<ref>), and we denote the derivatives with
respect to $\a$ and $z$ as

\[
\wh{p_\a}:=\frac{\partial p}{\partial \a}, \quad
\wh{p_z}:=\frac{\partial p}{\partial z}, \com{and}
\wh{p_{z\a}}:=\frac{\partial^2 p}{\partial z\,\partial \a},
\]

respectively, and similarly for $u$.  Differentiating (<ref>) in
$\a$, we get

\[
\wh{p_\a}_x  +  \wh{u_\a}_t = 0, \qquad
\wh{u_\a}_x - \big(v_p(p)\,\wh{p_\a}\big)_t = 0,
\]

which are the linearized equations.  Now differentiating in $z$ yields

\begin{equation}
\label{D2Eeq}
\wh{p_{z\a}}_x + \wh{u_{z\a}}_t=0, \qquad
\wh{u_{z\a}}_t - \Big(v_p(p)\,\wh{p_{z\a}} +
v_{pp}(p)\,\wh{p_\a}\,\wh{p_{z}} \Big)_t = 0,
\end{equation}

which captures the second derivative.  In the context we are working
in, we have $\wh{p_{z}}\big|_{(0,0)} = 1$, and the second derivative
equation is linear inhomogeneous, with varying coefficients
$v_p\big(\ol p,s(x)\big)$ and $v_{pp}\big(\ol p,s(x)\big)\ne 0$ which
encode the leading order effects of the nonlinear interaction of
acoustic waves with the varying entropy field.

The second derivative (<ref>) of the evolution operator
acting on $y^0$ given by (<ref>) is given by the
solution of the linear inhomogeneous SL system

\begin{equation}
\label{ODE}
\begin{aligned}
\dot{\wh\vp} + \w\,\wh\psi &= 0,\\
\dot{\wh\psi} - \sg^2\,\w\,\wh\vp &= - v_{pp}\,\w\,\vp_k,
\end{aligned}
\end{equation}

with vanishing initial data $\wh\vp(0) = \wh\psi(0) = 0$,
with coefficient $\sg^2 = - v_p(\ol p,s)$, and where
$v_{pp} = v_{pp}(\ol p,s)\ne 0$.  More precisely, we have

\begin{equation}
\label{dEdadz}
\frac{\del^2}{\del z\,\del\a} \mc E^\ell y^0\Big|_{(0,0)} =
\wh\vp(\ell)\,\c(\w t)+\wh\psi(\ell)\,\s(\w t),
\end{equation}

where $\w = \w_k$ is given by (<ref>).

For the data as given by (<ref>), (<ref>), we write the
corresponding solution of (<ref>) as

\begin{equation}
\label{pusol}
\begin{aligned}
p(x,t) &= \ol p + z + \a\,\vp_k(x)\,\c(\w t) + \wh p(x,t),\\
u(x,t) &= \a\,\psi_k(x)\,\s(\w t) + \wh u(x,t),
\end{aligned}
\end{equation}

where $\w = k \piotk$, and $\wh p$, $\wh u = o(|\a|)$.  Here we
have scaled the eigenfunction by $\vp_k(0)=1$, and we have

\[
\wh p(0,\cdot) + \wh u(0,\cdot) = W(\a,z).
\]

To rederive equation (<ref>), we make the ansatz
(<ref>), expand the nonlinear (classical) solution, and
differentiate, first in $\a$ and setting $\a=0$, then in $z$ and
setting $z=0$, to get the derivatives $\wh{p_{\a z}}$ and
$\wh{u_{\a z}}$.  This is allowed because our solutions are small
amplitude and locally defined, so do not suffer gradient blowup in
the region $0\le x\le \ell$.  In this notation, it follows that

\[
\frac{\del^2}{\del z\,\del\a}
\mc E^\ell y^0\Big|_{(0,0)} =
\wh{p_{\a z}}(\ell,\cdot) + \wh{u_{\a z}}(\ell,\cdot),
\]

where $\cdot$ denotes a function of $t$.

We use (<ref>) to expand $v(p,s)$ as

\[
v(p,s) = v(\ol p+z,s) + v_p(\ol p+z,s)\,
\big[\a\,\vp_k(x)\,\c(\w t) + \wh p\big] + O(|\a|^2),
\]

and substitute into (<ref>) to get

\[
\begin{gathered}
\a\,\dot\vp_k\,\c(\w t) + \wh p_x
+ \a\,\w\,\psi_k\,\c(\w t)+\wh u_t = 0,\\
\a\,\dot\psi_k\,\s(\w t) + \wh u_x - v_p(\ol p+z,s)\,
\big(-\a\,\w\,\vp_k\,\s(\w t)+\wh p_t\big) = O(\a^2).
\end{gathered}
\]

Differentiating with respect to $\a$ and setting $\a=0$, we get

\[
\begin{gathered}
\dot\vp_k\,\c(\w t) + \wh{p_\a}_x
+ \w\,\psi_k\,\c(\w t)+\wh{u_\a}_t = 0,\\
\dot\psi_k\,\s(\w t) + \wh{u_\a}_x - v_p(\ol p+z,s)\,
\big(-\w\,\vp_k\,\s(\w t) + \wh{p_\a}_t\big) = 0.
\end{gathered}
\]

We now differentiate this in $z$ and set $z=0$, to get

\begin{equation}
\label{bifDaz}
\begin{aligned}
\wh{p_{\a z}}_x + \wh{u_{\a z}}_t &= 0,\\
\wh{u_{\a z}}_x + \sg^2\,\wh{p_{\a z}}_t &=
- v_{pp}(\ol p,s)\,\w\,\vp_k\,\s(\w t),
\end{aligned}
\end{equation}

where we have used (<ref>) and the fact that
$\wh{p_\a}\big|_{(0,0)}=0$.  This is a restatement of the second
derivative equation (<ref>).  According to
Lemma <ref>, and refering to our ansatz (<ref>),

\[
y^0 := \ol p + z+\a\,\c(\w\,t) + W(\a,z),
\]

we have $W(\a,z) = o(|\a|)$, which yields

\[
\frac{\partial W}{\partial \a}\bigg|_{(0,0)} = 0, \com{so also}
\frac{\partial^2 W}{\partial \a\,\partial z}\bigg|_{(0,0)} = 0.
\]

This implies that the initial conditions for (<ref>) are

\[
\wh{p_{\a z}}(0,t) = 0, \quad \wh{u_{\a z}}(0,t) = 0,
\]

and we wish to integrate to $x=\ell$.

Finally, we make the ansatz

\[
\wh{p_{\a z}} = \wh\vp(x)\,\c(\w t), \quad
\wh{u_{\a z}} = \wh\psi(x)\,\s(\w t), \com{with}
\w = k\piotk,
\]

and separate variables in (<ref>) to get the inhomogeneous
system (<ref>), as required.

We solve (<ref>) using Duhamel's principle with the fundamental
solution $\Psi(x;\w)$ given in (<ref>): it is straight-forward to
check that the solution of (<ref>) is

\begin{equation}
\label{duhamel}
\(\wh\vp(x)\\\wh\psi(x)\) = -\w\,\int_0^x
\Psi(x-x';\w)\(0\\1\)\vp_k(x')\,v_{pp}\;dx',
\end{equation}

and where $v_{pp}=v_{pp}\big(\ol p,s(x')\big)$.  We carry out this
calculation explicitly in Lemma <ref> below.

This last integrand requires us to find the full fundamental solution
$\Psi(x;\w)$ of the linear system (<ref>).  To do so, we consider
initial data of the form

\[
\wt\vp(0) = 0 \com{and} \wt\psi(0) = \wt c_0,
\]

solution of which provides the second column of $\Psi$.  As in
(<ref>), we again use modified Prüfer coordinates, and so we
make the ansatz

\begin{equation}
\label{pruf2}
\wt\vp(x) := - \wt r(x)\,\frac1{\wt\rho(x)}\,\s\big(\wt\t(x)\big), \qquad
\wt\psi(x) := \wt r(x)\,\wt\rho(x)\,\c\big(\wt\t(x)\big),
\end{equation}

As above, plugging in (<ref>) into (<ref>) and simplifying

\[
\begin{aligned}
- \frac{\dot{\wt r}}{\wt r}\,\s(\wt\t)
+ \frac{\dot{\wt\rho}}{\wt\rho}\,\s(\wt\t)
- \c(\wt\t)\,\dot{\wt\t}
+ \w\,\wt\rho^2\,\c(\wt\t) &= 0, \\
\frac{\dot{\wt r}}{\wt r}\,\c(\wt\t)
+ \frac{\dot{\wt\rho}}{\wt\rho}\,\c(\wt\t)
- \s(\wt\t)\,\dot{\wt\t}
+ \w\,\frac{\sg^2}{\wt\rho^2}\,\s(\wt\t) &= 0,
\end{aligned}
\]

with initial conditions

\[
\wt\t(0) = 0 \com{and} \wt r(0) = 1,
\com{so} \wt c_0 = \frac1{\wt\rho(0)}.
\]

Again choosing $\wt\rho := \sqrt\sg = \rho(x)$, after simplifying we get

\begin{equation}
\label{rtwt}
\frac{\dot{\wt r}}{\wt r}
= - \frac{\dot{\sg}}{2\sg}\,\c(2\wt\t), \qquad
\dot{\wt\t} = \frac{\dot{\sg}}{2\sg}\,\s(2\wt\t)
+ \w\,\sg,
\end{equation}

and the first of these can again be integrated, giving

\[
\wt r(x) = \exp\Big\{ -\int_0^x
\c\big(2\wt\t(y)\big)\;d\log\sqrt\sg(y)\Big\}.
\]

We summarize the foregoing, which gives a full description of a
fundamental solution of the SL system.

A fundamental matrix of the system (<ref>) is

\[
\Psi(x;\w) = \(\vp&\wt\vp\\\psi&\wt\psi\) =
\( r(x)\,\frac1{\rho(x)}\,\c\big(\t(x)\big)
& -\wt r(x)\,\frac1{\rho(x)}\,\s\big(\wt\t(x)\big)\\[2pt]
& \wt r(x)\,\rho(x)\,\c\big(\wt\t(x)\big) \),
\]

\[
\rho(x) = \sqrt{\sg(x)} \com{and}
\Psi(0;\w) = \(\frac1{\rho(0)}&0\\[2pt]0&\rho(0)\),
\]

where $\t$ and $\wt\t$ solve

\[
\dot\t = \w\,\sg - \frac{\dot\sg}{2\sg}\,\s(2\t), \qquad
\dot{\wt\t} = \w\,\sg + \frac{\dot\sg}{2\sg}\,\s(2\wt\t),
\]

with $\t(0) = \wt\t(0) = 0$ respectively, and $r$ and $\wt r$ are
given explicitly by

\[
\begin{aligned}
r(x) &= \exp\Big\{ \int_0^x
\c\big(2\t(y)\big)\;d\log\sqrt\sg(y)\Big\}, \\
\wt r(x) &= \exp\Big\{ -\int_0^x
\c\big(2\wt\t(y)\big)\;d\log\sqrt\sg(y)\Big\},
\end{aligned}
\]

respectively.  Moreover, we have

\[
\det\Psi(x;\w) = r(x)\,\wt r(x)\,\c\big(\t(x)-\wt\t(x)\big) = 1
\]

for all $x$, and in particular

\[
\big|\t(x)-\wt\t(x)\big| <\frac\pi2 \com{for all $x$.}
\]

In this representation of the fundamental solution, the first column
describes the linearized evolution of the even modes $\c(\w\,t)$, and
the second column describes evolution of the odd modes $\s(\w\,t)$.

Combining (<ref>), (<ref>) and (<ref>) with
(<ref>) yields the fundamental solution, and $\Psi(0;\w)$
follows from our choices $\t(0)=\wt\t(0)=0$ and $r(0)=\wt r(0)=1$.
Abel's theorem implies the determinant is constant, and the last
inequality follows by continuity of $\t$ and $\wt\t$.

Having calculated the fundamental matrix $\Psi$ of (<ref>), we now
solve the inhomogeneous system (<ref>).  Rather than use Duhamel's
principle (<ref>) directly, we rederive it explicitly in the
Prüfer variables using variation of parameters.

The solution of (<ref>) can be written as

\begin{equation}
\label{hatpsi}
\(\wh\vp(x)\\[2pt]\wh\psi(x)\) = \Psi(x;\w)\,\(a(x)\\b(x)\),
\end{equation}

for appropriate functions $a$ and $b$, and we have $b(x) < 0$ for
all $x > 0$.

Because $\Psi$ is a fundamental solution, using the ansatz
(<ref>) in (<ref>) yields the simplified system

\begin{equation}
\label{abODE}
\(a\\b\)^{\dot{}} = - v_{pp}\,\w\,\vp\,\Psi^{-1}\,\(0\\1\),
\qquad a(0) = b(0) = 0.
\end{equation}

Next, since $\det\Psi \equiv 1$, we have

\[
\Psi^{-1} = \(\wt\psi&-\wt\vp\\-\psi&\vp\),
\]

and the second component of (<ref>) simplifies to

\[
\dot b = -v_{pp}\,\w\,\vp^2 < 0.
\]

It follows that $b(x) < 0$ for all $x > 0$, as required.

Lemma <ref> gives the second derivative of the evolution in
terms of the fundamental solution, which is calculated in
Lemma <ref>.  In Lemma <ref>, we explicitly
calculate a nonvanishing term due to genuine nonlinearity, as seen by
the $v_{pp}$ term in (<ref>).  This in turn allows us to solve
the bifurcation equation as stated in Lemma <ref> above.

We substitute (<ref>) into (<ref>), to get

\[
\Big\langle \s\big(\w\,t -k\,{\TS\frac\pi2}\big),
\frac{\del^2}{\del z\,\del\a}\mc E^\ell y^0
\Big|_{(0,0)}\Big\rangle =
\begin{cases}
\wh\vp(\ell), & k\text{ odd},\\
\wh\psi(\ell), & k\text{ even},
\end{cases}
\]

respectively, where $(\wh\vp,\wh\psi)$ solve (<ref>).

Evaluating (<ref>) at $x=\ell$, we get

\[
\(\wh\vp(\ell)\\[2pt]\wh\psi(\ell)\) =
\(\vp(\ell)&\wt\vp(\ell)\\\psi(\ell)&\wt\psi(\ell)\)
\,\(a(\ell)\\b(\ell)\),
\com{with} b(\ell)<0.
\]

By our choice (<ref>) of $\w$, for $k$ odd we have
(<ref>), so that $\vp(\ell)=0$, which implies that

\[
\wh\vp(\ell) = \wt\vp(\ell)\,b(\ell) \ne 0,
\]

and similarly for $k$ even, we have (<ref>), which is
$\psi(\ell)=0$, so we must have

\[
\wh\psi(\ell) = \wt\psi(\ell)\,b(\ell) \ne 0,
\]

and the proof is complete.

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