subsection

# Propagation of polarization sets for systems of MHD type

Rayhana Darwich

(May 2023)

###### Abstract

Polarization sets were introduced by Dencker (1982) as a refinement of
wavefront sets to the vector-valued case. He also clarified the propagation of
polarization sets when the characteristic variety of the pseudodifferential
system under study consists of two hypersurfaces intersecting tangentially
(1992), or transversally (1995). In this paper, we consider the case of more
than two intersecting characteristic hypersurfaces that are interesting
transversally (and we give a note on the tangential case).

Mainly, we consider two types of systems which we name ”systems of generalized
transverse type” and ”systems of MHD type”, and we show that we can get a
result for the propagation of polarization set similar to Dencker’s result for
systems of transverse type. Furthermore, we give an application to the MHD
equations.

##  1 Introduction

In [9], Hörmander defined the wavefront set of a distribution $u$, denoted by
$\operatorname{WF}(u)$, which is a refinement of the singular support of a
distribution. The wavefront sets does not only show the location of
singularity, but also the direction in which the singularity occurs.
Concerning the propagation of the wavefront sets, many results were given. For
example, in [9] Hörmander gave the result for the propagation of the wavefront
set for the solutions of partial differential equations, when considering the
partial differential operator to be of real principal type, where he stated
that the wavefront set is invariant under the bicharacteristic flow. Moreover,
in [4], Dencker studied the propagation of singularities for
pseudodifferential operator $P$ on a smooth manifold $X$, having
characteristics of variable multiplicity. He considered the characteristic set
to be union of hypersurfaces $S_{j}$, $j=1,...,r_{0}$ tangent at
$\cap_{j=1}^{r_{0}}S_{j}$. Under some assumptions he proved that the wavefront
set of the solution of the considered pseudodifferential operator, is
invariant under the union of the Hamilton flows on $S_{j}$, $j=1,...,r_{0}$,
given that $Pu$ is smooth on $X$.

In [9], Hörmander defined locally the wavefront set of distributional sections
$u\in\mathcal{D}^{\prime}(X;E)$, where $E\rightarrow X$ is a vector bundle
over the smooth manifold $X$. He defined the wavefront set of $u$ locally as
$\bigcup\operatorname{WF}(u_{j})$ where $(u_{1},...,u_{N})$ are the components
of $u$ with respect to a local trivialization of $E$. However, this definition
does not specify in which components $u$ is singular, that is why Dencker
defined in [1] the polarization set for vector-valued distribution $u$ which
we will denote by $\operatorname{Pol}(u)$. The polarization set also shows the
location and the direction of the singularity as the wavefront set, but it
additionally shows the most singular components of a distribution. Hence, the
polarization set of a distribution is a refinement of the wavefront set, and
the projection of $\operatorname{Pol}(u)\setminus 0$ on the cotangent bundle
$T^{*}X$ gives the wavefront set of $u$. Similarly, the $H^{s}$-polarization
set is defined as a refinement of the $H^{s}$-wavefront set, where $H^{s}$
denotes the usual Sobolev space.

In [1], Dencker defined systems of pseudodifferential operators of real
principal type; note that the definition of systems of pseudodifferential
operators of real principal type differs from the case of scalar
pseudodifferential operators of real principal type, and he defined Hamilton
orbits for systems of real principal type which are certain line bundles, and
then he proved that the polarization set of a solution $u$ of systems of real
principal type $P$ will be union of Hamilton orbits, given that $Pu$ is
smooth. In [7], Gérard pointed out that the above result also holds for
$H^{s}$-polarization sets.

Moreover, in [5], Dencker considered pseudodifferential system having its
characteristic set is union of two non-radial hypersurfaces intersecting
tangentially at an involutive manifold of exactly order $k_{0}\geq 1$. He also
assumed that the principal symbol vanishes of first order on the two-
dimensional kernel at the intersection, and he assumed a Levi type of
condition. Then, he defined systems satisfying these conditions to be systems
of uniaxial type. Outside the intersection of the hypersurfaces the system
will be of real principal type, hence the propagation result of the
polarization set is already known there. In this article, Dencker has also
proved a propagation result of the polarization set at the intersection. In
[6], Dencker considered pseudodifferential system having its characteristic
set is union of two non-radial hypersurfaces intersecting transversally at an
involutive manifold of codimension $2$. He also assumed that the principal
symbol vanishes of first order on the two-dimensional kernel at the
intersection. Systems satisfying these conditions are systems of transverse
type. In this article, Dencker has proved a propagation result of the
polarization set at the intersection. Outside the intersection the system is
of real principal type.

We worked on extending Dencker’s result stated above to pseudodifferential
systems having their characteristic sets is union of several non-radial
hypersurfaces intersecting transversally at an involutive manifold of
codimension $d_{0}\geq 2$; not necessary just two hypersurfaces as in the case
of systems of transverse type and systems of uniaxial type. Note that even if
we assumed that the hypersurfaces are intersecting tangentially of exactly
order $k_{0}\geq 1$ instead of intersecting transversally, we get a similar
result, and for the proof we use the same weight and metric introduced by
Dencker in [5] for the symbol classes $S(\vartheta,g)$ of the Weyl calculus.
We have considered two cases for that: the first case is the case where we
have $r_{0}$ hypersurfaces, and we assumed that the $r_{0}$th-differential of
the determinant of the principal symbol is different than zero at the
intersection, and the $i$th-differential of the determinant of the principal
symbol vanishes at the intersection for $i<r_{0}$. Moreover, we assumed that
the dimension of the kernel of the principal symbol to be $r_{0}$ at the
intersection, and we assumed a condition similar to the Levi type condition
considered in [5]. We called systems satisfying the above conditions systems
of generalized transverse type, and we proved that we have a similar
propagation result of the polarization set as that for systems of transverse
type. The second case, is the case where we also have $r_{0}$ hypersurfaces,
and a condition similar to the Levi type condition considered in [5], but here
we assumed that the $(r_{0}+1)$th-differential of the determinant of the
principal symbol is different than zero at the intersection, and the $i$th-
differential of the determinant of the principal symbol vanishes at the
intersection for $i<r_{0}+1$. Moreover, we assumed that the dimension of the
kernel of the principal symbol to be $r_{0}+1$ at the intersection. We also
assumed some additional conditions that we did not assume in the case of
systems of generalized transverse type. We defined systems satisfying these
conditions to be systems of MHD type. We named them systems of MHD type
because we have first noticed such systems when we considered the linearized
ideal MHD equations. Thus we will have a section in which we study the
propagation of polarization sets for the linearized ideal MHD equations.

In our work, we will assume that we have $P$ an $N\times N$ system of
pseudodifferential operators. Let $p=\sigma(P)$ be the principal symbol, $\det
p$ the determinant of $p$, and $\Sigma=(\det p)^{-1}(0)$ the characteristics
of $P$. We consider $\Sigma$ to be union of several non-radial hypersurfaces
intersecting transversally at an involutive manifold $\Sigma_{2}$. Now, we
state our main theorem in this paper regarding the propagation of polarization
sets for systems of generalized transverse type, and systems of MHD type, but
its proof will be postponed to Sections 3, and 4 to prove it for systems of
generalized transverse type, and systems of MHD type respectively. Let

$\displaystyle r^{*}_{u}(\nu)=\sup\\{r\in\mathbb{R}:u\in H^{r}\ \text{at}\
\nu\\}\ \ \ \ \nu\in T^{*}X\setminus 0$ (1.1)

be the regularity function.

######  Theorem 1.1.

Let $P\in\Psi^{m}_{phg}$ be an $N\times N$ system of generalized transverse
type (or of MHD type) at $\nu_{0}\in\Sigma_{2}$, and let $A\in\Psi^{0}_{phg}$
be an $N\times N$ system such that the dimension of
$\mathcal{N}_{A}\cap\mathcal{N}_{P}$ is equal to $1$ at $\nu_{0}$, and
$M_{A}=\pi_{1}(\mathcal{N}_{A}\cap\mathcal{N}_{P}\setminus 0)$ is a
hypersurface near $\nu_{0}$, where
$\pi_{1}:T^{*}X\times\mathbb{C}^{N}\rightarrow T^{*}X$ is the projection along
the fibers. Assume that $u\in\mathcal{D}^{\prime}(X,\mathbb{C}^{N})$ satisfies
$\min(r^{*}_{Pu}+m-1,r^{*}_{Au})>r$ at $\nu_{0}$. Then,
$\operatorname{Pol}^{r}(u)$ is a union of $\mathcal{C}^{\infty}$ line bundles
in $\mathcal{N}_{A}\cap\mathcal{N}_{P}$ over bicharacteristics of
$M_{A}=\pi_{1}(\mathcal{N}_{A}\cap\mathcal{N}_{P}\setminus 0)$ near $\nu_{0}$.

The plan of this paper is as follows: in Section 2, we mention previous
results on the propagation of polarization sets. More precisely, we will state
Dencker’s propagation result for systems of real principal type which was
proven in [1], and Dencker’s propagation result for systems of uniaxial type
that was proven in [5]. Also, we state Dencker’s result for the propagation of
polarization sets for systems of transverse type; see [6]. Note that in [5]
and [6], Dencker proved several results for the propagation of polarization
sets under different conditions. Here we just mention the result which is
similar to the result in our main theorem. In Sections 3 and 4, we define
systems of generalized transverse type, and systems of MHD type, and we prove
Theorem 1.1 for both types of systems. In Section 5, we give an application
for the results in [8], and for the propagation of polarization sets for
systems of MHD type, so we divide it into two subsections. First, we give the
set of equations describing the ideal MHD, and we linearize it. In Section
5.1, we write the linearized ideal MHD equations in the form of a wave
equation, and we give the characteristic variety of this wave equation under
some assumptions which was calculated in [14]. Then, we calculate the
transport equation under these assumptions as an application to Hansen’s and
Röhrig’s results in [8]. In Section 5.2, we return to the linearized ideal MHD
equations, and we calculate the eigenvalues and their multiplicities which are
not constant. Then, we study the propagation of polarization sets, where we
observe different cases, some in which our system is of real principal type,
some in which our system is of uniaxial type, and one where our system is of
MHD type.

Acknowledgments. This work was part of my PhD thesis at Georg-August
University, Göttingen, supervised by Prof. Dr. Ingo Witt, and involved with
the RTG 2491. My PhD studies was funded by a DAAD scholarship.

##  2 Previous results

In this section, we state some previous results regarding the propagation of
polarization sets. More precisely, we state the results for systems of real
principal type, systems of uniaxial type, and systems of transverse type
proven in [1], [5], and [6] respectively. First, we will state the definition
of polarization sets given by Dencker in [1].

######  Definition 2.1.

For $u\in\mathcal{D}^{\prime}(X,\mathbb{C}^{N})$, the polarization set of $u$
is given by

$\displaystyle\operatorname{Pol}(u)=\bigcap\mathcal{N}_{B}\subseteq(T^{*}X\setminus
0)\times\mathbb{C}^{N},$ (2.1)

where $\mathcal{N}_{B}=\ker\sigma(B)$, and the intersection is taken over
those $1\times N$ systems $B\in\Psi^{0}_{phg}$ such that
$Bu\in\mathcal{C}^{\infty}$.

The $H^{s}$-polarization set, where $H^{s}$ is the usual Sobolev space is
defined similarly.

######  Definition 2.2.

For $u\in\mathcal{D}^{\prime}(X,\mathbb{C}^{N})$, the $H^{s}$-polarization set
is given by

$\displaystyle\operatorname{Pol}^{s}(u)=\bigcap\mathcal{N}_{B}\subseteq(T^{*}X\setminus
0)\times\mathbb{C}^{N},$ (2.2)

where $\mathcal{N}_{B}=\ker\sigma(B)$, and the intersection is taken over
those $1\times N$ systems $B\in\Psi^{0}_{phg}$ such that $Bu\in H^{s}$.

###  2.1 Systems of real principal type

First, we want to state Dencker’s result regarding the propagation of
polarization sets for systems of real principal type. For the definition of
real principal type, we will differentiate between two cases, the scalar case,
and the case of system of pseudodifferential operators.

######  Definition 2.3.

We say that $P\in\Psi^{m}(X)$ is of real principal type if the principal
symbol $\sigma(P)=p$ is real and the Hamilton field
$H_{p}=\sum\partial_{\xi_{j}}p\partial_{x_{j}}-\partial_{x_{j}}p\partial_{\xi_{j}}$
is non-vanishing, and does not have the radial direction when $p=0$.

######  Definition 2.4 (Case of system of pseudodifferential operators).

An $N\times N$ system $P$ of pseudodifferential operators on $X$ with
principal symbol $p(x,\xi)$ is of real principal type at $(y,\eta)\in
T^{*}X\setminus 0$ if there exists an $N\times N$ symbol $\tilde{p}(x,\xi)$
such that

$\displaystyle\tilde{p}(x,\xi)p(x,\xi)=q(x,\xi)\cdot\operatorname{Id}_{N}$

in a neighborhood of $(y,\eta)$ where $q(x,\xi)$ is a scalar symbol of real
principal type and $\operatorname{Id}_{N}$ is the identity in
$\mathbb{C}^{N}$.

Assume $P(x,D)$ to be an $N\times N$ system of classical pseudodifferential
operators on an $n$-dimensional smooth manifold $X$ of order $m$. The symbol
of $P$ is an asymptotic sum of homogeneous terms:
$p(x,\xi)+p_{m-1}(x,\xi)+p_{m-2}(x,\xi)+...$ where $p$ is the principal symbol
of $P$ and $p_{j}$ is homogeneous of degree $j$. Assume $P$ to be of real
principal type, and let

$\displaystyle\Sigma=\\{(x,\xi):\det p(x,\xi)=0\\}$ (2.3)

be the characteristic set of $P$. To state the result of the propagation of
polarization set given by Dencker in [1], we have to introduce first the
connection he defined, and give the definition of the Hamilton orbit. Let

$\displaystyle
D_{p}w=H_{q}w+\frac{1}{2}\\{\tilde{p},p\\}w+i\tilde{p}p^{s}_{m-1}w,$ (2.4)

where $w$ is $\mathcal{C}^{\infty}$ function on $T^{*}X\setminus 0$ with
values in $\mathbb{C}^{N}$, $\\{,\\}$ is the Poisson bracket, that is
$\\{\tilde{p},p\\}=H_{\tilde{p}}p$, and $p^{s}_{m-1}$ is the subprincipal of
$P$ defined by

$\displaystyle
p^{s}_{m-1}=p_{m-1}-(2i)^{-1}\sum\partial_{x_{j}}\partial_{\xi_{j}}p.$ (2.5)

$D_{P}$ is a connection on $\mathcal{N}_{P}$ over $\Sigma$, that is, if
$w\in\ker p$ at one point of a bicharacteristic of $\Sigma$, then
$D_{P}w\in\ker p$ along the bicharacteristic if and only if $w\in\ker p$
there. Hence, each parallel section (that is $w$ such that $D_{P}w=0$) is
uniquely determined by one point. $D_{P}$ depends on the choice of $\tilde{p}$
and $q$, however Dencker showed that different choices of $\tilde{p}$ and $q$
only change the solution of $D_{P}w=0$ in $\mathcal{N}_{P}$ by a scalar
factor. This motivated him to define the following.

######  Definition 2.5 (Hamilton Orbit).

A Hamilton orbit of a system $P$ of real principal type is a line bundle
$L\subseteq\mathcal{N}_{P}|_{\gamma}$, where $\gamma$ is an integral curve of
the Hamilton field of $\Sigma$, and $L$ is spanned by $\mathcal{C}^{\infty}$
section $w$ satisfying $D_{p}w=0$.

######  Theorem 2.6 (Dencker’s propagation result).

Let $P$ be an $N\times N$ system of pseudodifferential operators on a manifold
$X$ and let $u\in\mathcal{D}^{\prime}(X,\mathbb{C}^{N})$. Assume that $P$ is
of real principal type at $(y,\eta)\in\Sigma$, and that
$(y,\eta)\notin\operatorname{WF}(Pu)$. Then, over a neighborhood of $(y,\eta)$
in $\Sigma$, $\operatorname{Pol}(u)$ is a union of Hamilton orbits of $P$.

In [7], Gérard stated that we have similar propagation result for the
$H^{s}$-polarization sets for systems of real principal type.

######  Theorem 2.7.

Let $P$ be an $N\times N$ system of pseudodifferential operators on a manifold
$X$ of order $m$, and let $u\in\mathcal{D}^{\prime}(X,\mathbb{C}^{N})$. Assume
that $P$ is of real principal type at $(y,\eta)\in\Sigma$, and that
$(y,\eta)\notin\operatorname{WF}^{s}(Pu)$. Then over a neighborhood of
$(y,\eta)\in\Sigma$, $\operatorname{Pol}^{s+m-1}(u)$ is a union of Hamilton
orbits of $P$.

###  2.2 Systems of uniaxial type

Let $P\in\Psi^{m}_{phg}(X)$ be an $N\times N$ system of classical
pseudodifferential operators on a smooth manifold $X$. Let $p$ be the
principal symbol of $P$. Let $\Sigma=(\det p)^{-1}(0)$ be the characteristics
of $P$, and let

$\displaystyle\Sigma_{2}=\\{(x,\xi)\in\Sigma:d(\operatorname{det}p)=0\
\text{at}\ (x,\xi)\\},$ (2.6)

and $\Sigma_{1}=\Sigma\setminus\Sigma_{2}$. Assume that we have

$\displaystyle\begin{split}&\Sigma=S_{1}\cup S_{2},\ \text{where}\ S_{1}\
\text{and}\ S_{2}\ \text{are non-radial hypersurfaces tangent at}\\\
&\Sigma_{2}=S_{1}\cap S_{2}\ \text{of exactly order}\ k_{0}\geq 1,\end{split}$
(2.7)

microlocally near $\nu_{0}\in\Sigma_{2}$. This means that the Hamilton field
of $S_{j}$ does not have the radial direction
$\langle\xi,\partial_{\xi}\rangle$, and it means also that the $k_{0}$th jets
of $S_{1}$ and $S_{2}$ coincide on $\Sigma_{2}$, but no $(k_{0}+1)$th jet
does. Note that we have $P$ is of real principal type at $\Sigma_{1}$, since
$d(\det p)\neq 0$ there; see Definition 2.4. $(\ref{2.1})$ gives us that
$\Sigma_{2}$ has to be a manifold of codimension $\geq 2$. We assume that

$\displaystyle\Sigma_{2}\ \text{is an involutive manifold of codimension}\
d_{0}\geq 2.$ (2.8)

Moreover, we assume that

$\displaystyle\text{the (complex) dimension of the fiber of}\ \mathcal{N}_{p}\
\text{is equal to}\ 2\ \text{at}\ \Sigma_{2},$ (2.9)

and

$\displaystyle d^{2}(\det p)\neq 0\ \text{at}\ \Sigma_{2},$ (2.10)

that is, $p$ vanishes of first order on the kernel. We want to consider the
limits of $\mathcal{N}_{P}|_{\Sigma_{1}}$ when we approach $\Sigma_{2}$, so
let

$\displaystyle\mathcal{N}_{p}^{j}=\mathcal{N}_{p}|_{S_{j}\setminus\Sigma_{2}},$
(2.11)

$T_{\Sigma_{2}}\Sigma:=T_{\Sigma_{2}}S_{1}=T_{\Sigma_{2}}S_{2}$ (note here
$\Sigma$ is not a manifold), and
$\partial\Sigma_{1}:=T_{\Sigma_{2}}\Sigma/T\Sigma_{2}$. Here
$\partial\Sigma_{1}$ is the normal bundle of $\Sigma_{2}$ in $S_{1}$ which is
equal to the normal bundle of $\Sigma_{2}$ in $S_{2}$. Let
$i_{0}:\Sigma_{2}\rightarrow\partial\Sigma_{1}$ denotes the zero section of
$\partial\Sigma_{1}$. By the tubular neighborhood theorem we know that there
exists a diffeomorphism $\Phi$ from some neighborhood $\mathcal{U}\subset
S_{j}$ of $\Sigma_{2}$ to a neighborhood
$\mathcal{U}_{0}\in\partial\Sigma_{1}$ of the zero section of
$\partial\Sigma_{1}$, and $\Phi$ identifies $\Sigma_{2}$ itself with the zero
section.

Before giving the definition of systems of uniaxial type, we need to give the
definition of the limit polarizations.

######  Definition 2.8.

For $j=1,2$, we define the limit polarizations

$\displaystyle\partial\mathcal{N}_{P}^{j}=\\{(\nu,\rho,z)\in\partial\Sigma_{1}\times\mathbb{C}^{N}:\rho\neq
0\ \text{and}\ z=\underset{k\rightarrow\infty}{\lim}z_{k}\\},$ (2.12)

where $z_{k}\in\ker p(\nu_{k})$ and $\nu_{k}\in S_{j}\setminus\Sigma_{2}$
satisfy
$(\nu-\nu_{k})/\lvert\nu-\nu_{k}\rvert\rightarrow\rho/\lvert\rho\rvert$ when
$k\rightarrow\infty$.

$\partial\mathcal{N}_{P}^{j}$ is conical in $\xi$ and $\rho$, and homogeneous
in the fiber, but it may have (complex) dimension $>1$ at $(\nu,\rho)$. We
assume that the fiber of

$\displaystyle\partial\mathcal{N}_{P}^{1}\cap\partial\mathcal{N}_{P}^{2}=\\{0\\}\
\text{over}\ \partial\Sigma_{1}\setminus(\Sigma_{2}\times 0).$ (2.13)

This condition means that no element in $\mathcal{N}_{P}|_{\Sigma_{2}}$ can be
the limit of polarization vectors on both characteristic surfaces, along the
same direction. Dencker showed that $(\ref{2.7})$ implies that
$\partial\mathcal{N}_{P}^{j}$ is a complex line bundle over
$\partial\Sigma_{1}\setminus(\Sigma_{2}\times 0)$, if we assume
$(\ref{2.1})$-$(\ref{2.4})$. Now, we give the definition of systems of
uniaxial type.

######  Definition 2.9.

The system $P$ is of uniaxial type at $\nu_{0}\in\Sigma_{2}$, if
$(\ref{2.1})$-$(\ref{2.4})$ and $(\ref{2.7})$ hold microlocally near
$\nu_{0}$.

If $P\in\Psi^{m}_{phg}$ is of uniaxial type and $Pu\in H^{r}$ near
$\nu\in\Sigma_{1}$, then we already know the result as Dencker showed in [1]
that $\operatorname{Pol}^{r+m-1}(u)$ is a union of Hamilton orbits in
$\mathcal{N}_{P}$ near $\nu$ because $P$ is of real principal type at
$\Sigma_{1}$. Now, we want to give Dencker’s result for the propagation of the
polarization set when we approach $\Sigma_{2}$. Here $S_{1}$ and $S_{2}$ are
tangent at $\Sigma_{2}$, so their Hamilton fields are parallel on
$\Sigma_{2}$. Since $\Sigma_{2}$ is involutive, the Hamilton fields are
tangent to $\Sigma_{2}$. Therefore, $\Sigma$ and $\Sigma_{2}$ are foliated by
the bicharacteristics of $\Sigma$. Also, Dencker proved that
$\partial\Sigma_{1}\setminus(\Sigma_{2}\times 0)$ is foliated by limit
bicharactersitcs, which are liftings of bicharactersitics in $\Sigma_{2}$, and
that $\partial\mathcal{N}_{P}^{1}\cup\partial\mathcal{N}_{P}^{2}$ is foliated
by limit Hamilton orbits, which are liftings of limits of Hamilton orbits, and
are unique line bundles over limit bicharacteristics.

######  Theorem 2.10.

Let $P\in\Psi^{m}_{phg}$ be an $N\times N$ system of uniaxial type at
$\nu_{0}\in\Sigma_{2}$, and let $A\in\Psi^{0}_{phg}$ be a $1\times N$ system
such that the dimension of $\mathcal{N}_{A}\cap\mathcal{N}_{P}$ is equal to
$1$ at $\nu_{0}$. Assume that $u\in\mathcal{D}^{\prime}(X,\mathbb{C}^{N})$
satisfies $\min(r^{*}_{Pu}+m-1,r^{*}_{Au})>r$ at $\nu_{0}$. Then,
$\operatorname{Pol}^{r}(u)$ is a union of $\mathcal{C}^{\infty}$ line bundles
in $\mathcal{N}_{A}\cap\mathcal{N}_{P}$ over bicharacteristics of $\Sigma$ in
$M_{A}=\pi_{1}(\mathcal{N}_{A}\cap\mathcal{N}_{P}\setminus 0)$ near $\nu_{0}$,
where $\pi_{1}:T^{*}X\setminus 0\times\mathbb{C}^{N}\rightarrow T^{*}X$ is the
projection along the fibers.

Moreover, in [5], Dencker showed under what assumptions we get
$\operatorname{Pol}^{r}(u)$ is union of limits of Hamilton orbits in
$\mathcal{N}_{A}\cap\mathcal{N}_{P}$ near $\nu_{0}\in\Sigma_{2}$.

###  2.3 Systems of transverse type

Finally, we want to state Dencker’s result regarding the propagation of
polarization sets for systems of transverse type; see [6]. Let
$P\in\Psi^{m}_{phg}$ be an $N\times N$ system of classical pseudodifferential
operators on a smooth manifold $X$, $p=\sigma(P)$ be the principal symbol, and
$\Sigma=(\det p)^{-1}(0)$ be the characteristics of $P$. Let

$\displaystyle\Sigma_{2}=\\{(x,\xi)\in\Sigma:d(\det p)=0\ \text{at}\
(x,\xi)\\},$ (2.14)

and $\Sigma_{1}=\Sigma\setminus\Sigma_{2}$. For systems of transverse type we
have $\Sigma$ is a union of two non radial hypersurfaces intersecting
transversally at $\Sigma_{2}$. More precisely, the systems of transverse type
is defined as the following

######  Definition 2.11.

The system $P$ is of transverse type at $\nu_{0}\in\Sigma_{2}$ if

$\displaystyle\Sigma_{2}\ \text{is a non-radial involutive manifold of
codimension}\ 2,$ (2.15) $\displaystyle\det p=e\cdot q,\ \text{where}\ e\neq
0\ \text{and}\ q\ \text{is real valued with Hessian having rank}\ 2\ \text{and
positivity}\ 1,$ (2.16) $\displaystyle\dim\ker p=2\ \text{on}\ \Sigma_{2},$
(2.17)

microlocally near $\nu_{0}$.

Similar to the case of systems of uniaxial type, if $P\in\Psi^{m}_{phg}$ is of
transverse type and $Pu\in H^{r}$ near $\nu\in\Sigma_{1}$, then $P$ is of real
principal type at $\nu$. Let $\mathcal{N}_{P}^{j}$ be as in $(\ref{Np at
Sj})$. In [6], Dencker modified slightly the definition of limit
polarizations.

######  Definition 2.12.

For $j=1,2$, the limit polarizations is defined by

$\displaystyle\partial\mathcal{N}_{P}^{j}=\\{(\nu,z)\in\Sigma_{2}\times\mathbb{C}^{N}:z=\underset{k\rightarrow\infty}{\lim}z_{k}\\},$
(2.18)

where $z_{k}\in\ker p(\nu_{k})$ and
$S_{j}\setminus\Sigma_{2}\ni\nu_{k}\rightarrow\nu$.

$\partial\mathcal{N}_{P}^{j}$ is conical in $\xi$ and linear in the fibers.
Dencker showed that $\partial\mathcal{N}_{P}^{j}$ is a $\mathcal{C}^{\infty}$
line bundle over $\Sigma_{2}$, $j=1,2$, and that

$\displaystyle\partial\mathcal{N}_{P}^{1}\cap\partial\mathcal{N}_{P}^{2}=\\{0\\}\
\ \ \text{over}\ \Sigma_{2}.$ (2.19)

Here, $S_{1}$ and $S_{2}$ are transverse at $\Sigma_{2}$, so their Hamilton
fields are non-parallel on $\Sigma_{2}$. Since $\Sigma_{2}$ is involutive of
codimension $2$, the Hamilton fields of $S_{j}$ are tangent to $\Sigma_{2}$
and generate the two-dimensional foliation of $\Sigma_{2}$. Moreover,
$\partial\mathcal{N}_{P}^{j}$ is foliated by limit Hamilton orbits which are
limits of Hamilton orbits in $\mathcal{N}_{P}^{j}$, and are unique line
bundles over bicharacteristics in $S_{j}$ at $\Sigma_{2}$ for $j=1,2$.

######  Theorem 2.13.

Let $P\in\Psi^{m}_{phg}$ be an $N\times N$ system of transverse type at
$\nu_{0}\in\Sigma_{2}$, and let $A\in\Psi^{0}_{phg}$ be a $1\times N$ system
such that the dimension of $\mathcal{N}_{A}\cap\mathcal{N}_{P}$ is equal to
$1$ at $\nu_{0}$, and
$M_{A}=\pi_{1}(\mathcal{N}_{A}\cap\mathcal{N}_{P}\setminus 0)$ is a
hypersurface near $\nu_{0}$. Assume that
$u\in\mathcal{D}^{\prime}(X,\mathbb{C}^{N})$ such that $Pu\in H^{r-m+1}$ and
$Au\in H^{r}$ at $\nu_{0}$. Then $\operatorname{Pol}^{r}(u)$ is a union of
(limit) Hamilton orbits in $\mathcal{N}_{A}\cap\mathcal{N}_{P}$. Here
$\pi_{1}:T^{*}X\times\mathbb{C}^{N}\rightarrow T^{*}X$ is the projection along
the fibers.

Note that in this case $M_{A}=S_{j}$ for some $j$, and
$\mathcal{N}_{A}\cap\mathcal{N}_{P}$ is a union of (limit) Hamilton orbits.

##  3 Propagation of polarization sets for systems of generalized transverse
type

In this section, we generalize Dencker’s result stated in Section 2.3 by
considering the system to have its characteristic set is union of $r_{0}$
hypersurfaces intersecting transversally at an involutive manifold of
codimension $d_{0}\geq 2$, with $r_{0}\geq 2$. Let $P\in\Psi^{m}_{phg}(X)$ be
an $N\times N$ system of classical pseudodifferential operators on a smooth
manifold $X$. Let $p=\sigma(P)$ be the principal symbol and $\Sigma=(\det
p)^{-1}(0)$ the characteristic set. Assume microlocally near
$(x_{0},\xi_{0})\in\Sigma$ that

$\displaystyle\begin{split}\Sigma=&\bigcup_{j=1}^{r_{0}}S_{j},\ r_{0}\geq 2,\
\text{where}\ S_{j}\ \text{are non-radial hypersurfaces intersecting
transversally at}\\\ \ &\Sigma_{2}=\bigcap_{j=1}^{r_{0}}S_{j},\end{split}$
(3.1) $\displaystyle\Sigma_{2}\ \text{is an involutive manifold of
codimension}\ d_{0}\geq 2.$ (3.2)

Moreover, assume that

$\displaystyle\text{the dimension of the fiber of}\ \mathcal{N}_{P}\ \text{is
equal to}\ r_{0}\ \text{at}\ \Sigma_{2},$ (3.3)

and

$\displaystyle d^{i}(\det p)=0\ \text{for}\ i<r_{0}\ \text{and}\
d^{r_{0}}(\det p)\neq 0\ \ \ \ \text{at}\ \Sigma_{2}.$ (3.4)

We want to consider the limits of $\mathcal{N}_{P}|_{\Sigma_{1}}$ when we
approach $\Sigma_{2}$, so let
$T_{\Sigma_{2}}\Sigma:=\bigcup_{j=1}^{r_{0}}T_{\Sigma_{2}}S_{j}$ (note that
$\Sigma$ is not a manifold), and
$\partial\Sigma_{1}:=T_{\Sigma_{2}}\Sigma/T\Sigma_{2}$. Before giving the
definition of systems of generalized transverse type, we need to give the
definition of the limit polarizations.

######  Definition 3.1.

For $j=1,...,r_{0}$, we define the limit polarizations

$\displaystyle\partial\mathcal{N}_{p}^{j}=\\{(\nu,\rho,z)\in\partial\Sigma_{1}\times\mathbb{C}^{N}:\rho\neq
0\ \text{and}\ z=\underset{k\rightarrow\infty}{\lim}z_{k}\\},$ (3.5)

where $z_{k}\in\ker p(\nu_{k})$ and $\nu_{k}\in S_{j}\setminus\Sigma_{2}$
satisfy
$(\nu-\nu_{k})/\lvert\nu-\nu_{k}\rvert\rightarrow\rho/\lvert\rho\rvert$ when
$k\rightarrow\infty$.

$\partial\mathcal{N}_{P}^{j}$ is conical in $\xi$ and $\rho$, and homogeneous
in the fiber. We will assume that the fiber of

$\displaystyle\mathcal{\partial}N_{P}^{1}\cap...\cap\mathcal{\partial}N_{P}^{r_{0}}=\\{0\\}\
\ \ \ \text{over}\ \ \partial\Sigma_{1}\setminus(\Sigma_{2}\times 0).$ (3.6)

This condition means that no element in $\mathcal{N}_{P}|_{\Sigma_{2}}$ can be
the limit of polarization vectors on all characteristic surfaces along the
same direction.

######  Definition 3.2.

The system $P$ is of generalized transverse type at $\nu_{0}\in\Sigma_{2}$, if
$(\ref{2.1 double refraction})$-$(\ref{2.4 double refraction})$, and
$(\ref{2.7 double refraction})$ hold microlocally near $\nu_{0}$.

######  Proposition 3.3.

Let $P\in\Psi^{1}_{phg}$ be an $N\times N$ system of generalized transverse
type at $\nu_{0}\in\Sigma_{2}$. Then by choosing suitable symplectic
coordinates, we may assume that $X=\mathbb{R}\times\mathbb{R}^{n-1}$,
$\nu_{0}=(0;(0,...,1))$, and

$\displaystyle S_{j}=\\{(t,x;\tau,\xi)\in
T^{*}(\mathbb{R}\times\mathbb{R}^{n-1}):\tau+\beta_{j}(t,x,\xi)=0\\},\ \
j=1,...,r_{0},$ (3.7)

microlocally near $\nu_{0}$. Here $\beta_{j}$ are real and homogeneous of
degree $1$ in $\xi$; with $\beta_{1}\equiv 0$, satisfies in a conical
neighborhood of $\nu_{0}$

$\displaystyle c\lvert\xi^{\prime}\rvert\leq\lvert\beta_{j}\rvert\leq
C\lvert\xi^{\prime}\rvert,\ \ \ j=2,...,r_{0},\ \ 0<c<C,$ (3.8)

where
$(\tau,\xi^{\prime},\xi^{\prime\prime})\in\mathbb{R}\times\mathbb{R}^{d_{0}-1}\times\mathbb{R}^{n-d_{0}}$,
which gives $\Sigma_{2}=\\{\tau=0,\xi^{\prime}=0\\}$. By conjugating $P$ with
an elliptic, scalar Fourier integral operators, and multiplying with elliptic
$N\times N$ systems of order $0$, we may assume that

$\displaystyle P\cong\begin{pmatrix}F&0\\\ 0&E\end{pmatrix}\ \
\operatorname{mod}\ \mathcal{C}^{\infty},$ (3.9)

microlocally near $\nu_{0}$, where $E\in\Psi^{1}_{phg}$ is an elliptic
$(N-r_{0})\times(N-r_{0})$ system and

$\displaystyle F\cong\operatorname{Id}_{r_{0}}D_{t}+K(t,x,D_{x})\ \
\operatorname{mod}\ \mathcal{C}^{\infty}.$ (3.10)

Here $K(t,x,D_{x})\in\mathcal{C}^{\infty}(\mathbb{R},\Psi^{1}_{phg})$ is an
$r_{0}\times r_{0}$ system, such that $k=\sigma(K)$ has $0$,
$\beta_{2}$,…,$\beta_{r_{0}}$ as eigenvalues.

###### Proof.

We will prove it in a way similar to how Dencker proved the normal form for
systems of uniaxial type. Since the result is local, we may assume
$X=\mathbb{R}^{n}$. Because $\Sigma_{2}$ is involutive, we may choose
symplectic, homogeneous coordinates $(x,\xi)\in T^{*}\mathbb{R}^{n}$ near
$\nu_{0}\in\Sigma_{2}$, so that $\nu_{0}=(0;(0,...,1))$ and

$\displaystyle\Sigma_{2}=\\{(x,\xi)\in T^{*}\mathbb{R}^{n}:\xi^{\prime}=0\\},$
(3.11)

where
$\xi=(\xi^{\prime},\xi^{\prime\prime})\in\mathbb{R}^{d_{0}}\times\mathbb{R}^{n-d_{0}}$.
We may also assume

$\displaystyle S_{1}=\\{(x,\xi)\in T^{*}\mathbb{R}^{n}:\xi_{1}=0\\},$ (3.12)

near $\nu_{0}$. Now, we rename $x_{1}=t$, $(x_{2},..,x_{d_{0}})=x^{\prime}$,
and $(x_{d_{0}+1},...,x_{n})=x^{\prime\prime}$. Since $S_{j}$ is intersecting
transversally with $S_{1}$ at $\Sigma_{2}$, we obtain

$\displaystyle S_{j}=\\{(t,x;\tau,\xi)\in
T^{*}(\mathbb{R}\times\mathbb{R}^{n-1}):\tau+\beta_{j}(t,x,\xi)=0\\},$ (3.13)

with $\beta_{j}$ real and homogeneous of degree $1$ in $\xi$, $\beta_{1}\equiv
0$, and

$\displaystyle
c\lvert\xi^{\prime}\rvert\leq\lvert\beta_{j}-\beta_{k}\rvert\leq
C\lvert\xi^{\prime}\rvert,\ \ \ j\neq k,$ (3.14)

in a conical neighborhood of $\nu_{0}$. By taking $k=1$, we obtain

$\displaystyle c\lvert\xi^{\prime}\rvert\leq\lvert\beta_{j}\rvert\leq
C\lvert\xi^{\prime}\rvert,\ \ \ j=2,...,r_{0}.$ (3.15)

Using that $\dim\mathcal{N}_{P}=r_{0}$ at $\Sigma_{2}$, we can find an
$N\times N$ elliptic matrix $b$ homogeneous of degree $0$ in the $\xi$
variables which maps $\operatorname{Im}p$ to $\\{z\in\mathbb{C}^{N};z_{j}=0,\
j\leq r_{0}\\}$ over $\Sigma_{2}$ near $\nu_{0}$, and we can choose an
$N\times N$ matrix $a$ homogeneous of degree $0$ in the $\xi$ variables such
that $a^{-1}$ maps $\ker p$ onto $\\{z\in\mathbb{C}^{N};z_{j}=0,\ j>r_{0}\\}$
over $\Sigma_{2}$ near $\nu_{0}$. Then we have

$\displaystyle bpa=\begin{pmatrix}s_{11}&s_{12}\\\ s_{21}&e\end{pmatrix}$
(3.16)

such that $e$ is an $(N-r_{0})\times(N-r_{0})$ matrix which is elliptic at
$\nu_{0}$, and $s_{11}$, $s_{12}$, $s_{21}$, vanish on $\Sigma_{2}$ near
$\nu_{0}$.

Now, we choose $N\times N$ systems of pseudodifferential operators $A$ and $B$
with principal symbols $a$, and $b$ respectively. Then

$\displaystyle BPA=\begin{pmatrix}S_{11}&S_{12}\\\ S_{21}&E\end{pmatrix}$
(3.17)

where its principal symbol is given by $(\ref{matrix principal symbol})$. As
$E$ is a system of order $1$ which is elliptic at $\Sigma_{2}$, choose $J$ to
be its microlocal parametrix of order $-1$. Multiply $BPA$ from the left with

$\displaystyle B_{1}=\begin{pmatrix}\operatorname{Id}_{r_{0}}&-S_{12}J\\\
0&\operatorname{Id}_{N-r_{0}}\end{pmatrix}.$ (3.18)

Multiply also $B_{1}BPA$ from the right by

$\displaystyle A_{1}=\begin{pmatrix}\operatorname{Id}_{r_{0}}&0\\\
-JS_{21}&\operatorname{Id}_{N-r_{0}}\end{pmatrix}.$ (3.19)

Hence, we get

$\displaystyle P\cong\begin{pmatrix}F&0\\\ 0&E\end{pmatrix}\ \
\operatorname{mod}\ \mathcal{C}^{\infty},$ (3.20)

microlocally near $\nu_{0}$, where $E\in\Psi^{1}_{phg}$ is an elliptic
$(N-r_{0})\times(N-r_{0})$ system. If $f$ is the principal symbol for $F$,
then conditions $(\ref{2.4 double refraction})$ and $(\ref{2.3
singularities})$ imply

$\displaystyle\det f=c\tau\prod_{i=2}^{r_{0}}(\tau+\beta_{i}),\ 0\neq c\in
S^{-1},$ (3.21)

thus $\partial_{\tau}^{r_{0}}(\det f)=\det(\partial_{\tau}f)\neq 0$ at
$\Sigma_{2}$. By Theorem A.3 in [5], and homogeneity, we may find homogeneous
system $C_{0}\in S^{0}$ such that

$\displaystyle f=C_{0}(\tau\operatorname{Id}_{r_{0}}+k(t,x,\xi)),$ (3.22)

where $\det C_{0}\neq 0$ at $\Sigma_{2}$. By multiplication with an elliptic
system, we may assume $C_{0}\equiv\operatorname{Id}_{r_{0}}$. Thus, $\det
f=\tau\prod_{i=2}^{r_{0}}(\tau+\beta_{j})$, which implies that $k(t,x,\xi)$
has the eigenvalues $0$, $\beta_{2}$,…,$\beta_{r_{0}}$. If $f_{0}\in S^{0}$ is
the term homogeneous of degree $0$ in the expansion of $F$, then Theorem A.4
in [5], and homogeneity give

$\displaystyle f_{0}=B_{-1}f+B_{0},$ (3.23)

where $B_{0}\in\mathcal{C}^{\infty}(\mathbb{R},S^{0})$ is independent of
$\tau$, and $B_{-1}\in S^{-1}$. By multiplying $f$ with an operator with
symbol $\operatorname{Id}_{r_{0}}-B_{-1}$, we may assume $B_{-1}\equiv 0$. By
induction over lower order terms we obtain $(\ref{3.4})$. ∎

We want to introduce symbol classes adapted to the functions $\beta_{j}$
defined in $(\ref{betaj})$ for $j=2,...,r_{0}$. Let

$\displaystyle\vartheta(\xi)=\langle\xi^{\prime}\rangle,$ (3.24)

where $\langle\xi^{\prime}\rangle=(1+\lvert\xi^{\prime}\rvert^{2})^{1/2}$,
thus $\vartheta\approx 1+\lvert\beta_{j}\rvert$. Consider the metric

$\displaystyle g(dx,d\xi)=\lvert dx\rvert^{2}+\lvert
d\xi^{\prime}\rvert^{2}\langle\xi^{\prime}\rangle^{-2}+\lvert
d\xi^{\prime\prime}\rvert^{2}\langle\xi\rangle^{-2},$ (3.25)

and $h^{2}=\sup g/g^{\sigma}=\langle\xi^{\prime}\rangle^{-2}$. We get that $g$
is $\sigma$ temperate, and $\beta_{j}\in S(\vartheta,g)$. Check [11, Chapter
XVIII] to know more about the symbol classes $S(\vartheta,g)$ of the Weyl
calculus. Moreover, using Taylor’s formula we can write

$\displaystyle\beta_{j}=a_{j}\xi^{\prime},$ (3.26)

with $a_{j}\in S^{0}$ is homogeneous of degree $0$ in $\xi$.

######  Proposition 3.4.

Let

$\displaystyle P=\operatorname{Id}_{r_{0}}D_{t}+K(t,x,D_{x})$ (3.27)

be an $r_{0}\times r_{0}$ system with
$K\in\mathcal{C}^{\infty}(\mathbb{R},\Psi^{1}_{phg})$, such that the
eigenvalues of $k=\sigma(K)$ are $0$, $\beta_{2}$,…,$\beta_{r_{0}}$. Then $P$
is of generalized transverse type if and only if
$k\in\mathcal{C}^{\infty}(\mathbb{R},S(\vartheta,g))$.

###### Proof.

First, let $k=(k_{ij})_{1\leq i,j\leq r_{0}}$. Let
$\alpha=(\alpha_{1},...,\alpha_{1r_{0}})\in\mathcal{C}^{\infty}(\mathbb{R},S^{1})$,
with $\alpha_{i}=(k_{i1},...,k_{ir_{0}})$ for $i=1,...,r_{0}$ homogeneous of
degree $1$ in $\xi$. By homogeneity,

$\displaystyle k\in\mathcal{C}^{\infty}(\mathbb{R},S(\vartheta,g))\
\Leftrightarrow\alpha=O(\beta_{j}),$ (3.28)

for every $j=2,...,r_{0}$.

Assume that $\alpha=O(\beta_{j})$ and $(\nu,\rho)\in\partial\Sigma_{1}$,
$\rho\neq 0$. Choose $\Sigma\setminus\Sigma_{2}\ni\nu_{l}\rightarrow\nu$ such
that
$(\nu-\nu_{l})\lvert\nu-\nu_{l}\rvert^{-1}\rightarrow\rho/\lvert\rho\rvert$,
$l\rightarrow\infty$. Let us define

$\displaystyle\gamma_{s}^{ij}(\nu,\rho):=\underset{\nu_{l}\rightarrow\nu}{\lim}\frac{k_{ij}}{\beta_{s}}(\nu_{l})\
\ \text{for}\ \ s\in\\{2,...,r_{0}\\}\ \ \text{and}\ \ 1\leq i,j\leq r_{0}.$
(3.29)

Since $\alpha=O(\beta_{j})$ does not depend on $\tau$, so the above definition
is independent of the choice of $\nu_{l}$. We get

$\displaystyle\partial\mathcal{N}_{P}^{1}(\nu,\rho)=\ker((\gamma_{s}^{ij}(\nu,\rho))_{1\leq
i,j\leq r_{0}}),\ \forall s\in\\{2,...,r_{0}\\},$ (3.30)

where $(\gamma_{s}^{ij}(\nu,\rho))_{1\leq i,j\leq r_{0}}$ denote the matrix
with entries $\gamma_{s}^{ij}(\nu,\rho)$ for $1\leq i,j\leq r_{0}$.

$\displaystyle\partial\mathcal{N}_{P}^{s}(\nu,\rho)=\ker(-\operatorname{Id}_{r_{0}}+(\gamma_{s}^{ij}(\nu,\rho))_{1\leq
i,j\leq r_{0}})\ \text{for}\ s\in\\{2,...,r_{0}\\}.$ (3.31)

It is easy to see that the condition $(\ref{2.7 double refraction})$ is
satisfied.

On the other hand, assume that $\alpha\neq O(\beta_{j})$ at
$\nu\in\Sigma_{2}$. Then, there exists a sequence
$\nu_{l}=(t_{l},x_{l};0,\xi_{l})\rightarrow\nu$, such that

$\displaystyle\lvert\alpha(\nu_{l})\rvert>l\lvert\beta_{j}(\nu_{l})\rvert,\ \
\forall l\in\mathbb{N}.$ (3.32)

It is no restriction to assume that
$\\{(\nu-\nu_{l})\lvert\nu-\nu_{l}\rvert^{-1}\\}$ has a limit
$0\neq\rho\in\partial\Sigma_{1}|_{\nu}$ as $l\rightarrow\infty$, and that

$\displaystyle\varepsilon^{i,j}=\underset{l\rightarrow\infty}{\lim}k_{ij}(\nu_{l})/\lvert\alpha(\nu_{l})\rvert\
\ \text{exists}.$ (3.33)

Since $\beta_{j}(\nu_{l})/\lvert\alpha(\nu_{l})\rvert\rightarrow 0$, we get
that

$\displaystyle\partial\mathcal{N}_{P}^{s}(\nu,\rho)\supseteq\ker((\varepsilon^{i,j})_{1\leq
i,j\leq r_{0}})\ \ \ \text{for}\ s=1,...,r_{0}.$ (3.34)

Now, we want to show that $\ker((\varepsilon^{i,j})_{1\leq i,j\leq
r_{0}})\neq\\{0\\}$. Since, we have $\det k=0$, we get
$\det\big{(}(\varepsilon^{i,j})_{1\leq i,j\leq r_{0}}\big{)}=0$. Hence, we get
that the rank of $(\varepsilon^{i,j})_{1\leq i,j\leq r_{0}}$ is less than or
equal to $r_{0}-1$, which gives in turn that
$\dim\ker((\varepsilon^{i,j})_{1\leq i,j\leq r_{0}})$ is greater than or equal
to $1$. ∎

We will introduce two spaces that we are going to use. These spaces were also
given by Dencker in [6] and he showed the relation between these two spaces by
a lemma that we will also state. Let $H^{r,s}$ be the space of
$u\in\mathcal{S}^{\prime}$ (here $u$ depends on $t$ and $x$) satisfying

$\displaystyle\lVert
u\rVert^{2}_{r,s}=(2\pi)^{-n}\int\lvert\hat{u}(\tau,\xi)\lvert^{2}\langle(\tau,\xi)\rangle^{2r}\langle(\tau,\xi^{\prime})\rangle^{2s}d\tau
d\xi<\infty.$ (3.35)

We say that $u\in H^{r,s}$ at $\nu\in T^{*}\mathbb{R}^{n}\setminus 0$, that
is, $\nu\notin\operatorname{WF}^{r,s}(u)$, if $u=u_{1}+u_{2}$, where $u_{1}\in
H^{r,s}$ and $\nu\notin\operatorname{WF}(u_{2})$.

Similarly, when we have $u$ depends only on $x$ then the norm becomes

$\displaystyle\lVert
u\rVert^{2}_{r,s}=(2\pi)^{-n}\int\lvert\hat{u}(\xi)\lvert^{2}\langle\xi\rangle^{2r}\langle\xi^{\prime}\rangle^{2s}d\xi<\infty.$
(3.36)

We have $S^{0}\subset S(1,g)$. Note that the spaces $H^{r,s}$ is a particular
case of the spaces $B_{p,k}$ introduced by Hörmander; check [10], where $p=2$
and
$k(\tau,\xi)=\langle\tau,\xi\rangle^{r}\langle\tau,\xi^{\prime}\rangle^{s}$.

######  Proposition 3.5.

Assume that $P$ is an $r_{0}\times r_{0}$ system of pseudodifferential
operators of order $1$ on $\mathbb{R}^{n}$, on the form $(\ref{form normal})$,
with $K\in\mathcal{C}^{\infty}(\mathbb{R},\operatorname{Op}(S(\vartheta,g)))$
near $\nu_{0}\in\Sigma_{2}$. Let
$u\in\mathcal{S}^{\prime}(\mathbb{R}^{n},\mathbb{C}^{r_{0}})$ and assume
$Pu\in H^{r,s}$ at $\nu_{0}$. Then, for every $\delta>0$ we can find
$c_{\delta}$ and $C_{\delta,N}>0$ and $v_{\delta}\in H^{r,s+1}$ at $\nu_{0}$,
such that $u_{\delta}=u-v_{\delta}$ satisfies

$\displaystyle\lvert\hat{u}_{\delta}(\tau,\xi)\rvert\leq
C_{\delta,N}\langle(\tau,\xi)\rangle^{-N},\ \ \ \forall N,$ (3.37)

when
$\lvert\tau\rvert>c_{\delta}(\langle\xi\rangle^{\delta}+\langle\xi^{\prime}\rangle)$.

###### Proof.

Follow the proof of Proposition $A.1$ in [6]. ∎

Let $H_{*}^{r,s}$ be the Banach space of $u\in\mathcal{S}^{\prime}$,
satisfying

$\displaystyle(\lVert
u\rVert^{*}_{r,s})^{2}=(2\pi)^{-n}\int\lvert\hat{u}(\tau,\xi)\lvert^{2}\langle\xi\rangle^{2r}\langle\xi^{\prime}\rangle^{2s}d\tau
d\xi<\infty.$ (3.38)

If we have $u$ depends only on $x$ then the norm becomes

$\displaystyle(\lVert
u\rVert^{*}_{r,s})^{2}=(2\pi)^{-n}\int\lvert\hat{u}(\xi)\lvert^{2}\langle\xi\rangle^{2r}\langle\xi^{\prime}\rangle^{2s}d\xi<\infty.$
(3.39)

Hence, $\lVert u\rVert^{*}_{r,s}=\lVert u\rVert_{r,s}$ when $u$ depends only
on $x$.

If $u\in H^{*}_{r,s}$ then we get $u|_{t=\rho}\in H^{r,s}$ for almost all
$\rho$, by Fubini’s theorem. If $u\in\mathcal{S}^{\prime}$ satisfies
$(\ref{3.30})$, then

$\displaystyle\lVert u\rVert^{*}_{r-\delta s_{-},s}\leq C_{r,s}(\lVert
u\rVert_{r,s}+1)\leq C^{\prime}_{r,s}(\lVert u\rVert^{*}_{r+\delta
s_{+},s}+1),\ \forall r,s\in\mathbb{R},$ (3.40)

where $s_{\pm}=\max(\pm s,0)$. Hence, we lose only $O(\delta)$ derivatives
when taking restriction to $\\{t=r\\}$, for almost all $r$.

######  Definition 3.6.

Let $u\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$, and assume $\xi\neq 0$ in
$\operatorname{WF}(u)$. We say that $u\in H_{*}^{r,s}$ at
$(t_{0},x_{0},\xi_{0})$, that is,
$(t_{0},x,_{0},\xi_{0})\notin\operatorname{WF}_{*}^{r,s}(u)$, if there exists
$\phi(t,x,\xi)\in\mathcal{C}^{\infty}(\mathbb{R},S^{0}_{1,0})$ such that
$\phi(t,x,D_{x})u\in H_{*}^{r,s}$ and
$\underline{\lim}_{\lambda\rightarrow\infty}\lvert\phi(t_{0},x_{0},\lambda\xi_{0})\rvert\neq
0$.

We have

$\displaystyle(t_{0},x_{0},\xi_{0})\notin\operatorname{WF}_{*}^{r,s}(u)\Rightarrow(x_{0},\xi_{0})\notin\operatorname{WF}^{r,s}(u_{\rho}),$
(3.41)

for almost all $\rho$ close to $t_{0}$, where $u_{\rho}=u|_{t=\rho}$. If
$\xi\neq 0$ in $\operatorname{WF}(u)$, then from [2, Lemma 2.3], we get that

$\displaystyle\pi_{0}(\operatorname{WF}^{r,0})(u)=\operatorname{WF}_{*}^{r,0}(u),$
(3.42)

where $\pi_{0}(t,x;\tau,\xi)=(t,x,\xi)$. The following lemma gives the result
for the more general wavefront sets:

######  Lemma 3.7.

Assume that $u\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ satisfies
$(\ref{3.30})$. Then $Au$ satisfies $(\ref{3.30})$, for any
$A\in\mathcal{C}^{\infty}(\mathbb{R},\Psi^{m}_{\delta,0})$ $\forall\ m$ and
$\forall\ \delta>0$. We also obtain

$\displaystyle\operatorname{WF}_{*}^{r-\delta
s_{-},s}(u)\subseteq\pi_{0}(\operatorname{WF}^{r,s}(u))\subseteq\operatorname{WF}_{*}^{r+\delta
s_{+},s}(u),$ (3.43)

where $s_{\pm}=\max(\pm s,0)$ and $\pi_{0}(t,x;\tau,\xi)=(t,x,\xi)$. Since
$u\in\mathcal{C}^{\infty}$ in $\pi_{0}^{-1}(\Sigma_{2})\setminus\Sigma_{2}$ by
$(\ref{3.30})$, we find

$\displaystyle\iota_{0}(\operatorname{WF}_{*}^{r-\delta
s_{-},s}(u))\subseteq\operatorname{WF}^{r,s}(u)\subseteq\iota_{0}(\operatorname{WF}_{*}^{r+\delta
s_{+},s}(u))\ \ \ \ \text{on}\ \Sigma_{2},$ (3.44)

where $\iota_{0}(t,x,\xi)=(t,x;0,\xi)$.

###### Proof.

See [6, Lemma A.3]. ∎

Note that $H^{r}=H^{r,0}$ is the usual Sobolev space. Changing the notation,
let $x_{1}=t$, $x^{\prime}=(x_{2},...,x_{d_{0}})$, then
$x=(x_{1},x^{\prime},x^{\prime\prime})\in\mathbb{R}\times\mathbb{R}^{d_{0}-1}\times\mathbb{R}^{n-d_{0}}$.
Introduce the symbol classes $S^{r,s}=S(\langle\xi\rangle^{r}h^{-s},g)$ where
$h^{-2}=1+\lvert\xi_{1}\rvert^{2}+\lvert\xi^{\prime}\rvert^{2k_{0}+2}/\langle\xi\rangle^{2k_{0}}$
and $\langle\xi\rangle$ are weight for the metric $g$ defined by

$\displaystyle g_{x,\xi}(dx,d\xi)=\lvert dx\rvert^{2}+\lvert
d\xi\rvert^{2}h^{2}.$ (3.45)

Let $\Psi^{r,s}=\operatorname{Op}S^{r,s}$ be the corresponding
pseudodifferential operators, which maps $H^{r,s}$ into $L^{2}$. Returning to
the old notation where using $t$ instead of $x_{1}$, and assume that
$P\in\mathcal{C}^{\infty}(\mathbb{R},\operatorname{Op}(S(\vartheta,g)))$ be as
in $(\ref{form normal})$, we get $P\in\Psi^{0,1}$.

In order to prove the main theorem, we need to study the regularity of a
Cauchy problem that we will state. Consider the following $N\times N$ system

$\displaystyle Q=q\operatorname{Id}_{N}+\sum_{i=0}^{r_{0}-1}Q_{i}.$ (3.46)

Here $q$ is a scalar operator with symbol

$\displaystyle q(t,x;\tau,\xi)=\tau\prod_{j=2}^{r_{0}}(\tau+\beta_{j}),$
(3.47)

where $\beta_{j}\in S(\vartheta,g)$ is homogeneous and satisfies
$(\ref{betaj})$. We will assume

$\displaystyle Q_{i}=\sum_{k=0}^{i}A_{i-k}^{i}D_{t}^{k},$ (3.48)

with $A_{k^{\prime}}^{i}\in\operatorname{Op}S(\vartheta^{k^{\prime}},g)$. We
are going to study the following Cauchy Problem:

$\displaystyle\begin{split}Qu&=f\\\ D_{t}^{k}u|_{t=0}&=u_{k},\ \text{for}\
k=0,...,r_{0}-1.\end{split}$ (3.49)

We are going to assume that the $\xi\neq 0$ in $\operatorname{WF}(u)$. Hence,
the restrictions are well defined.

######  Proposition 3.8.

Assume that $u\in\mathcal{D}^{\prime}(\mathbb{R}^{n},\mathbb{C}^{N})$
satisfies $(\ref{cauchy1})$, and $\xi\neq 0$ in $\operatorname{WF}(u)$. If
$u_{k}\in H^{r,s-k}$ at $(x_{0},\xi_{0})$ for $k=0,...,r_{0}-1$, $f\in
H_{*}^{r,s-r_{0}+1}$ at $(t,x_{0},\xi_{0})$ for $0\leq t\leq t_{0}$, and
$\xi_{0}^{\prime}=0$, then $u\in H_{*}^{r,s}$ at $(t_{0},x_{0},\xi_{0})$.

###### Proof.

By conjugating with an elliptic, scalar operator with symbol in
$S(\langle\xi\rangle^{r}\vartheta^{s},g)$, we may assume that $r=s=0$. We will
reduce to a first order symmetric system. Let
$v_{k}=\lambda^{k-1}D_{t}^{k-1}u$ for $k=1,...,r_{0}$, where
$\lambda\in\operatorname{Op}S(\vartheta^{-1},g)$ has symbol $\vartheta^{-1}$.
Hence, $v_{k}=\lambda D_{t}v_{k-1}$ for $k=2,...,r_{0}$. Then $V=\
^{t}(v_{1},...,v_{r_{0}})\in\mathcal{D}^{\prime}(\mathbb{R}^{n},\mathbb{C}^{r_{0}N})$,
$\xi\neq 0$ in $\operatorname{WF}(V)$, and $V$ satisfies

$\displaystyle\begin{split}PV&=F\\\ V|_{t=0}&=V_{0}.\end{split}$ (3.50)

Here $P=\operatorname{Id}_{r_{0}N}D_{t}+K$, $F=(0,...,0,\lambda^{r_{0}-1}f)$,
$V_{0}=(u_{0},\lambda u_{1},...,\lambda^{r_{0}-1}u_{r_{0}-1})$, and
$K\in\operatorname{Op}S(\vartheta,g)$ has principal symbol $k=(k_{ij})_{1\leq
i,j\leq r_{0}}$ such that

$\displaystyle k_{ii+1}=-\vartheta\operatorname{Id}_{N}\ \text{for}\ 1\leq
i\leq r_{0}-1,$ (3.51) $\displaystyle
k_{r_{0}j}=\sum\beta_{i_{1}}...\beta_{i_{r_{0}-j+1}}\vartheta^{j-r_{0}}\
\text{for}\ 2\leq j\leq r_{0},$ (3.52) $\displaystyle k_{ij}=0\
\text{elsewhere}$ (3.53)

where the sum in $(\ref{kr0j})$ is such that $2\leq i_{1}\leq j$ and
$i_{k-1}+1\leq i_{k}\leq j+k-1$ for $2\leq k\leq r_{0}-j+1$.

We find $V_{0}\in H^{0,0}=L^{2}$ at $(x_{0},\xi_{0})$ and $F\in H_{*}^{0,0}$
at $(t,x_{0},\xi_{0})$ for $0\leq t\leq t_{0}$. Thus the result follows from
the next proposition, which we will state after the following Lemma. ∎

######  Lemma 3.9.

When the above assumptions are satisfied we get

$\displaystyle\int_{0}^{\varepsilon}\lVert V\rVert(t)dt\leq
C\varepsilon^{1/2}\bigg{(}\int_{0}^{\varepsilon}\lVert
D_{t}V+KV\rVert(t)dt+\lVert V\rVert(0)\bigg{)}$ (3.54)

for $V\in\mathcal{C}^{\infty}_{0}(\mathbb{R}^{n+1},\mathbb{C}^{r_{0}N})$, if
$\varepsilon>0$ is sufficiently small. Here $\lVert.\rVert(t)$ denotes the
$L^{2}$ norm in the $x$ variables, depending on $t$.

###### Proof.

We are going to prove it in a similar way as the proof of Lemma $5.2$ in [3].
As $k$ is diagonalizable in $S(1,g)$, we get that
$k=\sum_{j=1}^{r_{0}}\beta_{j}\pi_{j}$, with $\beta_{1}=0$, where
$\pi_{j}(t,x,\xi)\in S(1,g)$ is the projection on the eigenvectors
corresponding to the eigenvalues $\beta_{j}$ along the others when
$\xi^{\prime}\neq 0$. $k$ is symmetrizable with symmetrizer
$M=\sum\pi_{j}^{*}\pi_{j}$, that is $M>0$ amd $Mk$ is symmetric. Note that $k$
is symmetrizable means there exists symmetric $N\times N$ system
$M(t,x,\xi)\in S(1,g)$ such that $0<c\leq M$ and $Mk-(Mk)^{*}\in S(1,g)$. If
we put $\lVert V\rVert(t)$ to be the $L^{2}$ norm in the $x$-variables,
depending on $t$, and we put

$\displaystyle\lVert V\rVert_{M}^{2}(t)=\langle MV,V\rangle=\int\langle
MV(t,x),\overline{V(t,x)}\rangle dx$ (3.55)

then

$\displaystyle c\leq\frac{\lVert V\rVert_{M}^{2}(t)}{\lVert
V\rVert^{2}(t)}\leq C$ (3.56)

If $D_{t}V+KV=F$, we obtain

$\displaystyle\begin{split}\partial_{t}\lVert
V\rVert_{M}^{2}(t)&=\langle(\partial_{t}M-i(MK-K^{*}M))V,V\rangle(t)+\langle
MF,V\rangle(t)+\langle MV,F\rangle(t)\\\ &\leq C(\lVert
V\rVert_{M}^{2}(t)+\lVert F\rVert_{M}^{2}(t)).\end{split}$ (3.57)

By Grönwall’s inequality we get, for bounded $t$,

$\displaystyle\lVert V\rVert_{M}^{2}(t)\leq C\bigg{(}\lVert
V\rVert_{M}^{2}(0)+\int_{0}^{t}\lVert F\rVert_{M}^{2}(s)ds\bigg{)},$ (3.58)

so $(\ref{5.7})$ and integration gives the result. ∎

######  Proposition 3.10.

Let $P=D_{t}\operatorname{Id}_{N}+K$ where
$K\in\operatorname{Op}S(\vartheta,g)$ has symbol which is diagonalizable in
$S(1,g)$ with eigenvalues $0,\beta_{2},...,\beta_{r_{0}}$ $\operatorname{mod}$
$S(1,g)$, and $\beta_{j}\in\mathcal{C}^{\infty}$ is homogeneous, satisfying
$(\ref{betaj})$ for $j=2,...,r_{0}$. Assume that
$V\in\mathcal{D}^{\prime}(\mathbb{R}^{n},\mathbb{C}^{N})$, $\xi\neq 0$ in
$\operatorname{WF}(V)$ and $V$ satisfies $(\ref{0.23})$. If $V_{0}\in L^{2}$
at $(x_{0},\xi_{0})$, $F\in H_{*}^{0,0}$ at $(t,x_{0},\xi_{0})$ for $0\leq
t\leq t_{0}$, and $\xi_{0}^{\prime}=0$, then $V\in H_{*}^{0,0}$ at
$(t_{0},x_{0},\xi_{0})$.

The condition on $k$ means that there exists a basis of eigenvectors
$\\{v_{j}\\}\in S(1,g)$, with eigenvalues $0$, $\beta_{2}$,…,$\beta_{r_{0}}$
$\operatorname{mod}$ $S(1,g)$.

The above proposition is similar to Proposition $7.2$ in [5]. To prove
Proposition $7.2$ in [5], Dencker used the parametrix constructed in [4] for
$P=D_{t}\operatorname{Id}_{N}+K$, where $K\in\operatorname{Op}S(\vartheta,g)$
has principal symbol $k$ satisfying the conditions in Proposition $7.2$, with
$\vartheta$, and $g$ are given as $(3.8)$, and $(3.9)$ in [5] respectively,
and he used the microlocal uniqueness; see [4]. In our case; case of
generalized transverse type, we are using different weight and metric, but we
can still construct a parametrix for the $N_{0}\times N_{0}$ matrix, where
$K\in\operatorname{Op}S(\vartheta,g)$ has principal symbol $k$ satisfying the
conditions in Proposition 3.10, and we can prove microlocal uniqueness as in
[4]. The steps are similar to that in [4], except some details are changed.
Thus, we will not write about the construction of the parametrix and the
microlocal uniqueness and we invite the readers to check [4].

###### Proof of Proposition 3.10.

The argument is all the same as in the proof of Proposition $7.2$ in [5],
except that to get
$U\in\mathcal{C}^{\infty}(\mathbb{R},\mathcal{D}^{\prime}(\mathbb{R}^{n-1},\mathbb{C}^{N}))\cap
L^{2}$, we need to prove $(5.6)$ in [3] for our case, which we proved in Lemma
3.9. For the parametrix and the microlocal uniqueness part we have mentioned
above that it can be proven in our case in a similar way. ∎

Note that here $S_{j}$ are transverse at $\Sigma_{2}$, so their Hamiton fields
are non-parallel on $\Sigma_{2}$. That is why we considered in the main
theorem, Theorem 1.1, that $M_{A}$ is a hypersurface near $\nu_{0}$, so one
can consider $M_{A}=S_{j}$ for some $j$. Now, we will prove Theorem 1.1 for
the case of systems of generalized transverse type.

###### Proof.

By multiplication and conjugation with elliptic, scalar pseudodifferential
operators, we may assume that $m=1$ and $r=0$, and using the normal form we
can assume that $N=r_{0}$, and $P$ is of the form in Proposition 3.4. By using
Theorem A.4 in [5] for all the terms in the expansion of $A$, we obtain that
$A\in\mathcal{C}^{\infty}(\mathbb{R},\Psi^{0}_{phg})$. As the dimension of the
fiber of $\mathcal{N}_{A}\cap\mathcal{N}_{P}$ is $1$ at
$\nu_{0}\in\Sigma_{2}$, and the dimension of the fiber of $\mathcal{N}_{P}$ is
$r_{0}$ at $\Sigma_{2}$, we get $\operatorname{rank}\sigma(A)=r_{0}-1$ at
$\Sigma_{2}$. Hence, we can conjugate by suitable elliptic systems in
$\mathcal{C}^{\infty}(\mathbb{R},\Psi^{0}_{phg})$ to get that
$Au\cong(u_{1},...u_{r_{0}-1},0)\in H^{\epsilon}$ in a conical neighborhood
$U$ of $\nu_{0}$, for some $\epsilon>0$. Then, we find
$\pi_{1}(\operatorname{Pol}^{0}(u))=\operatorname{WF}^{0}(u_{r_{0}})$ in $U$.
By shrinking $U$ and decreasing $\epsilon$, we may assume $Pu\in H^{\epsilon}$
in $U$. Remember that we have $P\in\Psi^{0,1}$, hence we get $Qu=\
^{t}P^{\operatorname{co}}Pu\in H^{\epsilon,-r_{0}+1}$ there, where $\
{}^{t}P^{\operatorname{co}}$ is the adjugate matrix of $P$. Let
$Q=(q_{ij})_{i,j=1}^{r_{0}}$, and $P=(P_{ij})_{i,j=1}^{r_{0}}$. Since
$q_{r_{0}i}$ are in
$\mathcal{C}^{\infty}(\mathbb{R},\operatorname{Op}S(\vartheta^{r_{0}-1},g))$
for $i=1,...r_{0}-1$, we find that $q_{r_{0}r_{0}}u_{r_{0}+1}\in
H^{\epsilon,-r_{0}+1}$ in $U$. Similarly, we find that
$P_{r_{0}r_{0}}u_{r_{0}}\in H^{\epsilon,-1}$, which in turns gives
$D^{k-1}_{t}P_{r_{0}r_{0}}u_{r_{0}}\in H^{\epsilon,-k}$ for $k=1,...,r_{0}-1$.
We want to prove that $u_{r_{0}}\in H^{0}$ at $(t,x_{0};0,\xi_{0})\in
U\cap\Sigma_{2}$ for $t<t_{0}$, implies $u_{r_{0}}\in H^{0}$ at
$(t_{0},x_{0};0,\xi_{0})=\nu_{0}$.

Thus assume that $u_{r_{0}}\in H^{0}$ at $(t,x_{0};0,\xi_{0})\in
U\cap\Sigma_{2}$ when $t<t_{0}$. We may assume that $\delta\leq 1$, then Lemma
3.7 gives that $u_{r_{0}}$, $P_{r_{0}r_{0}}u_{r_{0}}$,
$D^{k-1}_{t}P_{r_{0}r_{0}}u_{r_{0}}$ for $k=2,...,r_{0}-1$ and
$q_{r_{0}r_{0}}u_{r_{0}}$ satisfies $(\ref{3.30})$. Then $\xi\neq 0$ in
$\operatorname{WF}(u_{r_{0}})$, and assuming that $\delta\leq\epsilon$ in
$(\ref{3.30})$ we find that $P_{r_{0}r_{0}}u_{r_{0}}\in H_{*}^{0,-1}$,
$D^{k-1}_{t}P_{r_{0}r_{0}}u_{r_{0}}\in H_{*}^{0,-k}$ for $k=1,...,r_{0}-1$ and
$q_{r_{0}r_{0}}u_{r_{0}}\in H_{*}^{0,-r_{0}+1}$ in $\pi_{0}(U\cap\Sigma_{2})$,
and $u_{r_{0}}\in H_{*}^{0}$ at $(t,x_{0},\xi_{0})\in\pi_{0}(U\cap\Sigma_{2})$
for $t<t_{0}$. Since $P_{r_{0}r_{0}}\cong D_{t}$ $\operatorname{mod}$
$\mathcal{C}^{\infty}(\mathbb{R},\operatorname{Op}S(\vartheta,g))$, we get
$P_{r_{0}r_{0}}u_{r_{0}}\cong D_{t}u_{r_{0}}\in H_{*}^{0,-1}$ which implies,
using $D^{k-1}_{t}P_{r_{0}r_{0}}u_{r_{0}}\in H_{*}^{0,-k}$, that
$D_{t}^{k}u_{r_{0}}\in H_{*}^{0,-k}$ for $k=1,...,r_{0}-1$. This gives

$\displaystyle(D_{t}^{k}u_{r_{0}})|_{t=r}\in H_{*}^{0,-k}\ \ \ \text{at}\ \
(x_{0},\xi_{0}),\ \text{for}\ k=0,...,r_{0}-1,$ (3.59)

for almost all $r<t_{0}$, close to $t_{0}$. Proposition 3.8 (with $N=1$ and
$Q=q_{r_{0}r_{0}}$) gives $u_{r_{0}}\in H_{*}^{0}$ at $(t_{0},x_{0},\xi_{0})$,
and Lemma 3.7 gives $u_{r_{0}}\in H^{0}$ at $(t_{0},x_{0};0,\xi_{0})$. ∎

##  4 Propagation of polarization sets for systems of MHD type

In this section, we define systems of MHD type which are also systems having
their characteristic sets are union of $r_{0}$ hypersufaces intersecting
transversally at an involutive manifold of codimension $d_{0}\geq 2$, with
$r_{0}\geq 2$. However, they satisfy some assumptions different than that in
case of systems of generalized transverse type. We named them systems of MHD
type because we first noticed these types of systems when we considered
studying the propagation of polarization sets of the linearized ideal MHD
equations. Let $P\in\Psi^{m}_{phg}(X)$ be an $N\times N$ system of classical
pseudodifferential operators of order $m$ on a smooth manifold $X$. Let
$p=\sigma(P)$ be the principal symbol of $P$, $\det p$ the determinant of $p$
and $\Sigma=(\det p)^{-1}(0)$ the characteristic set of $P$. Assume
microlocally near $\nu_{0}=(x_{0},\xi_{0})\in\Sigma$, that

$\displaystyle\begin{split}\Sigma=&\bigcup_{j=1}^{r_{0}}S_{j},\ r_{0}\geq 2,\
\text{where}\ S_{j}\ \text{are non-radial hypersurfaces intersecting
transversally at}\\\ \ &\Sigma_{2}=\bigcap_{j=1}^{r_{0}}S_{j}.\end{split}$
(4.1)

We are interested in finding the propagation of polarization set at
$\Sigma_{2}$, as in our application; see Section 5, we know the result on
$\Sigma\setminus\Sigma_{2}$. We assume that

$\displaystyle\Sigma_{2}\ \text{is an involutive manifold of codimension}\
d_{0}\geq 2,$ (4.2) $\displaystyle d^{j}(\det p)=0\ \ \text{for}\ j\leq r_{0}\
\text{and}\ d^{r_{0}+1}(\det p)\neq 0\ \ \ \ \text{at}\ \Sigma_{2}.$ (4.3)

Moreover, assume that

$\displaystyle\text{the dimension of the fiber of}\ \mathcal{N}_{P}\ \text{is
equal to}\ r_{0}+1\ \text{at}\ \Sigma_{2},$ (4.4) $\displaystyle d(\det p)=0\
\text{and}\ d^{2}(\det p)\neq 0\ \ \text{at}\ S_{i_{0}}\setminus\Sigma_{2},\
\text{for only one}\ i_{0}\in\\{1,...r_{0}\\},$ (4.5)

and

$\displaystyle d(\det p)\neq 0\ \text{at}\ S_{j}\setminus\Sigma_{2}\ \text{for
each}\ j\in\\{1,...,r_{0}\\},\ \text{such that}\ j\neq i_{0}.$ (4.6)

Moreover, assume that $\ {}^{t}P^{\operatorname{co}}$ the adjugate matrix of
$P$ can be written as

$\displaystyle^{t}P^{\operatorname{co}}=RL_{1}+L_{2},$ (4.7)

with $R$ being a scalar pseudodifferential operator of order $m$, with
$\sigma(R)$ vanishing on $S_{i_{0}}$ to the first order. $L_{1}$, and $L_{2}$
are $N\times N$ system of pseudodifferential operators of order $m(N-2)$, and
$m(N-3)$ respectively. Assume also that

$\displaystyle\sigma(L_{1})p=f\operatorname{Id}_{N},$ (4.8)

with $\Sigma=\\{f=0\\}$. We are using same notation as the previous section.

######  Definition 4.1.

The system $P$ is of MHD type at $\nu_{0}\in\Sigma_{2}$, if $(\ref{sigma and
sigma2})$-$(\ref{principal symbol of L1})$, and $(\ref{2.7 double
refraction})$ hold microlocally near $\nu_{0}$.

We want to write systems of MHD in a normal form.

######  Proposition 4.2.

Let $P\in\Psi^{1}_{phg}$ be an $N\times N$ system of MHD type at
$\nu_{0}\in\Sigma_{2}$. Then by choosing suitable symplectic coordinates, we
may assume that $X=\mathbb{R}\times\mathbb{R}^{n-1}$, $\nu_{0}=(0;(0,...,1))$,
and

$\displaystyle S_{j}=\\{(t,x;\tau,\xi)\in
T^{*}(\mathbb{R}\times\mathbb{R}^{n-1}):\tau+\beta_{j}(t,x,\xi)=0\\},\ \
j=1,...,r_{0},$ (4.9)

microlocally near $\nu_{0}$. Here $\beta_{j}$ are real and homogeneous of
degree $1$ in $\xi$; with $\beta_{1}\equiv 0$, satisfies in a conical
neighborhood of $\nu_{0}$

$\displaystyle c\lvert\xi^{\prime}\rvert\leq\lvert\beta_{j}\rvert\leq
C\lvert\xi^{\prime}\rvert\ \ \ j=2,...,r_{0},\ \ 0<c<C,$ (4.10)

where
$(\tau,\xi^{\prime},\xi^{\prime\prime})\in\mathbb{R}\times\mathbb{R}^{d_{0}-1}\times\mathbb{R}^{n-d_{0}}$,
which gives $\Sigma_{2}=\\{\tau=0,\xi^{\prime}=0\\}$. By conjugating $P$ with
elliptic, scalar Fourier integral operators, and multiplying with elliptic
$N\times N$ systems of order $0$, we may assume that

$\displaystyle P\cong\begin{pmatrix}F&0\\\ 0&E\end{pmatrix}\ \
\operatorname{mod}\ \mathcal{C}^{\infty},$ (4.11)

microlocally near $\nu_{0}$, where $E\in\Psi^{1}_{phg}$ is an elliptic
$(N-r_{0}-1)\times(N-r_{0}-1)$ system, and

$\displaystyle F\cong\operatorname{Id}_{r_{0}+1}D_{t}+K(t,x,D_{x})\ \
\operatorname{mod}\ \mathcal{C}^{\infty}.$ (4.12)

Here $K(t,x,D_{x})\in\mathcal{C}^{\infty}(\mathbb{R},\Psi^{1}_{phg})$ is an
$(r_{0}+1)\times(r_{0}+1)$ system, and the eigenvalues of $k(t,x,\xi)$; the
principal symbol of $K(t,x,D_{x})$, are $0$ (double),
$\beta_{2}$,…$\beta_{r_{0}}$.

###### Proof.

In a similar way as in the proof of Proposition 3.3, we get $(\ref{2.3
singularities MHD})$, $(\ref{betaj MHD})$, and

$\displaystyle P\cong\begin{pmatrix}F&0\\\ 0&E\end{pmatrix}\ \
\operatorname{mod}\ \mathcal{C}^{\infty},$ (4.13)

microlocally near $\nu_{0}$, where $E\in\Psi^{1}_{phg}$ is an elliptic
$(N-r_{0}-1)\times(N-r_{0}-1)$ system. If $f$ is the principal symbol for $F$,
then conditions $(\ref{detp at Si0})$, $(\ref{detp at Sj})$, and $(\ref{2.3
singularities MHD})$ imply

$\displaystyle\det f=c\tau^{2}\prod_{i=2}^{r_{0}}(\tau+\beta_{i}),\ 0\neq c\in
S^{-1},$ (4.14)

thus $\partial_{\tau}^{r_{0}+1}(\det f)=\det(\partial_{\tau}f)\neq 0$ at
$\Sigma_{2}$. Same as before we get $(\ref{3.4 MHD})$. ∎

We use the same weight and metric introduced in the previous section; see
$(\ref{w})$, and $(\ref{metric generalized uniaxial})$. Thus, we have
$\beta_{j}\in S(\vartheta,g)$.

######  Proposition 4.3.

Let

$\displaystyle P=\operatorname{Id}_{r_{0}+1}D_{t}+K(t,x,D_{x})$ (4.15)

be an $(r_{0}+1)\times(r_{0}+1)$ system with
$K\in\mathcal{C}^{\infty}(\mathbb{R},\Psi^{1}_{phg})$, such that the
eigenvalues of $k=\sigma(K)$ are $0$ (double), $\beta_{2}$,…,$\beta_{r_{0}}$.
Then $P$ satisfies $(\ref{2.7 double refraction})$ if and only if
$k\in\mathcal{C}^{\infty}(\mathbb{R},S(\vartheta,g))$.

###### Proof.

Same proof as in Proposition 3.4, just we replace $r_{0}$ by $r_{0}+1$ when
needed. ∎

######  Proposition 4.4.

Assume that $P$ is an $(r_{0}+1)\times(r_{0}+1)$ system of pseudodifferential
operators of order $1$ on $\mathbb{R}^{n}$, on the form $(\ref{3.10})$, with
$K$ is in $\mathcal{C}^{\infty}(\mathbb{R},\operatorname{Op}(S(\vartheta,g)))$
near $\nu_{0}\in\Sigma_{2}$, and $k(t,x,\xi)$ has the eigenvalues $0$
(double), $\beta_{1}$,…, $\beta_{r_{0}}$. Let
$u\in\mathcal{S}^{\prime}(\mathbb{R}^{n},\mathbb{C}^{r_{0}+1})$ and assume
$Pu\in H^{r,s}$ at $\nu_{0}$. Then, for every $\delta>0$ we can find
$c_{\delta}$ and $C_{\delta,N}>0$ and $v_{\delta}\in H^{r,s+1}$ at $\nu_{0}$,
such that $u_{\delta}=u-v_{\delta}$ satisfies

$\displaystyle\lvert\hat{u}_{\delta}(\tau,\xi)\rvert\leq
C_{\delta,N}\langle(\tau,\xi)\rangle^{-N},\ \ \ \forall N,$ (4.16)

when
$\lvert\tau\rvert>c_{\delta}(\langle\xi\rangle^{\delta}+\langle\xi^{\prime}\rangle)$.

###### Proof.

Same proof as for Proposition 3.5, with replacing $r_{0}$ by $r_{0}+1$ when
needed. ∎

Now, we are ready to prove Theorem 1.1 for systems of MHD type.

###### Proof of Theorem 1.1.

Note that as $M_{A}=\pi_{1}(\mathcal{N}_{A}\cap\mathcal{N}_{P}\setminus 0)$ is
a hypersurface near $\nu_{0}$, we have $M_{A}=S_{j}$ for some $j$. By
multiplication and conjugation with elliptic, scalar pseudodifferential
operators we may assume that $m=1$ and $r=0$, and using the normal form we can
assume that $N=r_{0}+1$, and $P$ is of the form $(\ref{3.10})$, with
$k\in\mathcal{C}^{\infty}(\mathbb{R},S(\vartheta,g))$. By using Theorem A.4 in
[5] for all the terms in the expansion of $A$, we obtain that
$A\in\mathcal{C}^{\infty}(\mathbb{R},\Psi^{0}_{phg})$. As the dimension of the
fiber of $\mathcal{N}_{A}\cap\mathcal{N}_{P}$ is $1$ at
$\nu_{0}\in\Sigma_{2}$, and the dimension of the fiber of $\mathcal{N}_{P}$ is
$r_{0}+1$ at $\Sigma_{2}$, we get $\operatorname{rank}\sigma(A)=r_{0}$ at
$\Sigma_{2}$. Hence, we can conjugate by suitable elliptic systems in
$\mathcal{C}^{\infty}(\mathbb{R},\Psi^{0}_{phg})$ to get that $Au\cong\
^{t}(u_{1},...u_{r_{0}},0)\in H^{\epsilon}$ in a conical neighborhood $U$ of
$\nu_{0}$, for some $\epsilon>0$. Then, we find
$\pi_{1}(\operatorname{Pol}^{0}(u))=\operatorname{WF}^{0}(u_{r_{0}+1})$ in
$U$. By shrinking $U$ and decreasing $\epsilon$, we may assume $Pu\in
H^{\epsilon}$ in $U$. Remember that we have $P\in\Psi^{0,1}$. Let
$L=L_{1}+L_{2}$, where $L_{1}$ and $L_{2}$ are given in $(\ref{adjugate
matrix})$, and let $Q=LP$. We have $Qu=LPu\in H^{\epsilon,-r_{0}+1}$ there.
Let $Q=(q_{ij})_{i,j=1}^{r_{0}+1}$, and $P=(P_{ij})_{i,j=1}^{r_{0}+1}$. By
$(\ref{adjugate matrix})$ and $(\ref{principal symbol of L1})$ we have
$q_{ij}$ are in
$\mathcal{C}^{\infty}(\mathbb{R},\operatorname{Op}S(\vartheta^{r_{0}-1},g))$
for $i\neq j$, we find that $q_{r_{0}+1r_{0}+1}u_{r_{0}+1}\in
H^{\epsilon,-r_{0}+1}$ in $U$. Similarly, we find that
$P_{r_{0}+1r_{0}+1}u_{r_{0}+1}\in H^{\epsilon,-1}$, which in turns gives
$D^{k-1}_{t}P_{r_{0}+1r_{0}+1}u_{r_{0}+1}\in H^{\epsilon,-k}$ for
$k=1,...,r_{0}-1$. We want to prove that $u_{r_{0}+1}\in H^{0}$ at
$(t,x_{0};0,\xi_{0})\in U\cap\Sigma_{2}$ for $t<t_{0}$, implies
$u_{r_{0}+1}\in H^{0}$ at $(t_{0},x_{0};0,\xi_{0})=\nu_{0}$.

Thus assume that $u_{r_{0}+1}\in H^{0}$ at $(t,x_{0};0,\xi_{0})\in
U\cap\Sigma_{2}$ when $t<t_{0}$. We may assume that $\delta\leq 1$, then Lemma
3.7 gives that $u_{r_{0}+1}$, $P_{r_{0}+1r_{0}+1}u_{r_{0}+1}$,
$D^{k-1}_{t}P_{r_{0}+1r_{0}+1}u_{r_{0}+1}$ for $k=2,...,r_{0}-1$ and
$q_{r_{0}+1r_{0}+1}u_{r_{0}+1}$ satisfies $(\ref{3.30 MHD})$. Then $\xi\neq 0$
in $\operatorname{WF}(u_{r_{0}+1})$, and assuming that $\delta\leq\epsilon$ in
$(\ref{3.30})$ we find that $P_{r_{0}+1r_{0}+1}u_{r_{0}+1}\in H_{*}^{0,-1}$,
$D^{k-1}_{t}P_{r_{0}+1r_{0}+1}u_{r_{0}+1}\in H_{*}^{0,-k}$ for
$k=1,...,r_{0}-1$ and $q_{r_{0}+1r_{0}+1}u_{r_{0}+1}\in H_{*}^{0,-r_{0}+1}$ in
$\pi_{0}(U\cap\Sigma_{2})$, and $u_{r_{0}+1}\in H_{*}^{0}$ at
$(t,x_{0},\xi_{0})\in\pi_{0}(U\cap\Sigma_{2})$ for $t<t_{0}$. Since
$P_{r_{0}+1r_{0}+1}\cong D_{t}$ $\operatorname{mod}$
$\mathcal{C}^{\infty}(\mathbb{R},\operatorname{Op}S(\vartheta,g))$, we get
$P_{r_{0}+1r_{0}+1}u_{r_{0}+1}\cong D_{t}u_{r_{0}+1}\in H_{*}^{0,-1}$ which
implies using $D^{k-1}_{t}P_{r_{0}+1r_{0}+1}u_{r_{0}+1}\in H_{*}^{0,-k}$ that
$D_{t}^{k}u_{r_{0}+1}\in H_{*}^{0,-k}$ for $k=1,...,r_{0}-1$. This gives

$\displaystyle(D_{t}^{k}u_{r_{0}+1})|_{t=r}\in H_{*}^{0,-k}\ \ \ \text{at}\ \
(x_{0},\xi_{0}),\ \text{for}\ k=0,...,r_{0}-1$ (4.17)

for almost all $r<t_{0}$, close to $t_{0}$. Proposition 3.8 (with $N=1$ and
$Q=q_{r_{0}+1r_{0}+1}$) gives $u_{r_{0}+1}\in H_{*}^{0}$ at
$(t_{0},x_{0},\xi_{0})$, and Lemma 3.7 gives $u_{r_{0}+1}\in H^{0}$ at
$(t_{0},x_{0};0,\xi_{0})$. ∎

##  5 Application

Magnetohydrodynamics, or MHD couples Maxwell’s equations with hydrodynamics to
describe the behavior of electrically conducting fluids under the influence of
electromagnetic fields. In this section, we want to consider the simplest form
of MHD, which is the Ideal MHD to study the propagation of polarization set of
the solution of the linearized ideal MHD equations. Ideal MHD, assumes that
the fluid has so little resistivity that it can be treated as a perfect
conductor. See [14] to know more about MHD equations.

We will show, under some assumptions, that the linearized ideal MHD equations
is of real principal type. As we mentioned before, systems of real principal
type were defined by Dencker; see [1], who studied the propagation of their
solutions, and showed that the propagation of polarization sets is governed by
a certain connection on sections of the kernel subbundle, $\ker p_{2}$ where
$p_{2}$ is the principal symbol of the system. In [8], Hansen and Röhrig
merged the theory of real principal type systems with the calculus of Fourier
integral operators and constructed a Fourier integral solution for system of
real principal type, and derived a transport equation for the principal symbol
of this solution (note that disregarding half densities this transport
equation is the connection introduced by Dencker).

We will divide this section into two subsections: first, we introduce the
ideal MHD equations and its linearization. In Section 5.1, we write the
linearized ideal MHD equations in the form of a wave equation $P\beta=0$ where
$\beta$ is the displacement vector and $P$ is a second order $3\times 3$
system; see [14, Lecture 20], and we show that under some assumptions, the
characteristic variety of $P$ is disjoint union of the Shear Alfvén wave, the
slow magnetosonic wave and the fast magnetosonic wave; see [14, Lecture 24].
Moreover, we show that, under the considered assumptions, $P$ is of real
principal type and we calculate the transport equation on
$\operatorname{Char}P$, as an application to the result in [8]. In Section
5.2, we return to the linearized ideal MHD equations, and we study the
propagation of polarization sets. It turns out that we can consider different
cases, some in which we have our system is of real principal type, some in
which our system is of uniaxial type, and we have a case where our system is
of MHD type.

The set of equations describing the ideal MHD are

$\displaystyle\begin{cases}\partial_{t}\rho+u\cdot\nabla\rho+\rho\operatorname{div}u=0,\\\
\rho(\partial_{t}u+u\cdot\nabla u)+\nabla p+H\times\operatorname{curl}H=0,\\\
\partial_{t}H-\nabla\times(u\times H)=0,\\\ \partial_{t}p+u\cdot\nabla
p+\gamma p\operatorname{div}u=0,\end{cases}$ (5.1)

where $\rho$, $p\in\mathbb{R}$ denotes the density and the pressure
respectively. $u\in\mathbb{R}^{3}$ is the fluid velocity, $H\in\mathbb{R}^{3}$
is the magnetic field, and $\gamma$ is the adiabatic index, see [14, Lecture
20].
Assuming a stationary equilibrium the linearized equations of $(\ref{1})$
about $(\rho,H,p)$ is:

$\displaystyle\partial_{t}\dot{\rho}=-\rho\operatorname{div}\dot{u}-\dot{u}\cdot\nabla\rho,$
(5.2a)
$\displaystyle\rho\partial_{t}\dot{u}=-\nabla\dot{p}-H\times\operatorname{curl}\dot{H}-\dot{H}\times\operatorname{curl}H,$
(5.2b) $\displaystyle\partial_{t}\dot{H}=\nabla\times(\dot{u}\times H),$
(5.2c) $\displaystyle\partial_{t}\dot{p}=-\gamma
p\operatorname{div}\dot{u}-\dot{u}\cdot\nabla p,$ (5.2d)

where $(\rho,H,p)$ are the values in the equilibrium state (that is the
solutions of the Ideal MHD equations when $\partial/\partial t=0$, and as we
assumed stationary equilibrium we have $u=0$). Note that we used that

$\displaystyle\nabla p+H\times\operatorname{curl}H=0,$ (5.3)

which we get from the stationary equilibrium assumption.

###  5.1 The ideal MHD wave equation and the transport equation

We can write the linearized ideal MHD equations (5.2) in the form of a wave
equation; see [14, Lecture 20]. Consider the displacement $\beta$ to be
defined by

$\displaystyle\dot{u}=\frac{\partial\beta}{\partial t}.$ (5.4)

Substituting (5.4) in (5.2a), (5.2c), and (5.2d) respectively, and integrating
with respect to $t$ we get

$\displaystyle\dot{\rho}=-\rho\operatorname{div}\beta-\beta\cdot\nabla\rho,$
(5.5a) $\displaystyle\dot{H}=\nabla\times(\beta\times H).$ (5.5b)
$\displaystyle\dot{p}=-\gamma p\operatorname{div}\beta-\beta\cdot\nabla p,$
(5.5c)

Replace now (5.5b), (5.5c), and $(\ref{4})$ in (5.2b) to get

$\displaystyle\rho\frac{\partial^{2}\beta}{\partial t^{2}}=$
$\displaystyle\gamma\nabla(p\operatorname{div}\beta)+\nabla(\beta\cdot\nabla
p)+(\nabla\times(\nabla\times(\beta\times H)))\times H+(\nabla\times
H)\times(\nabla\times(\beta\times H)).$ (5.6)

Equation (5.6) is the ideal MHD wave equation. Consider from now on $P$ where

$\displaystyle\begin{split}P\beta&=-\rho\frac{\partial^{2}\beta}{\partial
t^{2}}+\gamma\nabla(p\operatorname{div}\beta)+\nabla(\beta\cdot\nabla
p)+(\nabla\times(\nabla\times(\beta\times H)))\times H\\\ &+(\nabla\times
H)\times(\nabla\times(\beta\times H))\\\ &=0.\end{split}$ (5.7)

Now, we want to give the Characteristic variety of $P$ under some assumptions;
for this part we refer to [14, Lectures 23 and 24].

######  Lemma 5.1.

Assume $c^{2}=\gamma p/\rho>0$, $0<\lvert H\rvert^{2}\neq\rho c^{2}$,
$\xi\cdot H\neq 0$, and $\xi\times H\neq 0$ . The characteristic variety of
$P$ is disjoint union of the Shear Alfvén wave, the slow magnetosonic wave,
and the fast magnetosonic wave characteristic varieties $\\{q_{1}=0\\}$,
$\\{q_{2}=0\\}$, and $\\{q_{3}=0\\}$ respectively, where

$\displaystyle\begin{cases}&q_{1}=\rho\tau^{2}-(H.\xi)^{2},\\\
&q_{2}=\rho(\tau^{2}-c_{s}^{2}(x,\xi))\\\
&q_{3}=\rho(\tau^{2}-c_{f}^{2}(x,\xi)),\end{cases}$ (5.8)

with

$\displaystyle
c_{f}^{2}(x,\xi):=\frac{1}{2}\big{(}(c^{2}+h^{2})\xi^{2}+\sqrt{(c^{2}-h^{2})^{2}\xi^{4}+4b^{2}c^{2}\xi^{2}}\big{)},$
$\displaystyle
c_{s}^{2}(x,\xi):=\frac{1}{2}\big{(}(c^{2}+h^{2})\xi^{2}-\sqrt{(c^{2}-h^{2})^{2}\xi^{4}+4b^{2}c^{2}\xi^{2}}\big{)},$

where $c^{2}=\gamma p/\rho>0$, and $h^{2}=\lvert H\rvert^{2}/\rho$ are
considered to be the sound speed and the Alfvén speed respectively, and
$b^{2}=\lvert\xi\times H\rvert^{2}/\rho.$

###### Proof.

We have

$\displaystyle p_{2}\beta=\rho\tau^{2}\beta-\gamma
p\xi(\xi\cdot\beta)-(\xi\times(\xi\times(\beta\times H)))\times H,$ (5.9)

with $p_{2}$ is the principal symbol of $P$. Considering $v=H/{\sqrt{\rho}}$,
equation (5.9) can be written as

$\displaystyle\rho(\tau^{2}-(\xi\cdot
v)^{2})\beta=\rho((c^{2}+h^{2})(\xi\cdot\beta)-(v\cdot\beta)(\xi\cdot
v))\xi-\rho v(\xi\cdot\beta)(\xi\cdot v).$ (5.10)

Without loss of generality, let $v=\lvert H\rvert/\sqrt{\rho}\ \hat{e}_{z}$,
$\xi=\xi_{\perp}\hat{e}_{x}+\xi_{\parallel}\hat{e}_{z}$ with
$\xi^{2}=\xi_{\perp}^{2}+\xi_{\parallel}^{2}$, and
$\beta=\beta_{x}\hat{e}_{x}+\beta_{y}\hat{e}_{y}+\beta_{z}\hat{e}_{z}$ with
$\hat{e}_{x}$, $\hat{e}_{y}$, and $\hat{e}_{z}$ are unit vectors that points
in the direction of the x-axis, y-axis, and z-axis respectively. Substituting
this in equation (5.10), we find

$\displaystyle\text{x-component:}\ \ \rho\tau^{2}\beta_{x}=\rho
c^{2}\xi_{\parallel}\xi_{\perp}\beta_{z}+\rho(h^{2}\xi^{2}+c^{2}\xi_{\perp}^{2})\beta_{x},$
(5.11a) $\displaystyle\text{y-component:}\ \ \rho\tau^{2}\beta_{y}=\rho
h^{2}\xi_{\parallel}^{2}\beta_{y},$ (5.11b) $\displaystyle\text{z-component:}\
\ \rho\tau^{2}\beta_{z}=\rho(c^{2}\xi_{\parallel}\xi_{\perp})\beta_{x}+\rho
c^{2}\xi_{\parallel}^{2}\beta_{z}.$ (5.11c)

Notice that the y-component decouples from the x- and z-components. This
immediately gives

$\displaystyle\rho\tau^{2}=\rho h^{2}\xi_{\parallel}^{2},$ (5.12)

This is the shear Alfvén wave. The characteristic equation for the coupled
x-and z-component is

$\displaystyle\rho^{2}\tau^{4}-\rho^{2}\tau^{2}\xi^{2}(c^{2}+h^{2})+\rho^{2}c^{2}h^{2}\xi_{\parallel}^{2}\xi^{2}=0.$
(5.13)

Hence, we get

$\displaystyle\rho\tau^{2}=\frac{\rho}{2}\bigg{(}(c^{2}+h^{2})\xi^{2}\pm\sqrt{(c^{2}-h^{2})\xi^{4}+4c^{2}h^{2}\xi_{\perp}^{2}\xi^{2}}\bigg{)}.$
(5.14)

Still we want to prove that $\\{q_{1}=0\\}$, $\\{q_{2}=0\\}$, and
$\\{q_{3}=0\\}$ are disjoint. Dividing $(\ref{8})$ by $\rho^{2}$, it can be
written as

$\displaystyle(\tau^{2}-h^{2}\xi_{\parallel}^{2})(\tau^{2}-c^{2}\xi^{2})-\tau^{2}h^{2}\xi_{\perp}^{2}.$
(5.15)

Consider
$R(X)=(X^{2}-h^{2}\xi_{\parallel}^{2})(X^{2}-c^{2}\xi^{2})-X^{2}h^{2}\xi_{\perp}^{2}$,
$\\{R\leq 0\\}=[c_{s}^{2},c_{f}^{2}]$ and $R(X)\leq 0$ for
$X\in[\min(h^{2}\xi_{\parallel}^{2},c^{2}\xi^{2}),$
$\max(h^{2}\xi_{\parallel}^{2},c^{2}\xi^{2})]$. Thus,

$\displaystyle c_{f}^{2}\geq\max(h^{2}\xi_{\parallel}^{2},c^{2}\xi^{2})\geq
h^{2}\xi_{\parallel}^{2},$ (5.16a) $\displaystyle
c_{s}^{2}\leq\min(h^{2}\xi_{\parallel}^{2},c^{2}\xi^{2})\leq
h^{2}\xi_{\parallel}^{2}.$ (5.16b) As $h^{2}\xi_{\parallel}^{2}\neq 0$, we
have $R(h^{2}\xi_{\parallel}^{2})=-h^{2}\xi_{\parallel}^{2}c^{2}\xi^{2}\leq
0$. Hence, $c_{s}^{2}<h^{2}\xi_{\parallel}^{2}<c_{f}^{2}$.

∎

Suppose that the conditions of Lemma 5.1 are satisfied. Now, we are interested
in calculating the transport equation as in [8]. The full symbol of $P$ is
$p_{2}+p_{1}+p_{0}$, with $p_{2}$ is the principal symbol of $P$ homogeneous
of degree $2$, and $p_{1}$ and $p_{0}$ are homogeneous terms of degree $1$ and
$0$ respectively. One can check that the principal symbol of $P$ is

$\displaystyle p_{2}=(\rho\tau^{2}-(H.\xi)^{2})\operatorname{Id}_{3}-(\gamma
p+\lvert H\rvert^{2})\xi\otimes\xi+(H.\xi)(\xi\otimes H+H\otimes\xi),$ (5.17)

and

$\displaystyle p_{1}=$ $\displaystyle i\gamma(\nabla
p)\otimes\xi+i\xi\otimes(\nabla p)+i(\nabla(H\cdot\xi)\cdot
H+(H\cdot\xi)\operatorname{div}H)\operatorname{Id}_{3}+\frac{i}{2}(\xi\otimes\nabla\lvert
H\rvert^{2})+i(\nabla\lvert H\rvert^{2}\otimes\xi)$
$\displaystyle-2i(H\cdot\nabla)(H\otimes\xi)-i(\nabla(H\cdot\xi))\otimes
H-i(H\cdot\xi)(\nabla\otimes H)-i(\nabla\cdot H)(\xi\otimes H).$

Using $(\ref{condition equilibrium})$, we get

$\displaystyle\begin{split}p_{1}&=i\gamma(\nabla
p)\otimes\xi+i\xi\otimes(\nabla p)-i(\nabla
p)\otimes\xi+i(\nabla(H\cdot\xi)\cdot
H+(H\cdot\xi)\operatorname{div}H)\operatorname{Id}_{3}+\frac{i}{2}(\xi\otimes\nabla\lvert
H\rvert^{2})\\\ &+\frac{i}{2}(\nabla\lvert
H\rvert^{2}\otimes\xi)-i(H\cdot\nabla)(H\otimes\xi)-i(\nabla(H\cdot\xi))\otimes
H-i(H\cdot\xi)(\nabla\otimes H)-i(\nabla\cdot H)(\xi\otimes H).\end{split}$
(5.18)

One can check $(\ref{principal symbol mathematika})$, $(\ref{p1
mathematika})$, and the calculations given below by using ”mathematica” for
example.

Let $\Gamma_{1}$, $\Gamma_{2}$, and $\Gamma_{3}$ be disjoint conic
neighborhoods of $\\{q_{1}=0\\}$, $\\{q_{2}=0\\}$, and $\\{q_{3}=0\\}$
respectively. Set $q=q_{1}$ in $\Gamma_{1}$, $q=q_{2}$ in $\Gamma_{2}$, and
$q=q_{3}$ in $\Gamma_{3}$.

######  Proposition 5.2.

$P$ is of real principal type with respect to the Hamilton field $H_{q}$ of
$\operatorname{Char}P=\\{q=0\\}$.

###### Proof.

We have $q_{1}$, $q_{2}$, and $q_{3}$ are scalar real principal type. Let

$\displaystyle w^{2}:=\lvert
H\rvert^{2}\xi\otimes\xi+\lvert\xi\rvert^{2}H\otimes
H-(H\cdot\xi)(H\otimes\xi+\xi\otimes H)$ (5.19)

In $\Gamma_{1}$ we take

$\displaystyle\tilde{p}_{2}=\operatorname{Id}_{3}+\frac{q_{1}}{q_{2}q_{3}}(\gamma
p+\lvert H\rvert^{2})\xi\otimes\xi-\frac{q_{1}}{q_{2}q_{3}}($
$\displaystyle(H\cdot\xi)(\xi\otimes
H+H\otimes\xi))+\frac{(H\cdot\xi)^{2}w^{2}}{q_{2}q_{3}}$ (5.20)

to get $\tilde{p}_{2}p_{2}=q_{1}\operatorname{Id}_{3}$. In $\Gamma_{2}$ we
take

$\displaystyle\tilde{p}_{2}=\frac{q_{2}}{q_{1}}\operatorname{Id}_{3}+\frac{1}{q_{3}}(\gamma
p+\lvert H\rvert^{2})\xi\otimes\xi-\frac{1}{q_{3}}($
$\displaystyle(H\cdot\xi)(\xi\otimes
H+H\otimes\xi))+\frac{(H\cdot\xi)^{2}w^{2}}{q_{1}q_{3}}$ (5.21)

to get $\tilde{p}_{2}p_{2}=q_{2}\operatorname{Id}_{3}$. In $\Gamma_{3}$ we
take

$\displaystyle\tilde{p}_{2}=\frac{q_{3}}{q_{1}}\operatorname{Id}_{3}+\frac{1}{q_{2}}(\gamma
p+\lvert H\rvert^{2})\xi\otimes\xi-\frac{1}{q_{2}}($
$\displaystyle(H\cdot\xi)(\xi\otimes
H+H\otimes\xi))+\frac{(H\cdot\xi)^{2}w^{2}}{q_{1}q_{2}}$ (5.22)

to get $\tilde{p}_{2}p_{2}=q_{3}\operatorname{Id}_{3}$.∎

######  Remarks.

$\bullet$ The principal symbol of $P$ calculated before can be written as

$\displaystyle p_{2}=q_{1}\pi_{1}+q_{2}\pi_{2}+q_{3}\pi_{3},$ (5.23)

where

$\displaystyle\pi_{1}=\operatorname{Id}_{3}+\frac{w^{2}}{(H\cdot\xi)^{2}-\lvert
H\rvert^{2}\lvert\xi\rvert^{2}},$ (5.24) $\displaystyle\pi_{2}=\frac{1}{\rho
c_{s}^{2}(x,\xi)-\rho c_{f}^{2}(x,\xi)}\bigg{(}(\gamma p+\lvert
H\rvert^{2})\xi\otimes\xi-(H\cdot\xi)(\xi\otimes
H+H\otimes\xi)+\frac{(H\cdot\xi)^{2}w^{2}}{\rho
c_{s}^{2}(x,\xi)-(H\cdot\xi)^{2}}\bigg{)},$ (5.25)

and

$\displaystyle\pi_{3}=\frac{1}{\rho c_{f}^{2}(x,\xi)-\rho
c_{s}^{2}(x,\xi)}\bigg{(}(\gamma p+\lvert
H\rvert^{2})\xi\otimes\xi-(H\cdot\xi)(\xi\otimes
H+H\otimes\xi)+\frac{(H\cdot\xi)^{2}w^{2}}{\rho
c_{f}^{2}(x,\xi)-(H\cdot\xi)^{2}}\bigg{)},$ (5.26)

with $\pi_{1},\pi_{2}$ and $\pi_{3}$ are orthogonal projectors and
$\pi_{1}+\pi_{2}+\pi_{3}=\operatorname{Id}_{3}$
$\bullet$ In $\Gamma_{1}$,

$\displaystyle\tilde{p}_{2}=\pi_{1}+\frac{q_{1}}{q_{2}}\pi_{2}+\frac{q_{1}}{q_{3}}\pi_{3},$
(5.27)

and set $\pi=\pi_{1}$ .
In $\Gamma_{2}$,

$\displaystyle\tilde{p}_{2}=\pi_{2}+\frac{q_{2}}{q_{1}}\pi_{1}+\frac{q_{2}}{q_{3}}\pi_{3},$
(5.28)

and set $\pi=\pi_{2}$.
In $\Gamma_{3}$,

$\displaystyle\tilde{p}_{2}=\pi_{3}+\frac{q_{3}}{q_{1}}\pi_{1}+\frac{q_{3}}{q_{2}}\pi_{2},$
(5.29)

and set $\pi=\pi_{3}$.
On $\operatorname{Char}P$, $\tilde{p}_{2}=\pi$, $p_{2}\pi=0=\pi p_{2}$, and
$p_{2}a=0$ if and only if $a=\pi a$.

In what follows, let $X=\mathbb{R}\times\mathbb{R}^{3}$, $\Lambda\subset
T^{*}X\setminus 0$ be a closed Lagrangian submanifold of the characteristic
set of $P$, and let $\Omega_{\Lambda}^{1/2}$ denote the half-density bundle of
$\Lambda$. $S^{\mu+1}(\Lambda,(\Omega_{\Lambda}^{1/2})^{3})$ is the space of
symbols of the space of Lagrangian distributions
$I^{\mu}(X,\Lambda;(\Omega_{X}^{1/2})^{3})$; see [12, Section 25.1] to read
more about Lagrangian distributions.
From [8], we know that there is a first order differential operator
$\mathcal{T}_{P,H_{q}}$ on $\Lambda$, uniquely determined by $P$ and $H_{q}$
which maps $a$ a $3$-vector of half densities with $p_{2}a=0$ to $3$-vector of
half densities where

$\displaystyle\mathcal{T}_{P,H_{q}}a=\mathcal{L}_{H_{q}}a+\frac{1}{2}\\{\tilde{p}_{2},p_{2}\\}a+i\tilde{p}_{2}p^{s}a.$
(5.30)

Here

$\displaystyle
p^{s}=p_{1}-\frac{1}{2i}\sum_{j}\frac{\partial^{2}p_{2}}{\partial
x_{j}\partial\xi_{j}},$ (5.31)

is the subprincipal symbol of $P$, $\mathcal{L}_{H_{q}}$ is the Lie derivative
with respect to $H_{q}$, and $\\{.\\}$ is the Poisson bracket.

######  Lemma 5.3.

$\tilde{p}_{2}p^{s}\pi=0$ on $\operatorname{Char}P$.

###### Proof.

Differentiating $p_{2}$ we get

$\displaystyle\sum_{j=1}^{3}\frac{\partial^{2}p_{2}}{\partial
x_{j}\partial\xi_{j}}=$ $\displaystyle-2(\nabla(H\cdot\xi)\cdot
H+(H\cdot\xi)\operatorname{div}H)\operatorname{Id}_{3}-(\gamma\nabla
p+\nabla(\lvert H\rvert^{2}))\otimes\xi-\xi\otimes(\gamma\nabla p+\nabla\lvert
H\rvert^{2})$ $\displaystyle+(\operatorname{div}H+H\cdot\nabla)(\xi\otimes
H+H\otimes\xi)+(\nabla(H\cdot\xi))\otimes H+(H\cdot\xi)(\nabla\otimes
H+(\nabla\otimes H)^{\intercal})$ $\displaystyle+H\otimes(\nabla(H\cdot\xi))$

Therefore, the subprincipal is given as follows:

$\displaystyle 2ip^{s}=$ $\displaystyle\gamma(\xi\otimes\nabla p-\nabla
p\otimes\xi)+\operatorname{div}H(\xi\otimes
H-H\otimes\xi)+(H\cdot\xi)(-(\nabla\otimes H)^{\intercal}+(\nabla\otimes H))$
$\displaystyle+(H\cdot\nabla)(H\otimes\xi-\xi\otimes
H)+((\nabla(H\cdot\xi))\otimes H-H\otimes\nabla(H\cdot\xi))+2(\nabla
p\otimes\xi-\xi\otimes\nabla p).$

We have

$\displaystyle 2i\tilde{p}_{2}p^{s}\pi=2i\pi p^{s}\pi\ \ \ \
\text{on}\operatorname{Char}P.$ (5.32)

Using that $2ip^{s}$ is a $3\times 3$ skew-symmetric matrix with zero entries
on the diagonal, and $\pi$ is a symmetric matrix we get that $2i\pi p^{s}\pi$
is a $3\times 3$ skew-symmetric matrix. Therefore to prove that it vanishes,
it suffices to show that its rank is $<2$. Since $\pi$ is projection we have
rank $\pi$ =trace $\pi$. Calculating the trace of $\pi$ we get that rank
$\pi$=1 and hence we proved the lemma. ∎

######  Lemma 5.4.

Let $\Lambda\subset\operatorname{Char}P$ be a conic Lagrangian submanifold.
Let $a\in S^{\mu+1}(\Lambda,(\Omega_{\Lambda}^{1/2})^{3})$. Then,
$\\{\tilde{p}_{2},p_{2}\\}\pi a=-2(H\pi)a$ on $\operatorname{Char}P$.

###### Proof.

We will prove the result for $\pi=\pi_{1}$ and the same argument applies for
$\pi_{2}$ and $\pi_{3}$. We have in a conic neighborhood of $\\{q_{1}=0\\}$,
$p_{2}=q_{1}\pi_{1}+q_{2}\pi_{2}+q_{3}\pi_{3}$, and
$\tilde{p}_{2}=\pi_{1}+\frac{q_{1}}{q_{2}}\pi_{2}+\frac{q_{1}}{q_{3}}\pi_{3}$.
Using that $\pi^{2}=\pi$, and $\\{q_{1},q_{1}\\}=0$ we get

$\displaystyle\\{\tilde{p}_{2},p_{2}\\}\pi_{1}a=$
$\displaystyle\\{\pi_{1},q_{1}\\}\pi_{1}a+q_{2}\\{\pi_{1},\pi_{2}\\}\pi_{1}a+q_{3}\\{\pi_{1},\pi_{3}\\}\pi_{1}a+\pi_{2}\\{q_{1},\pi_{2}\\}\pi_{1}a+\frac{q_{3}}{q_{2}}\pi_{2}\\{q_{1},\pi_{3}\\}\pi_{1}a$
$\displaystyle+\frac{q_{2}}{q_{3}}\pi_{3}\\{q_{1},\pi_{2}\\}\pi_{1}a+\pi_{3}\\{q_{1},\pi_{3}\\}\pi_{1}a.$

Using that $H\pi_{j}=H\pi_{j}^{2}=(H\pi_{j})\pi_{j}+\pi_{j}(H\pi_{j})$ for
$j=2,3$, we get
$\pi_{j}\\{q_{1},\pi_{j}\\}\pi_{1}a=\\{q_{1},\pi_{j}\\}\pi_{1}a$ for $j=2,3$,
and using $0=H(\pi_{2}\pi_{3})=\pi_{2}(H\pi_{3})+(H\pi_{2})\pi_{3}$ and
$0=H(\pi_{3}\pi_{2})=\pi_{3}(H\pi_{2})+(H\pi_{3})\pi_{2}$ we get
$\pi_{2}\\{q_{1},\pi_{3}\\}\pi_{1}a=\pi_{3}\\{q_{1},\pi_{2}\\}\pi_{1}a=0$.
Using $\pi_{2}=\operatorname{Id}_{3}-\pi_{1}-\pi_{3}$,
$\\{\pi_{1},\operatorname{Id}_{3}\\}=\\{q_{1},\operatorname{Id}_{3}\\}=0$ and
$\\{\pi_{1},\pi_{1}\\}=0$ we get

$\displaystyle\\{\tilde{p}_{2},p_{2}\\}\pi_{1}a=-2(H\pi_{1})\pi_{1}a+(q_{3}-q_{2})\\{\pi_{1},\pi_{3}\\}\pi_{1}a.$

Now, we want to prove that $\\{\pi_{1},\pi_{3}\\}\pi_{1}a=0$. We have
$\partial_{\xi_{i}}\pi_{1}=\partial_{\xi_{i}}\pi_{1}^{2}=\pi_{1}\partial_{\xi_{i}}\pi_{1}+\partial_{\xi_{i}}\pi_{1}\pi_{1}$,
and similarly
$\partial_{x_{i}}\pi_{1}=\partial_{x_{i}}\pi_{1}^{2}=\pi_{1}\partial_{x_{i}}\pi_{1}+\partial_{x_{i}}\pi_{1}\pi_{1}$.
Also,
$0=\partial_{\xi_{i}}(\pi_{1}\pi_{3})=\pi_{1}\partial_{\xi_{i}}\pi_{3}+\partial_{\xi_{i}}\pi_{1}\pi_{3}$.
Hence, $\pi_{1}\partial_{\xi_{i}}\pi_{3}=-\partial_{\xi_{i}}\pi_{1}\pi_{3}$.
Similarly, we have
$\pi_{1}\partial_{x_{i}}\pi_{3}=-\partial_{x_{i}}\pi_{1}\pi_{3}$. Combining
these together we get

$\displaystyle\\{\pi_{1},\pi_{3}\\}\pi_{1}a=\pi_{1}\\{\pi_{1},\pi_{3}\\}\pi_{1}a.$

We have

$\displaystyle\pi_{1}\\{\pi_{1},\pi_{3}\\}\pi_{1}a=\sum_{i=1}^{3}\pi_{1}(\partial_{\xi_{i}}\pi_{1}\partial_{x_{i}}\pi_{3}-\partial_{x_{i}}\pi_{1}\partial_{\xi_{i}}\pi_{3})\pi_{1}a.$

Moreover,

$\displaystyle\pi_{1}\partial_{\xi_{i}}\pi_{1}\partial_{x_{i}}\pi_{3}\pi_{1}a$
$\displaystyle=\pi_{1}\partial_{\xi_{i}}\pi_{1}(\pi_{3}\partial_{x}\pi_{3}+\partial_{x_{i}}\pi_{3}\pi_{3})\pi_{1}a=\pi_{1}(\partial_{\xi_{i}}\pi_{1}\pi_{3})\partial_{x}\pi_{3}\pi_{1}a=-\pi_{1}\partial_{\xi_{i}}\pi_{3}(\partial_{x_{i}}\pi_{3}\pi_{1})a$
$\displaystyle=\pi_{1}(\partial_{\xi_{i}}\pi_{3}\pi_{3})\partial_{x_{i}}\pi_{1}a=\pi_{1}(\partial_{\xi_{i}}\pi_{3}-\pi_{3}\partial_{\xi_{i}}\pi_{3})\partial_{x_{i}}\pi_{1}a=\pi_{1}\partial_{\xi_{i}}\pi_{3}\partial_{x_{i}}\pi_{1}a.$

Using that $\pi_{1}a=a$ on $\operatorname{Char}P$, we get
$\pi_{1}\partial_{\xi_{i}}\pi_{1}\partial_{x_{i}}\pi_{3}\pi_{1}a=\pi_{1}\partial_{\xi_{i}}\pi_{3}\partial_{x_{i}}\pi_{1}\pi_{1}a$.
Therefore,

$\displaystyle\pi_{1}\\{\pi_{1},\pi_{3}\\}\pi_{1}a=\sum_{i=1}^{3}\pi_{1}(\partial_{\xi_{i}}\pi_{3}\partial_{x_{i}}\pi_{1}-\partial_{x_{i}}\pi_{1}\partial_{\xi_{i}}\pi_{3})\pi_{1}a.$

We have
$\partial_{\xi_{i}}\pi_{3}\partial_{x_{i}}\pi_{1}-\partial_{x_{i}}\pi_{1}\partial_{\xi_{i}}\pi_{3}$
is a $3\times 3$ skew-symmetric matrix with the entries in the diagonal equal
to zero. So same as before we get $\pi_{1}\\{\pi_{1},\pi_{3}\\}\pi_{1}a=0$ as
the rank of $\pi_{1}$ equal to $1$. Hence, the lemma is proved. ∎

######  Proposition 5.5.

Let $\Lambda\subset\operatorname{Char}P$ be a conic Lagrangian submanifold.
Let $a\in S^{\mu+1}(\Lambda,(\Omega_{\Lambda}^{1/2})^{3})$ with $p_{2}a=0$.
Then

$\displaystyle\mathcal{T}_{P,H}a=\mathcal{L}_{H}a-(H\pi)a\ \ \ \ on\ \
\Lambda.$ (5.33)

###  5.2 Propagation of polarization sets for the linearized ideal MHD
equations

Note that $(\ref{2})$ is hyperbolic symmetric with symmetrizer

$\displaystyle S=\begin{pmatrix}\gamma p&0&0&0&0&0&0&-\rho\\\
0&\rho&0&0&0&0&0&0\\\ 0&0&\rho&0&0&0&0&0\\\ 0&0&0&\rho&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&1&0\\\
-\rho&0&0&0&0&0&0&\frac{1+\rho^{2}}{\gamma p}\end{pmatrix}.$ (5.34)

The principal symbol of $(\ref{2})$ is

$\displaystyle\begin{cases}\tau\dot{\rho}+\rho(\xi\cdot\dot{u})=0,\\\
\tau\dot{u}+\rho^{-1}\dot{p}\xi+\rho^{-1}H\times(\xi\times\dot{H})=0,\\\
\tau\dot{H}+(\xi\cdot\dot{u})H-(H\cdot\xi)\dot{u}=0,\\\ \tau\dot{p}+\gamma
p(\xi\cdot\dot{u})=0.\end{cases}$ (5.35)

We use here the notation $\xi=(\xi_{1},\xi_{2},\xi_{3})$ for the spatial
frequencies and $\xi=\lvert\xi\rvert\hat{\xi}$,
$\dot{u}_{\parallel}=\hat{\xi}\cdot\dot{u}$,
$\dot{u}_{\perp}=\dot{u}-\dot{u}_{\parallel}\hat{\xi}=-\hat{\xi}\times(\hat{\xi}\times\dot{u})$.
We write $(\ref{4'})$ in the general form $\tau\dot{U}+A(U,\xi)\dot{U}=0$ with
parameters $U=(\rho,H,p)$, and $\dot{U}=(\dot{\rho},\dot{u},\dot{H},\dot{p})$.

We have the following result

######  Lemma 5.6.

Assume that $c^{2}=\gamma p/\rho>0$. The eigenvalues of $A(U,\xi)$ are

$\displaystyle\begin{cases}\lambda_{0}=\lambda_{4}=0,\\\ \lambda_{\pm 1}=\pm
c_{s}(\hat{\xi})\lvert\xi\rvert,\\\ \lambda_{\pm 2}=\pm(\xi\cdot
H)/\sqrt{\rho},\\\ \lambda_{\pm 3}=\pm c_{f}(\hat{\xi})\lvert\xi\rvert,\\\
\end{cases}$ (5.36)

with $\hat{\xi}=\xi/\lvert\xi\rvert$ and

$\displaystyle
c_{f}^{2}(\hat{\xi}):=\frac{1}{2}\big{(}(c^{2}+h^{2})+\sqrt{(c^{2}-h^{2})^{2}+4b^{2}c^{2}}\big{)},$
(5.37) $\displaystyle
c_{s}^{2}(\hat{\xi}):=\frac{1}{2}\big{(}(c^{2}+h^{2})-\sqrt{(c^{2}-h^{2})^{2}+4b^{2}c^{2}}\big{)},$
(5.38)

where $h^{2}=\lvert H\rvert^{2}/\rho$, $b^{2}=\lvert\hat{\xi}\times
H\rvert^{2}/\rho.$

Moreover, if we assume that $0<\lvert H\rvert^{2}\neq\rho c^{2}$, then we have
$(i)$ When $\xi\cdot H\neq 0$ and $\xi\times H\neq 0$:
$\lambda_{0}=\lambda_{4}$ is double eigenvalue of $A(U,\xi)$, and the
eigenvalues $\lambda_{\pm 1},\lambda_{\pm 2}$ and $\lambda_{\pm 3}$ are simple
eigenvalues of $A(U,\xi)$.

$(ii)$ When $\xi\cdot H=0$, $\xi\neq 0$:
$\lambda_{\pm 3}$ are simple eigenvalues, while $\lambda_{0}=\lambda_{\pm
1}=\lambda_{\pm 2}=\lambda_{4}$ is a multiple eigenvalue.

$(iii)$ When $\xi\times H=0$, $\xi\neq 0$:
when $\lvert H\rvert^{2}<\rho c^{2}$ $(\text{resp.}\ \lvert H\rvert^{2}>\rho
c^{2})$, $\lambda_{\pm 3}$ $(\text{resp.}\ \lambda_{\pm 1})$ are simple;
$\lambda_{+2}\neq\lambda_{-2}$ are double, equal to $\lambda_{\pm 1}$ $($resp.
$\lambda_{\pm 3})$ depending on $\xi\cdot H$, $\lambda_{0}$ is double equal to
$\lambda_{4}$.

The proof of this lemma is very similar to the explanation given in [13,
Appendix A] except here we have the additional eigenvalue $\lambda_{4}=0$.
Also, here we will not state all the eigenspaces as in [13].

###### Proof.

Let $\dot{U}=(\dot{\rho},\dot{u},\dot{H},\dot{p})$. The eigenvalue equation
$A(U,\xi)\dot{U}=\lambda\dot{U}$ reads

$\displaystyle\begin{cases}\lambda\dot{\rho}=\rho\dot{u}_{\parallel},\\\
\rho\lambda\dot{u}_{\parallel}=\dot{p}+H_{\perp}\cdot\dot{H}_{\perp},\\\
\lambda\rho\dot{u}_{\perp}=-H_{\parallel}\dot{H}_{\perp},\\\
\lambda\dot{H}_{\perp}=\dot{u}_{\parallel}H_{\perp}-H_{\parallel}\dot{u}_{\perp},\\\
\lambda\dot{H}_{\parallel}=0,\\\ \lambda\dot{p}=\gamma
p\dot{u}_{\parallel}.\end{cases}$ (5.39)

On
$\\{\dot{\rho}=0,\dot{u}=0,\dot{H}_{\perp}=0,\dot{p}=0\\}=\mathbb{E}_{0}(\xi)$,
$A$ is equal to $\lambda=0$. From now on we work on
$\mathbb{E}_{0}^{\perp}=\\{\dot{H}_{\parallel}=0\\}$ which is invariant by
$A(p,\xi)$.

Consider $v=H/\sqrt{\rho}$, $\dot{v}=\dot{H}/\sqrt{\rho}$,
$\dot{\alpha}=\dot{p}/\rho$, $\alpha=p/\rho$, and
$\dot{\sigma}=\dot{\rho}/\rho$. The characteristic system reads:

$\displaystyle\begin{cases}\lambda\dot{\sigma}=\dot{u}_{\parallel},\\\
\lambda\dot{u}_{\parallel}=\dot{\alpha}+v_{\perp}\cdot\dot{v}_{\perp},\\\
\lambda\dot{u}_{\perp}=-v_{\parallel}\dot{v}_{\perp},\\\
\lambda\dot{v}_{\perp}=\dot{u}_{\parallel}v_{\perp}-v_{\parallel}\dot{u}_{\perp},\\\
\lambda\dot{\alpha}=\gamma\alpha\dot{u}_{\parallel}.\end{cases}$ (5.40)

Take a basis of $\xi^{\perp}$ such that $v_{\perp}=(b,0)$ and let
$a=v_{\parallel}$. In such a basis, the matrix of the system is

$\displaystyle\lambda-\tilde{A}:=\begin{pmatrix}\lambda&-1&0&0&0&0&0\\\
0&\lambda&0&0&-b&0&-1\\\ 0&0&\lambda&0&a&0&0\\\ 0&0&0&\lambda&0&a&0\\\
0&-b&a&0&\lambda&0&0\\\ 0&0&0&a&0&\lambda&0\\\
0&-\gamma\alpha&0&0&0&0&\lambda\end{pmatrix}.$ (5.41)

The characteristic roots satisfy

$\displaystyle\lambda(\lambda^{2}-a^{2})((\lambda^{2}-a^{2})(\lambda^{2}-c^{2})-\lambda^{2}b^{2})=0$
(5.42)

Thus either

$\displaystyle\lambda=0\ \text{or}$ (5.43) $\displaystyle\lambda^{2}=a^{2}\
\text{or}$ (5.44)
$\displaystyle\lambda^{2}=c_{f}^{2}(\hat{\xi})=\frac{1}{2}(c^{2}+h^{2}+\sqrt{(c^{2}-h^{2})^{2}+4b^{2}c^{2}})\
\text{or}$ (5.45)
$\displaystyle\lambda^{2}=c_{s}^{2}(\hat{\xi})=\frac{1}{2}(c^{2}+h^{2}-\sqrt{(c^{2}-h^{2})^{2}+4b^{2}c^{2}}),$
(5.46)

with $h^{2}=a^{2}+b^{2}=\lvert H\rvert^{2}/\rho$.

As in Lemma 5.1, if we consider $R(X)=(X-a^{2})(X-c^{2})-b^{2}X$, $\\{R\leq
0\\}=[c_{s}^{2}(\hat{\xi}),c_{f}^{2}(\hat{\xi})]$, and $R(X)\leq 0$ for
$X\in[\min(a^{2},c^{2}),\max(a^{2},c^{2})]$. Thus,

$\displaystyle c_{f}^{2}(\hat{\xi})\geq\max(a^{2},c^{2})\geq a^{2},$ (5.47)
$\displaystyle c_{s}^{2}(\hat{\xi})\leq\min(a^{2},c^{2})\leq a^{2}.$ (5.48)

At the case $v_{\perp}\neq 0$ that is $w=\hat{\xi}\times v\neq 0$: we have the
basis such that (5.41) holds is smooth in $\xi$. In this basis, $w=(0,b)$,
$b=\lvert v_{\perp}\rvert>0$. Since $R(c^{2})=-b^{2}c^{2}<0$ there holds
$c_{s}^{2}(\hat{\xi})<c^{2}<c_{f}^{2}(\hat{\xi})$. Suppose that $a\neq 0$.
Then $R(a^{2})=-a^{2}c^{2}<0$ and
$c_{s}^{2}(\hat{\xi})<a^{2}<c_{f}^{2}(\hat{\xi})$. Moreover,
$c_{s}^{2}(\hat{\xi})c_{f}^{2}(\hat{\xi})=a^{2}c^{2}$ and
$c_{s}^{2}(\hat{\xi})>0$. However, when $a=0$, we get
$c_{s}^{2}(\hat{\xi})=0$, but $c_{f}^{2}(\hat{\xi})>c^{2}>0$.

When $a\neq 0$, and $b=0$, the eigenvalues of $\tilde{A}$ are $\pm c$
(simple), $0$ (simple), and $\pm h$ (double). Assume that $c^{2}\neq h^{2}$.
Note that when $b=0$, then $\lvert a\rvert=h$ and

$\displaystyle\text{when}\ c^{2}>h^{2}:c_{f}(\hat{\xi})=c,\
c_{s}(\hat{\xi})=h,$ $\displaystyle\text{when}\
c^{2}<h^{2}:c_{f}(\hat{\xi})=h,\ c_{s}(\hat{\xi})=c.$

∎

Let $Q$ be a pseudodifferential operator of order $1$, such that $Q\dot{U}=0$
be the system of the linearized ideal MHD equations, and $q=\sigma(Q)$ be its
principal symbol. We have
$\operatorname{det}q=\tau^{2}(\tau^{2}-c_{s}^{2}(\hat{\xi})\lvert\xi\rvert^{2})(\tau^{2}-c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2})(\tau^{2}-(\xi\cdot
H)^{2}/\rho)$.

######  Proposition 5.7.

When we have $\Sigma$ is disjoint union of the hypersurfaces
$S_{1}=\\{q_{1}=\tau=0\\}$, $S_{2}=\\{q_{2}=\tau-
c_{s}(\hat{\xi})\lvert\xi\rvert=0\\}$,
$S_{3}=\\{q_{3}=\tau+c_{s}(\hat{\xi})\lvert\xi\rvert=0\\}$,
$S_{4}=\\{q_{4}=\tau-(\xi\cdot H)/\sqrt{\rho}=0\\}$,
$S_{5}=\\{q_{5}=\tau+(\xi\cdot H)/\sqrt{\rho}=0\\}$, $S_{6}=\\{q_{6}=\tau-
c_{f}(\hat{\xi})\lvert\xi\rvert=0\\}$, and
$S_{7}=\\{q_{7}=\tau+c_{f}(\hat{\xi})\lvert\xi\rvert=0\\}$; that is when we
are outside the intersection of any of these hypersurfaces then $Q$ is of real
principal type. Note that we have this case when $\xi\cdot H\neq 0$, and
$\xi\times H\neq 0$.

###### Proof.

Let $\Gamma_{1}$,….$\Gamma_{7}$ be the disjoint conic neighborhoods of
$S_{1}$,…,$S_{7}$ respectively.

Let ${}^{t}q^{\operatorname{co}}$ be the adjugate matrix (transpose of the
cofactor matrix) of $q$. We can check by using ”Mathematica” for example that
${}^{t}q^{\operatorname{co}}$ can be written as

$\displaystyle^{t}q^{\operatorname{co}}=\tau M,$ (5.49)

with $M$ being an $8\times 8$ matrix.

In $\Gamma_{1}$, we choose

$\displaystyle\tilde{q}=\bigg{(}1/\prod_{i=2}^{7}q_{j}\bigg{)}M,$ (5.50)

so we get

$\displaystyle\tilde{q}q=\tau\operatorname{Id}_{8}.$ (5.51)

In $\Gamma_{j}$, for $j=2,...,7$ we choose

$\displaystyle\tilde{q}=\bigg{(}1/\bigg{(}q_{1}^{2}\prod_{\begin{subarray}{c}i=2\\\
i\neq j\end{subarray}}^{7}q_{i}\bigg{)}\bigg{)}\ ^{t}q^{\operatorname{co}},$
(5.52)

so we get

$\displaystyle\tilde{q}q=q_{j}\operatorname{Id}_{8}.$ (5.53)

As $q_{j}$ for $j=1,...7$ are of real principal type we get the result.

∎

Remember that from lemma 5.6, we know that $c_{f}^{2}(\hat{\xi})\neq
c_{s}^{2}(\hat{\xi})$ and that $c_{f}^{2}(\hat{\xi})\neq 0$.

######  Proposition 5.8.

If $\tau\neq 0$, and $\tau^{2}\neq c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$,
or if $\tau\neq 0$ and $\tau^{2}\neq c_{s}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$,
then our system is of uniaxial type at $\Sigma_{2}$.

###### Proof.

First case: If $\tau\neq 0$, and $\tau^{2}\neq
c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$, we have $\Sigma$ is union of two
hypersurfaces $S_{1}=\\{\tau-
c_{s}(\hat{\xi})\lvert\xi\rvert=0\\}\sqcup\\{\tau+c_{s}(\hat{\xi})\lvert\xi\rvert=0\\}$,
and $S_{2}=\\{\tau-(\xi\cdot H)/\sqrt{\rho}=0\\}\sqcup\\{\tau+(\xi\cdot
H)/\sqrt{\rho}=0\\}$ intersecting at $\Sigma_{2}=\\{\tau=\lvert\xi\rvert\lvert
H\rvert/\sqrt{\rho},\ \xi\times H=0,\lvert H\rvert^{2}<\rho c^{2},\xi\neq
0\\}\sqcup\\{\tau=-\lvert\xi\rvert\lvert H\rvert/\sqrt{\rho},\ \xi\times
H=0,\lvert H\rvert^{2}<\rho c^{2},\xi\neq 0\\}$.
Second case: If $\tau\neq 0$, and $\tau^{2}\neq
c_{s}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$, we have $\Sigma$ is union of two
hypersurfaces $S_{1}=\\{\tau-
c_{f}(\hat{\xi})\lvert\xi\rvert=0\\}\sqcup\\{\tau+c_{f}(\hat{\xi})\lvert\xi\rvert=0\\}$
and $S_{2}=\\{\tau-(\xi\cdot H)/\sqrt{\rho}=0\\}\sqcup\\{\tau+(\xi\cdot
H)/\sqrt{\rho}=0\\}$ intersecting at $\Sigma_{2}=\\{\tau=\lvert\xi\rvert\lvert
H\rvert/\sqrt{\rho},\xi\times H=0,\lvert H\rvert^{2}>\rho c^{2},\xi\neq
0\\}\sqcup\\{\tau=-\lvert\xi\rvert\lvert H\rvert/\sqrt{\rho},\xi\times
H=0,\lvert H\rvert^{2}>\rho c^{2},\xi\neq 0\\}$.

In the first and in the second case we have: $S_{1}$ and $S_{2}$ are tangent
of order $1$ at $\Sigma_{2}$, the codimension of $\Sigma_{2}$ is three, the
(complex) dimension of $\mathcal{N}_{Q}$ is equal to $2$ at $\Sigma_{2}$,
$d^{2}(\det q)\neq 0$ at $\Sigma_{2}$, and $d^{i}(\det q)=0$ at $\Sigma_{2}$
for $i<2$. Hence, the conditions $(\ref{2.1})$-$(\ref{2.4})$ are satisfied. It
remains only to prove $(\ref{2.7})$. In [5], Dencker mentioned that by
proposition $3.2$ in [5], we only have to verify

$\displaystyle\partial_{\rho}q:\ker q\mapsto\operatorname{Im}q\ \text{at}\
\Sigma_{2}$ (5.54)

when $\rho\in T_{\Sigma_{2}}\Sigma$, since the order of tangency of $S_{1}$
and $S_{2}$ is $1$. $T_{\Sigma_{2}}\Sigma$ is characterized as those $\rho\in
T_{\Sigma_{2}}X$ such that $\partial_{\rho}^{2}(\operatorname{det}q)=0$. Thus
$T_{\Sigma_{2}}\Sigma$ is spanned by
$D_{1}=\xi_{2}\partial_{\xi_{1}}-\xi_{1}\partial_{\xi_{2}}$,
$D_{2}=\xi_{3}\partial_{\xi_{1}}-\xi_{1}\partial_{\xi_{3}}$,
$D_{3}=\xi_{2}\partial_{\xi_{3}}-\xi_{3}\partial_{\xi_{2}}$,
$D_{4}=H_{2}\partial_{\xi_{1}}-H_{1}\partial_{\xi_{2}}$,
$D_{5}=H_{1}\partial_{\xi_{3}}-H_{3}\partial_{\xi_{1}}$,
$D_{6}=H_{2}\partial_{\xi_{3}}-H_{3}\partial_{\xi_{2}}$, $D_{7}=\partial_{t}$,
$D_{8}=\xi_{1}\tau\partial_{\tau}+\lvert\xi\rvert^{2}\partial_{\xi_{1}}$,
$D_{9}=\xi_{2}\tau\partial_{\tau}+\lvert\xi\rvert^{2}\partial_{\xi_{2}}$,
$D_{10}=\xi_{3}\tau\partial_{\tau}+\lvert\xi\rvert^{2}\partial_{\xi_{3}}$,
$D_{11}=\tau H_{1}\partial_{\tau}+(\xi\cdot H)\partial_{\xi_{1}}$,
$D_{12}=\tau H_{2}\partial_{\tau}+(\xi\cdot H)\partial_{\xi_{2}}$,
$D_{13}=\tau H_{3}\partial_{\tau}+(\xi\cdot H)\partial_{\xi_{3}}$. We can
check that if ${}^{t}(\nu_{1},...,\nu_{8})\in\ker q$ at $\Sigma_{2}$, then we
find

$\displaystyle D_{j}q^{t}(\nu_{1},...,\nu_{8})=\ ^{t}(0,...,0),\ \ \ \
j=1,...,13,$ (5.55)

so, $(\ref{2.7})$ is satisfied. For $D_{7}$ we clearly have $D_{7}q^{t}\nu=0$.
We will show how one can get $(\ref{Dj for the first and third case})$ for the
other $D_{j}$, in particular we show the proof for $D_{1}$, and one can prove
it for the other $D_{j}$ in a similar way. Let $\nu\in\ker q$ at $\Sigma_{2}$
so we have

$\displaystyle\begin{cases}\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{1}+\rho\xi_{1}\nu_{2}+\rho\xi_{2}\nu_{3}+\rho\xi_{3}\nu_{4}=0\\\
\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{2}-\rho^{-1}(H_{3}\xi_{3}+H_{2}\xi_{2})\nu_{5}+\rho^{-1}\xi_{1}H_{2}\nu_{6}+\rho^{-1}\xi_{1}H_{3}\nu_{7}+\rho^{-1}\xi_{1}\nu_{8}=0\\\
\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{3}+\rho^{-1}\xi_{2}H_{1}\nu_{5}-\rho^{-1}(H_{3}\xi_{3}+H_{1}\xi_{1})\nu_{6}+\rho^{-1}\xi_{2}H_{3}\nu_{7}+\rho^{-1}\xi_{2}\nu_{8}=0\\\
\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{4}+\rho^{-1}\xi_{3}H_{1}\nu_{5}+\rho^{-1}\xi_{3}H_{2}\nu_{6}-\rho^{-1}(H_{2}\xi_{2}+H_{1}\xi_{1})\nu_{7}+\rho^{-1}\xi_{3}\nu_{8}=0\\\
-(H_{2}\xi_{2}+H_{3}\xi_{3})\nu_{2}+\xi_{2}H_{1}\nu_{3}+\xi_{3}H_{1}\nu_{4}\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{5}=0\\\
\xi_{1}H_{2}\nu_{2}-(H_{1}\xi_{1}+H_{3}\xi_{3})\nu_{3}+\xi_{3}H_{2}\nu_{4}\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{6}=0\\\
\xi_{1}H_{3}\nu_{2}+\xi_{2}H_{3}\nu_{3}-(H_{1}\xi_{1}+H_{2}\xi_{2})\nu_{4}\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{7}=0\\\ \gamma p\xi_{1}\nu_{2}+\gamma
p\xi_{2}\nu_{3}+\gamma p\xi_{3}\nu_{4}\pm\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{8}=0.\end{cases}$ (5.56)

We have

$\displaystyle\begin{split}D_{1}q\nu&=D_{1}\begin{pmatrix}\tau\nu_{1}+\rho\xi_{1}\nu_{2}+\rho\xi_{2}\nu_{3}+\rho\xi_{3}\nu_{4}\\\
\tau\nu_{2}-\rho^{-1}(H_{3}\xi_{3}+H_{2}\xi_{2})\nu_{5}+\rho^{-1}\xi_{1}H_{2}\nu_{6}+\rho^{-1}\xi_{1}H_{3}\nu_{7}+\rho^{-1}\xi_{1}\nu_{8}\\\
\tau\nu_{3}+\rho^{-1}\xi_{2}H_{1}\nu_{5}-\rho^{-1}(H_{3}\xi_{3}+H_{1}\xi_{1})\nu_{6}+\rho^{-1}\xi_{2}H_{3}\nu_{7}+\rho^{-1}\xi_{2}\nu_{8}\\\
\tau\nu_{4}+\rho^{-1}\xi_{3}H_{1}\nu_{5}+\rho^{-1}\xi_{3}H_{2}\nu_{6}-\rho^{-1}(H_{2}\xi_{2}+H_{1}\xi_{1})\nu_{7}+\rho^{-1}\xi_{3}\nu_{8}\\\
-(H_{2}\xi_{2}+H_{3}\xi_{3})\nu_{2}+\xi_{2}H_{1}\nu_{3}+\xi_{3}H_{1}\nu_{4}+\tau\nu_{5}\\\
\xi_{1}H_{2}\nu_{2}-(H_{1}\xi_{1}+H_{3}\xi_{3})\nu_{3}+\xi_{3}H_{2}\nu_{4}+\tau\nu_{6}\\\
\xi_{1}H_{3}\nu_{2}+\xi_{2}H_{3}\nu_{3}-(H_{1}\xi_{1}+H_{2}\xi_{2})\nu_{4}+\tau\nu_{7}\\\
\gamma p\xi_{1}\nu_{2}+\gamma p\xi_{2}\nu_{3}+\gamma
p\xi_{3}\nu_{4}+\tau\nu_{8}\end{pmatrix}\\\
&=D_{1}\begin{pmatrix}\tau\nu_{1}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{1}\\\ \tau\nu_{2}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{2}\\\ \tau\nu_{3}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{3}\\\ \tau\nu_{4}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{4}\\\ \tau\nu_{5}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{5}\\\ \tau\nu_{6}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{6}\\\ \tau\nu_{7}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{7}\\\ \tau\nu_{8}\mp\frac{\lvert
H\rvert\lvert\xi\rvert}{\sqrt{\rho}}\nu_{8}\end{pmatrix}=\
^{t}(0,...,0),\end{split}$ (5.57)

for $\nu\in\ker q$ at $\Sigma_{2}$. ∎

######  Proposition 5.9.

If $\tau^{2}\neq c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$ then our system is
of MHD type at $\Sigma_{2}$.

###### Proof.

When $\tau^{2}\neq c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$, then $\Sigma$ is
union of the five hypersurfaces $S_{1}=\\{\tau=0\\}$, $S_{2}=\\{\tau-
c_{s}(\hat{\xi})\lvert\xi\rvert=0\\}$,
$S_{3}=\\{\tau+c_{s}(\hat{\xi})\lvert\xi\rvert=0\\}$, $S_{4}=\\{\tau-(\xi\cdot
H)/\sqrt{\rho}=0\\}$, and $S_{5}=\\{\tau+(\xi\cdot H)/\sqrt{\rho}=0\\}$,
intersecting at $\Sigma_{2}=\cap_{j=1}^{5}S_{j}=\\{\tau=0,\xi\cdot H=0,\xi\neq
0\\}$. We want to prove that our system is of MHD type at $\Sigma_{2}$. We
have $S_{j}$ intersect transversally at $\Sigma_{2}$, the codimension of
$\Sigma_{2}$ is equal to two, $d^{6}(\det q)\neq 0$, and $d^{i}(\det q)=0$ for
$i<6$ at $\Sigma_{2}$, and $\dim$ of the fiber of $\mathcal{N}_{Q}$ is equal
to $6$ at $\Sigma_{2}$. $(\ref{detp at Si0})$ is satisfied for $i_{0}=1$.
Hence, still we want to check $(\ref{2.7 double refraction})$. Again, we will
prove this by proving the following

$\displaystyle\partial_{\rho}q:\ker q\mapsto\operatorname{Im}q\ \text{at}\
\Sigma_{2},$ (5.58)

when $\rho\in T_{\Sigma_{2}}\Sigma$. $T_{\Sigma_{2}}\Sigma$ is spanned by
$D_{1}=\xi_{1}\partial_{\xi_{1}}+\xi_{2}\partial_{\xi_{2}}+\xi_{3}\partial_{\xi_{3}}$,
$D_{2}=\tau\partial_{\tau}$, $D_{3}=\partial_{t}$,
$D_{4}=\tau\partial_{\xi_{1}}$, $D_{5}=\tau\partial_{\xi_{2}}$,
$D_{6}=\tau\partial_{\xi_{3}}$, $D_{7}=\tau\partial_{x_{1}}$,
$D_{8}=\tau\partial_{x_{2}}$, $D_{9}=\tau\partial_{x_{3}}$, $D_{10}=(\xi\cdot
H)\partial_{x_{1}}$, $D_{11}=(\xi\cdot H)\partial_{x_{2}}$, $D_{12}=(\xi\cdot
H)\partial_{x_{3}}$, $D_{13}=(\xi\cdot H)\partial_{\xi_{1}}$, and
$D_{14}=(\xi\cdot H)\partial_{\xi_{3}}$ (note that we have not mentioned
$(\xi\cdot H)\partial_{\xi_{2}}$ as it can be written in terms of $D_{13}$,
$D_{14}$, and $D_{5}$). We have for $(\nu_{1},...,\nu_{8})\in\ker q$ at
$\Sigma_{2}$

$\displaystyle D_{i}q\ ^{t}(\nu_{1},...,\nu_{8})=\ ^{t}(0,...,0)\ \text{at}\
\Sigma_{2}\ \text{for}\ i=1,...,14.$ (5.59)

From $(\ref{adjugate app})$ we know that $(\ref{adjugate matrix})$, and
$(\ref{principal symbol of L1})$ are satisfied with $R=D_{t}$,
$\sigma(L_{1})=M$, and
$f=\tau(\tau^{2}-c_{s}^{2}(\hat{\xi})\lvert\xi\rvert^{2})(\tau^{2}-c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2})(\tau^{2}-(\xi\cdot
H)^{2}/\rho)$. ∎

######  Remarks.

$\bullet$ When $\tau^{2}\neq c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$, on
$\Sigma\setminus\Sigma_{2}$ we have either $\tau\neq 0$ and for this case we
have Proposition 5.8, or $\tau=0$ but $\xi\cdot H\neq 0$ so our system is of
real principal type in this case.
$\bullet$ If $\tau^{2}\neq c_{s}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$, but
$\tau=0$ then we know that $\tau^{2}\neq
c_{f}^{2}(\hat{\xi})\lvert\xi\rvert^{2}$ and again our system is of real
principal type in this case.

Note: As an application for systems of generalized transverse type, one can
consider the linearized isentropic MHD equations, which is $7\times 7$ matrix;
check [13, Appendic A] where the first order term of the linearized isentropic
MHD equations and its eigenvalues are given, and then we can easily check the
type of the system as we did in this section for linearized ideal MHD
equations.

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