Writing 2D linear system of balance laws in compact form I have three equations
$$\rho \frac{\partial }{\partial t}v = \frac{\partial}{\partial x}\sigma_{21} + \frac{\partial}{\partial z}\sigma_{23}$$
$$\frac{\partial}{\partial t}\sigma_{21} = a\frac{\partial}{\partial x}v + b\frac{\partial}{\partial z}v$$
$$\frac{\partial }{\partial t}\sigma_{23} = b\frac{\partial}{\partial x}v + c\frac{\partial}{\partial z}v$$
where 
$$v=v(t,x_1,x_3)$$
$$\sigma_{21} = \sigma_{21}(t,x_1,x_3)$$
$$\sigma_{23} = \sigma_{23}(t,x_1,x_3).$$
Is there a way to write these equations in a system, of the form:
$$\frac{\partial}{\partial z} \bigg(\begin{array}{c}
\sigma_{23}\\ 
v\end{array}\bigg) = \left(\begin{array}{cc} ... & ...\\ ... & ... \end{array}\right)\frac{\partial}{\partial t} \bigg(\begin{array}{c}
\sigma_{23}\\ 
v\end{array}\bigg)$$
or a similar form? I tried with fourier transforms but I feel like this is losing information. Any help would be greatly appreciated.
 A: I think it is more useful to write it as a system of two equations
$$
\rho \frac{\partial v}{\partial t} = \nabla \cdot \sigma
$$
$$
\frac{\partial \sigma}{\partial t} = A \nabla v
$$
with $A=\begin{pmatrix} a & b \\b & c\end{pmatrix}$.
A: One could write
$
\partial_t
\boldsymbol{u}
=
\boldsymbol{A}\,
\partial_x
\boldsymbol{u}
+
\boldsymbol{C}\,
\partial_z
\boldsymbol{u}
$,
where
$$
\boldsymbol{u} = \begin{pmatrix}
v \\
\sigma_{21} \\
\sigma_{23}
\end{pmatrix} ,
\qquad
\boldsymbol{A} = \begin{pmatrix}
0 & 1/\rho & 0 \\
a & 0 & 0 \\
b & 0 & 0 \\
\end{pmatrix} ,
\qquad
\boldsymbol{C} = \begin{pmatrix}
0 & 0 & 1/\rho \\
b & 0 & 0 \\
c & 0 & 0 \\
\end{pmatrix} .
$$
In Fourier domain, we have $\left(\kappa_x\boldsymbol{A} + \kappa_z\boldsymbol{C}- \omega\boldsymbol{I}\right) \hat{\boldsymbol{u}} = \boldsymbol{0}$, which has non-trivial solutions provided that the determinant $|\kappa_x\boldsymbol{A} + \kappa_z\boldsymbol{C}- \omega\boldsymbol{I}|$ equals zero, i.e. that the following dispersion relation is satisfied
$$a {\kappa_x}^2 + 2b {\kappa_x}{\kappa_z} + c {\kappa_z}^2 = \rho \omega^2 .$$
The dispersion relation may be interpreted in terms of conic sections.
