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.
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.