I firstly apologise for my English which is not my mothertongue.I hope this question is not too physiscs related so that my post fits well in here.
I am trying to understand a generalization of the wave equation. Consider all the unknown functions to be at least of class $C^2$ for every possible input.
I firstly introduce the wave equation for the unknown function $u : \mathbb{R}^{n} \times \mathbb{R}_{\geq 0} \to \mathbb{R}$ and $c \in \mathbb{R}_{>0}$, which in cartesian coordinates reads: \begin{align} \partial_{tt} u= c^2 \nabla^2 u = c^2 \sum_{1\leq i \leq n} \partial_{x_{i}x_{i}} u. \end{align}
Now for ease of notation we consider $n=2$. I am aware that the solution depends heavily on the Initial and Boundary Condition but let us consider a possible simple solution described [in Wikipedia][1] which has many properties like being radially symmetric.
Now we focus on the main question. I consider two generalizations.
For the first, we consider $c_{1} \neq c_{2}$ both in $\mathbb{R}_{>0}$ and consider the following PDE for the function $u: \mathbb{R}^2 \to \mathbb{R}$ \begin{align} \partial_{tt} u= c_{1}^2 \partial_{x_{1}x_{1}}u +c_{2}^2\partial_{x_{2}x_{2}}u. \end{align} My first question is... does this PDE still belongs to the class of "Wave Equations"? My first answer would be yes. We could think of $c_{1}$ as the speed of the propagation in $x_{1}$ and $c_{2}$ in $x_{2}$. I would expect, instead of a radially symmetric solution, an elliptic one. To be more precise the image from Wikipedia would propagate ellipses instead of circles. Is this interpretation correct or at least does it make sense?
The second generalization stems out of the first. Define for two matrices $A,B \in \text{Mat}_{3\times 3}(\mathbb{R})$ their anticommutator $$]A,B[ = AB+BA.$$ Now consider a vector-valued function $u: \mathbb{R}^2 \to \mathbb{R}^3$ and the following PDE \begin{align} \partial_{tt}u = (A \partial_{x_{1}}+B\partial_{x_{2}})^2u = A^2\partial_{x_{1}x_{1}}u+]A,B[\partial_{x_{1}x_{2}}u +B^2 \partial_{x_{2}x_{2}}u \end{align} where $]A,B[ \neq 0$.
Does this vector valued PDE still belongs to the broad class of wave equation (my doubts are related to the presence of a mixed derivative term)? If yes, how can I make sense of $A,B$? If not does anyone have any interpretation or any source that I can read?
Thanks in advance for anyone answering!
Edit: Specified regularity of $u$ [1]: https://en.wikipedia.org/wiki/Wave_equation#/media/File:Spherical_wave2.gif
Take $A$ to be 90 deg rotation around the$z$ axis, and $B=A^{-1}$. Then the first component of the pde is $$u_{1tt}=-u_{1xx}+2u_{1xy}-u_{1yy}.$$ I don't think you would call this a wave equation. For example any function of $x+y$ is a solution.
There might not be a formal definition of wave equation, but this pde is parabolic. See any text or Wikipedia on classification of pdes. Generally, hyperbolic equations are thought of as wave equations.