In my research work, I am dealing with a quasi-linear system of first order p.d.e.'s with two independent variables (say $x_1$ and $x_2$) and four dependent variables (say $u_1(x_1,x_2)$, $u_2(x_1,x_2)$, $u_3(x_1,x_2)$ and $u_4(x_1,x_2)$) of the form \begin{equation} \mathbf{A}(\mathbf{x},\mathbf{u}(\mathbf{x}))\frac{\partial \mathbf{u}}{\partial x_1}+\mathbf{B}(\mathbf{x},\mathbf{u}(\mathbf{x}))\frac{\partial \mathbf{u}}{\partial x_2}=\mathbf{f}(\mathbf{x}), \end{equation} where $\mathbf{x}=[x_1\,\,x_2]^{T}$ and $\mathbf{u}=[u_1\,\,u_2\,u_3\,u_4]^{T}$ are, respectively, the independent and dependent vector variables, constructed out of $x_\alpha$'s and $u_i$'s, $\alpha=1,2$ and $i=1,2,3,4$. $\mathbf{A}$ and $\mathbf{B}$ are two $4\times 4$ real matrices which are functions of $\mathbf{x}$ and $\mathbf{u}$ (dependence on $\mathbf{u}$ makes the system quasi-linear) and $\mathbf{f}$ is a known vector valued function of $\mathbf{x}$.

To determine the type of the system, we consider the eigenvalues of the matrix $\mathbf{A}^{-1}\mathbf{B}$ (assuming $\mathbf{A}$ to be invertible which it is in my case). Now there are 4 possibilities about the nature of these eigenvalues and there are corresponding "type" of the p.d.e. system defined:

All eigenvalues real and distinct, implying hyperbolic system. We have the well-studied system of hyperbolic system of conservation laws falling under this category.

All eigenvalues are complex (two pairs of complex conjugate eigenvalues). We have then an elliptic system. (I don't know about any physical example of this type.)

All eigenvalues are real but with unequal algebraic and geometric multiplicity (i.e. no set of 4 linearly independent eingenvectors). We then have a parabolic system. (Again, no knowledge of physical example.)

Two real and one pair of complex conjugate eigenvalues. This does not fall into the above categories. Probably these are called the quasi-linear system of mixed type.

My first question is about the the above classification scheme which, in fact, imitates the classification scheme of linear system of first order p.d.e.'s. Matrices $\mathbf{A}$ and $\mathbf{B}$ involve unknown function $\mathbf{u}$. So, how to determine the type of the system (from the nature of its eigenvalues) a priori to the determination of the solution?

My second question is about the type 4 systems, because my problem falls into this category. How to deal with this mixed type p.d.e. systems?

My major is mechanical engineering and I am working in theory of elasticity. Any help would be greatly appreciated as I know very little about analytical methods to solve quasi-linear p.d.e. systems of first order.

Just to mention, in a previous question I was concerned about a system of 3 linear first order pdes of mixed type (3 roots of the characteristic polynomial: either all are real, or one real and the others appearing as a complex conjugate pair) which is a special case of this problem. To emphasize, I come up with these systems in a physical problem and I know that there must exist real valued "well-behaved" solutions (bounded). Which I do not know is how to find those solutions which must exist and I am looking for possible methods in this forum.

Thanks in advance for any help.