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I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which decouples from the rest, determining $4$ dependent variables, which written in matrix notation is

$u_{t,i} + B_{i j}u_{x,j}+ C_i=0$.

Here $$B=\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{pmatrix}$$ so has doubly degenerate eigenvalues $\pm i$. I gather that because these eigenvalues are complex, this is an elliptic system of first-order linear PDE's. Can anybody recommend a good reference which would help me see/explain to me whether and how this elliptic system could be solved analytically?

So far I have come across few references which go into detail about how to solve a system of first-order linear PDE's, in particular ones that are not totally hyperbolic.

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    $\begingroup$ (1) Do you know that you can use LaTeX in the question? Click on 'help' at the top of the page for details. (2) What are the $C_j$? If these terms are linear in $u$ and have constant coefficients, then the best way to get an analytic solution is probably using the Fourier transform. $\endgroup$
    – Igor Khavkine
    Commented Nov 9, 2013 at 21:14
  • $\begingroup$ The C_j have terms linear in u and also source terms that are just functions of independent variables t and x. $\endgroup$
    – user42587
    Commented Nov 10, 2013 at 2:33

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If your equation can be written as $$ (\delta_{ij}\partial_t + B_{ij} \partial_x + c_{ij}) u_j = f_i , $$ where the $c_{ij}$ are constants and $f_i = f_i(t,x)$, then the most general solution can be obtained using the Fourier transform: \begin{gather} (\delta_{ij} (-i\omega) + B_{ij} (ik) + c_{ij}) \tilde{u}_j = A_{ij}(\omega,k) \tilde{u}_j = \tilde{f}_i(\omega,k) , \\ u_j = \int d\omega\,dk\, e^{-i\omega t + ikx} (A^{-1}_{ji}(\omega,k) \tilde{f}_i(\omega,k) + \tilde{g}_j(\omega,k)) , \end{gather} where $\tilde{g}_i(\omega,k)$ is any distribution that is supported only on set where $\det A_{ij}(\omega,k) = 0$, that is, where $A_{ij}$ is singular. Finally, the term $A^{-1}\tilde{f}$ may also have (non-integrable) singularities, so it has to be treated as a distribution as well (think of how the Fourier representation of propagators in quantum field theory is regulated using $+i\epsilon$ or some other prescription).

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    $\begingroup$ There's also a chance that this can be made simpler, since the first order terms appear to uncouple into two 2-by-2 systems, one with $u_1$ and $u_3$ and the other with $u_2$ and $u_4$. If the zero-th order term ($c_{ij}$) also uncouples in the same way, then you can solve each subsystem separately. $\endgroup$
    – Deane Yang
    Commented Nov 10, 2013 at 17:34
  • $\begingroup$ It turns out that I cannot make both matrices $B$ and $c$ constant, i.e. if I want B to be constant, $c \propto 1/t$, and if I let $c$ be constant, $B \sim e^{t}$. It seems the Fourier method is no longer explicity applicable. But is there a good book/reference with some exposition of the Fourier method you mention above, and possibly methods that might apply to my problem? Thanks. $\endgroup$
    – user42587
    Commented Nov 11, 2013 at 3:34
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    $\begingroup$ Sorry, but in that case, Fourier methods would not be directly helpful, unless the non-constant coefficient part can be treated as a perturbation. Any textbook on mathematical physics will discuss integral transforms, e.g., Arfken. More generally, you may have to face the fact that no analytic solution can be easily (or at all found). You then have to think about what qualitative information do you really need from the solution. $\endgroup$
    – Igor Khavkine
    Commented Nov 11, 2013 at 13:57

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