System of coupled first order PDE

[blue2script]

Hello everybody,

I have a problem here related to QFT in a research project. I end up with some Dirac equation with space-time dependent mass in 2 spatial dimensions.

More mathematically, the PDE to solve is

$$\left( {i\left( {\sigma ^i \otimes I_2 } \right)\partial _i + g_y \varphi ^a \left( {I_2 \otimes \sigma ^a } \right)} \right)\psi = 0$$

where $$\varphi = \varphi\left(x,y\right)$$ is a (given) function of x,y.

More explicit, the system looks like:
$$\left[ {i\left( {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} } \right)\partial _1 + i\left( {\begin{array}{*{20}c} 0 & { - i} & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & { - i} \\ 0 & 0 & i & 0 \\ \end{array} } \right)\partial _2 + g_y \left\{ {\varphi ^1 \left( {\begin{array}{*{20}c} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} } \right) + \varphi ^2 \left( {\begin{array}{*{20}c} 0 & 0 & { - i} & 0 \\ 0 & 0 & 0 & { - i} \\ i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ \end{array} } \right) + \varphi ^3 \left( {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & { - 1} & 0 \\ 0 & 0 & 0 & { - 1} \\ \end{array} } \right)} \right\}} \right]\left( {\begin{array}{*{20}c} {\psi _1 } \\ {\psi _2 } \\ {\psi _3 } \\ {\psi _4 } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right)$$

with $$\psi_i = \psi_i\left(x,y\right)$$. I tried some Fourier-method, but that wouldn't work out. I am completely stuck here and have no idea how to proceed. Could anyone give me a hint?

A big thanks in advance!

Blue2script

 

[blue2script]

Ok, sorry guys, I got it. Its pretty simple to decouple this linear PDE. I am left now with four second order PDEs.

Thanks!

Blue2script