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System of coupled first order PDE

  1. Feb 25, 2008 #1
    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

    [tex]
    \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
    [/tex]

    where [tex]\varphi = \varphi\left(x,y\right)[/tex] is a (given) function of x,y.

    More explicit, the system looks like:
    [tex]
    \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)
    [/tex]

    with [tex]\psi_i = \psi_i\left(x,y\right)[/tex]. 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
     
  2. jcsd
  3. Feb 25, 2008 #2
    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
     
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