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Question about a partial differential equation

  1. Aug 6, 2007 #1
    Hi all,

    I have a pair of partial differential equations which arose from a study of Dirac equation in a scalar background,
    I have tried some methods but still can't work out.

    [tex]-\partial_z\partial_{\bar{z}}u + 2i\partial_{\bar{z}}\theta\partial_z u + m^2u=0[/tex]

    [tex]-\partial_z\partial_{\bar{z}}v - 2i\partial_{z}\theta\partial_{\bar{z}} v + m^2v=0[/tex]

    Where [tex]\theta = 2\arctan[\exp(2m(ze^{i\phi} + \bar{z}e^{-i\phi}))][/tex] is a solution of the
    doubled elliptic sine-Gordon equation. [tex]m > 0[/tex] and [tex]\phi[/tex] is a real parameter.
    The domain is the whole complex plane.

    First of all, does any solution exist? And is there any method to solve it?

    I am not familiar with the theory of partial differential equation. Any help will be appreciated.

    Thanks.
     
    Last edited: Aug 6, 2007
  2. jcsd
  3. Aug 7, 2007 #2

    Astronuc

    User Avatar

    Staff: Mentor

    Please provide a reference for the two PDE's.
     
  4. Aug 7, 2007 #3
    Those are found by myself. I don't know any reference.
     
  5. Aug 7, 2007 #4

    Astronuc

    User Avatar

    Staff: Mentor

    For some background on sine-Gordon

    http://mathworld.wolfram.com/Sine-GordonEquation.html

    http://en.wikipedia.org/wiki/Sine-Gordon_equation

    http://eqworld.ipmnet.ru/en/solutions/npde/npde2106.pdf (not sure how useful)

    http://library.lanl.gov/cgi-bin/getfile?00285946.pdf

    http://www2.appmath.com:8080/site/Pan5/Pan5.html

    I must admit that I am not familiar with the del operations in complex variables, so am curious about the del operation with respect to the complex conjugate in
    [tex]-\partial_z\partial_{\bar{z}}u[/tex]
     
  6. Aug 7, 2007 #5
    Thank you Astronuc, I made some progress. I will show the solution later when I
    finished.
     
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