Question about a partial differential equation

  • Thread starter feng
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  • #1
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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.
 
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Answers and Replies

  • #2
Astronuc
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Please provide a reference for the two PDE's.
 
  • #3
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Those are found by myself. I don't know any reference.
 
  • #4
Astronuc
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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]
 
  • #5
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Thank you Astronuc, I made some progress. I will show the solution later when I
finished.
 

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