Question about a partial differential equation

[feng]

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.

$$-\partial_z\partial_{\bar{z}}u + 2i\partial_{\bar{z}}\theta\partial_z u + m^2u=0$$

$$-\partial_z\partial_{\bar{z}}v - 2i\partial_{z}\theta\partial_{\bar{z}} v + m^2v=0$$

Where $$\theta = 2\arctan[\exp(2m(ze^{i\phi} + \bar{z}e^{-i\phi}))]$$ is a solution of the
doubled elliptic sine-Gordon equation. $$m > 0$$ and $$\phi$$ 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.

 

[Astronuc]

Please provide a reference for the two PDE's.

 

[feng]

Those are found by myself. I don't know any reference.

 

[Astronuc]

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
$$-\partial_z\partial_{\bar{z}}u$$

 

[feng]

Thank you Astronuc, I made some progress. I will show the solution later when I
finished.