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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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# Question about a partial differential equation

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