Coupling by diffusion : continuous complex model with diffusion

(1) $latex \displaystyle{ \Large \frac{d}{dt} z(x,t) = (\mu +i \omega(x)) z - z |z|^2 + e^{i \theta} A(x,t)   }$  

(2) $latex \displaystyle{ \Large ~A_t = \alpha ~ z(x,t) ~-\gamma A } + D A_{xx}$

Pdf version : https://wiki.epfl.ch/synchro/documents/continuous/fieldaandselfcons.pdf

General case :

Searching for solution of the type :  $latex \displaystyle{ \Large  z(x,t) = R(x) e^{i~\Omega~t + i~\phi(x)}  }$  (Solutions observed in numerical solutions)

Then $latex \displaystyle{ \Large ~ A(x,t) = e^{i~\Omega~t} \frac{  \alpha } {2 D^{1/2} \sqrt{(i\Omega + \gamma)}} \int R(\tau) e^{i~\phi(\tau) }  e^{-\sqrt{(i \Omega + \gamma)/D  )} ~ |x-\tau| } ~ d\tau   }$

Replacing in (1) :

$latex \displaystyle{ \Large  \Omega  =   \omega(x)  + Im \left( \frac{1}{z(x,t)} e^{i \theta} A(x,t) \right)  =  \omega(x)  + Im \left( e^{i~\theta } \frac{1}{R(x)} e^{-i~\phi(x) } \frac{ \alpha } {2 D^{1/2} \sqrt{(i\theta + \gamma)}} \int R(\tau) e^{i~\phi(\tau) }  e^{-\sqrt{(i \Omega + \gamma)/D  )} ~ |x-\tau| } ~ d\tau \right) }$

$latex \displaystyle{ \Large  R(x)^2  =   \mu  + Re \left( \frac{1}{z(x,t)} e^{i \theta} A(x,t) \right)  =  \mu  + Re \left( e^{i~\theta } \frac{1}{R(x)} e^{-i~\phi(x) } \frac{  \alpha } {2 D^{1/2} \sqrt{(i\Omega + \gamma)}} \int R(\tau) e^{i~\phi(\tau) }  e^{-\sqrt{(i \Omega + \gamma)/D  )} ~ |x-\tau| } ~ d\tau \right) }$

The kernel in the convolution is : 

$latex \displaystyle{ \Large  e^{-\sqrt{(i \Omega + \gamma)/D  )} ~ |x| } = e^{ -\frac{1}{\sqrt{D}} (\Omega^2 + \gamma^2)^{1/4} \cos(\frac{1}{2} atan(\Omega / \gamma))~ |x| } ~ e^{-i \frac{1}{\sqrt{D}} (\Omega^2 + \gamma^2)^{1/4} \sin( \frac{1}{2} atan(\Omega / \gamma)) ~ |x| } }$

Linear phase :

Let's assume $latex \displaystyle{ \Large R(x)=R$ and $\phi(x)=\beta x }$ :