I want to know for which condition on "$a$" and "$k$", i.e. for which function of $a(k)$, the following Riccati equation, with the initial condition $u(0)=ia$ ($i^2=-1$), have periodic solution with period $L=2\pi/k$
$$\frac{du(z)}{dz}=u(z)^2+(a^2\cos^2(kz)-iak\sin(kz)-4)$$ $$i^2=-1$$
For example I know that if $a=10$ then if $k=1.2868$, we have a periodic solution with period $L=2\pi/k=0.7771$ , but, for $a=10$ and other $k$ values such as $2$, $3.5$, $...$ we have not periodic solution.
Thank you.