The Riemann power problem with constant exponents for doubly periodic functions with zeros on contour (Q1603099)

It is considered the following homogeneous nonlinear boundary value problem of power type \[ [\Phi^{+}(t)]^{\alpha} = G(t) [\Phi^{-}(t)]^{\beta},\quad t\in {\mathbf T} := \{z\in {\mathbb C} : |z|= 1\}, \tag{1} \] with periodicity condition on the sides \(l_k\), \(k=1,2,\) of the parallelogram \(\Pi := \{z\in {\mathbb C} : z = t_1 + s_1 \omega_1 + s_2 \omega_2\), \(0 < s_1 < 1\), \(0 < s_2 < 1\}\): \[ \Phi^{+}(t+\omega_k) = \Phi^{+}(t), \quad t\in l_k,\;k=1,2, \tag{2} \] \(\text{Im} (\frac{\omega_1}{\omega_2}) > 0\), \(|t_1|> 1\), \(\partial \Pi \cap {\mathbf T} = \emptyset\). The solvability conditions depending on the distribution of the internal and boundary zeros of the solution are given. Problem (1)-(2) is solved in the explicit form in terms of the Weierstrass \(\zeta\)-function. The results are illustrated by an example.