On analytic properties of periodic solutions for equation $\mathcal{H}u_x-u+u^p=0$

The paper focuses on 2L-periodic solutions, UL(x), of the nonlocal equation (p > 2 is integer, is the Hilbert transform). We give numerical evidence for the existence of a continuous family of these solutions parametrized by L for p = 3, 4, 5. The features of the solutions UL(x) are discussed. In particular, we offer strong numerical arguments that UL(x) can be continued analytically from the real axis to some strip in the complex plane, UL(x) ? UL(z). The singularities which arise under analytical continuation of these solutions into the complex plane are considered. It is proved (theorem 1) that UL(z) cannot be any power of a meromorphic function. The formula which describes the asymptotic behavior of UL(z) in a neighborhood of the singularity is given. It agrees with the numerical results; computations also allow us to locate the closest to the real axis singularity of UL(z) and estimate the width of the analyticity strip.