Finite-gap solutions of nonlocal equations in Ablowitz-Kaup-Newell-Segur hierarchy

Nonlinear nonlocal models exist in many fields of physics. The most known of them are models possessing $\mathcal{PT}$-symmetries. Apart of $\mathcal{PT}$-symmetric models, nonlocal models with inverse time and/or coordinates are actively studied. Other types of nonlocalities arise much rare. As a rule, in works devoted to nonlinear nonlocal equations, soliton or quasi-rational solutions to such equations are studied.
In the present work we consider nonlocal symmetries, to which all equations in the Ablowitz-Kaup-Newell-Segur hierarchy. On the base of the properties of solutions satisfying nonlocal reductions of the equations in the Ablowitz-Kaup-Newell-Segur hierarchy, we propose a modification of theta-functional formula for Baker-Akhiezer functions. We find the conditions for the parameters of spectral curves associated with multi-phase solutions possessing no exponential growth at infinity. We show that under these conditions, the variables separate. The most part of statement of our work remain true for soliton and quasi-rational solutions since they are limiting cases for the multi-phase solutions.