Domains of existence and asymptotics of complex periodic solutions of the stationary nonlinear Schrödinger equation

The domains of existence for complex periodic particular solutions of the stationary nonlinear Schrödinger equation are determined. It is shown that, if the solution of this class has at least one point in which the amplitude does not have a local extremum and is equal to zero, then the phase of the solution is constant. The asymptotics of such solutions are considered and a simple procedure (the replacement of one set of the "basis" elliptic functions by another) is described which allows one to construct the solution corresponding to the nonlinearity of one type (defocusing or focusing) from the known solution for the nonlinearity of another type (focusing or defocusing).