Modulation instability and periodical solutions of the nonlinear Schroedinger equation

The simplest exact analytical solution of the nonlinear Schroedinger equation is found which belongs to the class of periodic solutions. The solution describes the time evolution of a wave of constant amplitude with the small periodic perturbation. Formulas for the evolution of the spectrum of this equation are obtained and their qualitative analysis is performed. It is shown that there exists a certain class of periodic solutions whose real and imaginary parts are connected linearly with each other. The example of one-parametric family of such solutions is given