Modulation instability and periodic solutions of the nonlinear Schrödinger equation

A very simple exact analytic solution of the nonlinear Schrödinger equation is found in the class of periodic solutions. It describes the time evolution of a wave with constant amplitude on which a small periodic perturbation is superimposed. Expressions are obtained for the evolution of the spectrum of this solution, and these expressions are analyzed qualitatively. It is shown that there exists a certain class of periodic solutions for which the real and imaginary parts are linearly related, and an example of a one-parameter family of such solutions is given.