One-phase and two-phase solutions of the focusing nonlinear Schrodinger equation

A periodic boundary value problem for the focusing nonlinear Schrodinger equation is considered. This version of the equation has applications in nonlinear optics. The existence of single-phase solutions with the structure of traveling waves is shown. For such solutions, the question of their stability is considered. Three other types of single-phase solutions are found. Asymptotic formulas are obtained for these solutions. It is also shown that these solutions generate three types of already two-phase solutions of the main boundary value problem for the focusing Schrodinger equation. For this, the principle of self-similarity is used. Some results can be applied to the defocusing version of the nonlinear Schrodinger equation.