Breathers of the nonlinear Schrödinger equation are coherent self-similar solutions

We reveal and discuss the self-similar structure of breather solutions of the cubic nonlinear Schrödinger equation which describe the modulational instability of infinitesimal perturbations of plane waves. All the time of the evolution, the breather solutions are represented by fully coherent perturbations with self-similar shapes. The evolving modulations are characterized by constant values of the similarity parameter of the equation (i.e., the nonlinearity to dispersion ratio), just like classic solitons. The Peregrine breather is a self-similar solution in both the physical and Fourier domains. Due to the forced periodicity property, the Akhmediev breather losses the self-similar structure in the physical space, but exhibits it in the Fourier domain. Approximate breather-type solutions are obtained for non-integrable versions of the nonlinear Schrödinger equation with different orders of nonlinearity. They are verified by the direct numerical simulation of the modulational instability.