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Breathers of the nonlinear Schrodinger 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.