The IST spectral portraits of the first order doubly periodic solutions of the nonlinear Schrödinger equation

The spectra of the inverse scattering technique (IST) play a crucial role in the physics of nonlinear phenomena. They define the long term evolution of dynamical systems. We present the IST spectral portraits for the extensive three-parameter families of the first order doubly periodic solutions of the nonlinear Schrödinger equation that cover a wide range of physical phenomena such as modulation instability, rogue waves and many other problems with periodic boundary conditions. We relate these spectral portraits with the parameters of the family. We show that there are two qualitatively different types of spectral portraits. A-type spectra consist of two continuous bands: a band of purely imaginary eigenvalues within the interval [-i, i] and a finite band of complex eigenvalues. On the contrary, B-type spectra possess only continuous bands of imaginary eigenvalues all located within the interval [-i, i] and separated by a finite band gap. A physical interpretation of these results is given.