NLS breathers, rogue waves, and solutions of the Lyapunov equation for   Jordan blocks
  The infinite families of Peregrine, Akhmediev and Kuznetsov-Ma breather solutions of the focusing Nonlinear Schroedinger (NLS) equation are obtained via a matrix version of the Darboux transformation, with a spectral matrix of the form of a Jordan block. The structure of these solutions is essentially determined by the corresponding solution of the Lyapunov equation. In particular, regularity follows from properties of the Lyapunov equation. 
Oleksandr Chvartatskyi and Folkert M\"uller-Hoissen
nlin.SI math-ph math.MP