This talk is dedicated to the description of the modulation instabilities, leading to the appearance of phenomena of multiple rogue waves generation in a frame of integrable hierarchies. It is mainly concentrated on the AKNS hierarchy and some related systems.
An essential part of the results is exposed in our recent article: V.B. Matveev, A.O. Smirnov "AKNS and NLS hierarchies, MRW solutions, P_n breathers, and beyond", J.Math. Phys. 59, 091419(2018); doi: 10.1063/1.5049949