| Summary: | Complex instabilities are the major reason for drastic changes and extreme events in dynamical systems. Several modes of instability growing simultaneously with nonlinear interaction between them may lead to unforeseeable outcomes leading to catastrophic consequences. The most common examples of these instabilities are the modulation instability (MI). Studies show that an infinite number of instability modes remain active in a dynamical system. Although a one-mode MI can be analysed in the frame of a precise mathematical model, namely the Akhmediev breather, the dynamics of several concurrent MI modes referred to as the higher-order MI is very difficult to handle. We developed a unique geometrical approach that provides an entirely new and intuitive way to deal with higher-order MI. We apply this approach in description of higher-order modulation instability, multi-breather solutions, their degenerate versions and higher-order rogue waves of the nonlinear Schrödinger equation. For a system with infinitely many interacting instability modes, the band of the instability in this description is a hypercube, a multi-dimensional space of modulation frequencies. A large variety of special multi-breather and multi-rogue wave solutions of the nonlinear Schrödinger equation in this description corresponds to special points and lines within this hypercube.
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