Higher-order modulation instability in a fourth-order Nonlinear Schr\"{o}dinger Equation

We present a complete dynamical description of the higher-order modulation instability for a fourth-order nonlinear Schr\"{o}dinger equation. For two-breather solutions of this equation, we have identified the locus in a geometrical space where the growth rates for the breathers are equal in parameter space. We show that a circle bounds the entire parameter space for the nonlinear Schr\"{o}dinger equation. In contrast, it is bound by an intersecting circle and an ellipse for the fourth-order equation. We show that, for all the higher-order equations in the nonlinear Schr\"{o}dinger equation hierarchy, the parameter space follows a similar geometric interpretation as for the fourth-order equation.