  We report an exact link between Zakharov-Gelash super-regular (SR) breathers
(formed by a pair of quasi-Akhmediev breathers) with interesting different
nonlinear propagation characteristics and modulation instability (MI). This
shows that the absolute difference of group velocities of SR breathers
coincides exactly with the linear MI growth rate. This link holds for a series
of nonlinear Schr\"{o}dinger equations with infinite-order terms. For the
particular case of SR breathers with opposite group velocities, the growth rate
of SR breathers is consistent with that of each quasi-Akhmediev breather along
the propagation direction. Numerical simulations reveal the robustness of
different SR breathers generated from various non-ideal single and multiple
initial excitations. Our results provide insight into the MI nature described
by SR breathers and could be helpful for controllable SR breather excitations
in related nonlinear systems.
