We construct the multi-breather solutions of the focusing nonlinear
Schr\"odinger equation (NLSE) on the background of elliptic functions by the
Darboux transformation, and express them in terms of the determinant of theta
functions. The dynamics of the breathers in the presence of various kinds of
backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions
are presented, and their behaviors dependent on the effect of backgrounds are
elucidated. We also determine the asymptotic behaviors for the multi-breather
solutions with different velocities in the limit t→±∞, where the
solution in the neighborhood of each breather tends to the simple one-breather
solution. Furthermore, we exactly solve the linearized NLSE using the squared
eigenfunction and determine the unstable spectra for elliptic function
background. By using them, the Akhmediev breathers arising from these
modulational instabilities are plotted and their dynamics are revealed.
Finally, we provide the rogue-wave and higher-order rogue-wave solutions by
taking the special limit of the breather solutions at branch points and the
generalized Darboux transformation. The resulting dynamics of the rogue waves
involves rich phenomena: depending on the choice of the background and
possessing different velocities relative to the background. We also provide an
example of the multi- and higher-order rogue wave solution.Comment: 45 pages, 16 figure