Abstract
In this work, we study the superregular solitonic solutions of the
higher-order nonlinear Schr\”{o}dinger equation (HNLSE)
with nonvanishing boundary conditions. Solving the HNLSE with the
dressing method, we obtain the explicit form of one-and two-solitonic
solutions and discuss them in detail. These solutions can be used to
describe the nonlinear stage of the modulation instability (MI) of the
condensate. Moreover, we also find some novel features of the nonlinear
stage of the MI arising from higher-order effects. This family of novel
solutions include Peregrine soliton, Akhmediev breather, Kuznetsov-Ma
breather, a symmetrical quasi-Akhmediev breather, coexistence of two
symmetrical quasi-Akhmediev breather, coexistence of two nonsymmetrical
quasi-Akhmediev breather, coexistence of a quasi-Akhmediev breather and
a soliton, a bipolar-freak-wave etc. Finally, the main characteristics
of these rational solutions are discussed with some graphics. These
solutions would be of much importance in understanding and explaining
rogue wave phenomena arising in nonlinear and complex systems.