  We investigate the localized waves of the coupled two-mode nonlinear
Schr\"{o}dinger equations with a pair-tunneling term representing strongly
interacting particles can tunnel between the modes as a fragmented pair.
Facilitated by Darboux transformation, we have derived exact solution of
nonlinear vector waves such as bright solitons, Kuznetsov-Ma soliton, Akhmediev
breathers and rogue waves and demonstrated their interesting temporal-spatial
structures. The phase diagram that demarcates the parameter ranges of the
nonlinear waves is obtained.
  Our results have implications in such diverse fields as Bose-Einstein
condensate, nonlinear fibers and super fluids.
