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

Pair-Tunneling Induced Localized Waves in a Vector Nonlinear Schr\"{o}dinger Equation