Vector breathers with the negatively coherent coupling in a weakly birefringent fiber

We investigate the breathers and their interactions via the N th binary Darboux transformation for the coupled nonlinear Schrodinger equations with negatively coherent coupling in a weakly birefringent fiber, where N is a positive integer. For the two interacting optical modes, we obtain four types of the vector breathers with different structures, i.e., the single- and double-hump Kuznetsov–Ma breathers, double-hump Akhmediev breathers, kink-type breathers, as well as their interactions. Interaction with the varying background amplitude, i.e., interaction between the single-hump/double-hump Kuznetsov–Ma breather and kink-type breather, is pointed out. Interaction between the double-hump Kuznetsov–Ma breather and double-hump Akhmediev breather is elastic with the invariant background amplitude, while interaction between the single-hump Kuznetsov–Ma breather and double-hump Akhmediev breather is inelastic. Breathers and rogue waves can coexist and interact with each other. During the interaction between an eye-shaped rogue wave and a breather, eye-shaped rogue wave can split to a pair of the beak-type rogue waves. Those waves correspond to the slowly varying envelopes of two interacting optical modes in a weakly birefringent fiber.