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In this paper, we investigate the coupled Sasa-Satsuma equations, which describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Darboux-dressing transformation is applied to obtain the dark-bright soliton and semirational rogue-wave solutions. Dark-bright one solitons with the single-hump, double-hump, and even breather-like structures are presented. Interactions between the double-peak breather and different kinds of dark-bright solitons are studied. We show that the double-peak (or single-peak) rogue wave can coexist and interact with different kinds of dark-bright solitons. Coexistence of the solitons with different velocities and rogue waves is also found. Numerical stabilities of the dark-bright solitons and semirational rogue waves are exhibited. It is expected that those localized wave phenomena can be experimentally observed and have potential applications.

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A set of the coherently-coupled nonlinear Schrödinger equations with the positive coherent coupling terms, which are related to the optical fiber communication, are studied through the binary Darboux transformation with the dimensional reduction. Formalisms of the solutions appear as the mixtures of the polynomial functions with exponential functions. When the spectral parameter is real, we obtain different kinds of the solutions, such as the soliton, degeneratesoliton, periodic, and soliton-like rational solutions. When the spectral parameter is complex with a non-zero imaginary part, we obtain the rogue waves and twisted rogue-wave pairs, and show that an eye-shaped rogue wave splits into a twisted rogue-wave pair.

Under investigation in this paper is the Sasa-Satsuma equation, which describes the propagation of ultrashort pulses in a monomode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Based on the known bilinear forms, through the modified expanded formulas and symbolic computation, we construct the bright two-soliton solutions. Through classifying the interactions under different parameter conditions, we reveal six cases of interactions between the two solitons via an asymptotic analysis. With the help of the analytic and graphic analysis, we find that such interactions are different from those of the nonlinear Schrödinger equation and Hirota equation. When those solitons interact with each other, the singular-I soliton is shape-preserving, while the singular-II and nonsingular solitons may be shape preserving or shape changing. Such elastic and inelastic interaction phenomena in a scalar equation might enrich the knowledge of soliton behavior, which could be expected to be experimentally observed.