Vector localized and periodic waves for the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation

Dynamical properties of vector localized and periodic waves hold significant importance in the study of physical systems. In this work, we investigate the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation. For the two interacting components, we construct the binary Darboux transformation formulas, vector localized, and periodic wave solutions. Via those solutions, different kinds of nonlinear waves can be achieved, including rogue waves, solitons, positons, and periodic waves. When the imaginary part of the spectral parameter is not zero, eye-shaped rogue waves appear in one component, and the twisted rogue wave pairs in the other component. As the spectral parameter is real, we derive distinct forms of vector localized and periodic waves on the non-zero background, such as the vector solitons, positons, periodic waves, breathers on the periodic wave background, and rational solitons. These results may be valuable in this investigation of nonlinear waves in physical systems.