Lax pair, Darboux transformation, vector rational and semi-rational rogue waves for the three-component coupled Hirota equations in an optical fiber

.The optical fiber communication system is one of the supporting systems of the modern internet age. In this paper, we study the three-component coupled Hirota equations, which govern the simultaneous propagation of three fields in the normal dispersion regime of an optical fiber. We derive a Lax pair and construct the corresponding Darboux transformation. Via the Darboux transformation, rogue wave solutions with the corresponding characteristic polynomial admiting a quadruple root and two/one double roots are obtained. Via such solutions, we depict the first-order vector rational rogue wave with the two components containing the four-petaled rogue wave, and the other component containing one eye-shaped rogue wave; increasing the value of the real parameter which denotes the integrable perturbation, we observe that the range of the first-order vector rational rogue wave along an axis increases; we display the first-order vector rational rogue waves with each component containing two/three merged and separated rogue waves. The second-order rogue waves are graphically displayed, with each component containing five, seven or nine rogue waves, which form the pentagon, triangle, clawlike, hexagon, arrow, line or trapezoid structures. The first- and second-order vector rational/semi-rational rogue waves are graphically exhibited. Two types of the vector semi-rational rogue waves are presented: the one with each component containing the rogue waves and line breathers, and the other with each component containing the rogue waves and Y-shaped breathers.