  In this paper, We extend the two-component coupled Hirota equation to the
three-component one, and reconstruct the Lax pair with $4\times4$ matrixes of
this three-component coupled system including higher-order effects such as
third-order dispersion, self-steepening and delayed nonlinear response.
Combining the generalized Darboux transformation and a specific vector solution
of this $4\times4$ matrix spectral problem, we study higher-order localized
nonlinear waves in this three-component coupled system. Then, the semi-rational
and multi-parametric solutions of this system are derived in our paper. Owing
to these more free parameters in the interactional solutions than those in
single- and two-component Hirota equation, this three-component coupled system
has more abundant and fascinating localized nonlinear wave solutions
structures. Besides, in the first- and second-order localized waves, we get a
variety of new and appealing combinations among these three components $q_1,
q_2$ and $q_3$. Instead of considering various arrangements of the three
potential functions, we consider the same combination as the same type
solution. Moreover, the phenomenon that these nonlinear localized waves merge
with each other observably, may appears by increasing the absolute values of
two free parameters $\alpha, \beta$. These results further uncover some
striking dynamic structures in multi-component coupled system.
