Localized nonlinear waves of the three-component coupled Hirota equation by the generalized Darboux transformation

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.