High-order rogue waves for the Hirota equation

•The determinant representation of the NN-fold Darboux transformation of the Hirota equation.•Properties of the fundamental pattern of the higher order rogue wave.•Ring structure and triangular structure of the higher order rogue waves.

The Hirota equation is better than the nonlinear Schrödinger equation when approximating deep ocean waves. In this paper, high-order rational solutions for the Hirota equation are constructed based on the parameterized Darboux transformation. Several types of this kind of solutions are classified by their structures.