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.