The Hirota equation: Darboux transform of the Riemann–Hilbert problem and higher-order rogue waves

Abstract

The matrix Riemann–Hilbert problem of the Hirota equation with non-zero boundary conditions is investigated. Based on the matrix Riemann–Hilbert problem, the n-fold Darboux transformation is established for the Hirota equation such that (2n−1,2n)th-order rogue waves can be found simultaneously. Particularly, we exhibit the first-, second-, third-, and fourth-order rogue waves, and first- and second-order temporal–spatial and spatial periodic breathers for some parameters, respectively.