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 ( 2 n − 1 , 2 n ) 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.

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