A Hermitian symmetric space Fokas–Lenells equation: Solitons, breathers, rogue waves

Abstract

Based on the zero-curvature representation, we propose a Hermitian symmetric space Fokas–Lenells equation associated with a 4 × 4 matrix spectral problem. Resorting to the gauge transformation between spectral problems, we find a restrictive condition of the spectral functions that can ensure the symmetry of the new potential matrix, from which n-fold classical Darboux transformations and N-fold generalized Darboux transformations of this equation are constructed with the help of the limit technique and an iterative procedure. Utilizing the Darboux transformations and Mathematica software, we obtain various solutions for the Hermitian symmetric space Fokas–Lenells equation, including the multi-soliton solution, kink–breather solution, eye-shaped rogue wave solution, one-peaky shaped rogue wave solution, and dark two-peaky shaped rogue wave solution. By varying the values of the free parameters, we obtain different types of spatial–temporal distribution structures for the solutions, including the fundamental, line and triangular patterns. Furthermore, the dynamical behaviour of these solutions, including the fusion and fission processes of rogue waves, is analysed in detail.