The Hermitian symmetric space Fokas-Lenells equation: spectral analysis and long-time asymptotics

Based on the inverse scattering transformation, we carry out spectral analysis of the $4\times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas-Lenells equation, by which the solution of the Cauchy problem of the Hermitian symmetric space Fokas-Lenells equation is transformed into the solution of a Riemann-Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann-Hilbert problem, from which the various Deift-Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann-Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space Fokas-Lenells equation with the aid of the parabolic cylinder function. Keywords: Hermitian symmetric space Fokas-Lenells equation; Nonlinear steepest descent method; Spectral analysis; Long-time asymptotics.