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

Abstract

Abstract 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 (FL) equation, by which the solution of the Cauchy problem of the Hermitian symmetric space FL 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 FL equation with the aid of the parabolic cylinder function.