The coupled Fokas-Lenells equations by a Riemann-Hilbert approach

In this paper, we use the unified transform method to consider the initial-boundary value problem for the coupled Fokas-Lenells equations on the half-line, assuming that the solution $\{q(x,t),r(x,t)\}$ of the coupled Fokas-Lenells equations exists, we show that $\{q_x(x,t),r_x(x,t)\}$ can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\lambda$. Thus, the solution $\{q(x,t),r(x,t)\}$ can be obtained by integration with respect to $x$.