  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
$x$.
