An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a $4\times 4$ Lax pair

In this paper we investigate the coupled focusing-defocusing complex short pulse equation, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the unified transform method, the initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with $4\times 4$ Lax pair on the half-line are to be analyzed. Assuming that the solution $\{q_1(x,t),q_2(x,t)\}$ of the coupled focusing-defocusing complex short pulse equation exists, we show that $\{q_{1,x}(x,t),q_{2,x}(x,t)\}$ can be expressed in terms of the unique solution of a $4\times 4$ matrix Riemann-Hilbert problem formulated in the complex $\lambda$-plane. Thus, the solution $\{q_1(x,t),q_2(x,t)\}$ can be obtained by integration with respect to $x$. Moreover, we also get that some spectral functions are not independent and satisfy the so-called global relation.