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In this paper we investigate the coupled focusing-defocusing complex shortpulse equation, which describe the propagation of ultra-short optical pulses incubic nonlinear media. Through the unified transform method, theinitial-boundary value problem for the coupled focusing-defocusing complexshort pulse equation with $4\times 4$ Lax pair on the half-line are to beanalyzed. Assuming that the solution $\{q_1(x,t),q_2(x,t)\}$ of the coupledfocusing-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 uniquesolution of a $4\times 4$ matrix Riemann-Hilbert problem formulated in thecomplex $\lambda$-plane. Thus, the solution $\{q_1(x,t),q_2(x,t)\}$ can beobtained by integration with respect to $x$. Moreover, we also get that somespectral functions are not independent and satisfy the so-called globalrelation.

An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a $4\times4$ Lax pair