On the Riemann–Hilbert problem of the matrix Lakshmanan–Porsezian–Daniel system with a $4\times4$ AKNS-type matrix Lax pair

The initial-boundary value problems for the matrix Lakshmanan–Porsezian–Daniel system are studied by utilizing the Fokas unified transform approach. First, the spectral analysis of the $4\times4$ Ablowitz–Kaup–Newell–Segur-type matrix Lax pair is performed. Second, solutions of the matrix Lakshmanan–Porsezian–Daniel system are reconstructed from a $4\times4$ matrix Riemann–Hilbert problem. It is proved in addition that the spectral functions are not independent but are related by the so-called global relation.