Integrable semi-discretization of complex and multi-component coupled dispersionless systems and their solutions

An integrable semi-discretization of complex and multi-component coupled dispersionless systems via Lax pairs is presented. A Lax pair is proposed for the complex sdCD system. We derive the Lax pair for the multi-component sdCD system through generalizing the $2 \times 2$ Lax matrices to the case of $2^{N} \times 2^{N}$ Lax matrices. A Darboux transformation (DT) is applied to the complex and multi-component sdCD systems and is used to compute soliton solutions of the systems. It is also shown that the soliton solutions of the semi-discrete systems reduce to the continuous systems by applying continuum limit.