Quasi-Grammian soliton and kink dynamics of an $M$-component semidiscrete coupled integrable system

We investigate the standard binary Darboux transformation (SBDT) for an $M$-component sdC integrable system. For this, we construct the Darboux matrix using specific eigenvector solutions associated to the Lax pair, not only in the direct space but also in the adjoint space, resulting in the binary Darboux matrix. By the iterative application of the SBDT, we derive quasi-Grammian soliton solutions of the $M$-component sdC integrable system. We also examine the Darboux transformation (DT) applied to matrix solutions of the sdC integrable system, expressing solutions using quasideterminants. Additionally, we thoroughly discuss the DT applied to scalar solutions of the system, expressing solutions as ratios of determinants. Furthermore, we investigate the SBDT and its application to obtaining quasi-Grammian multikink and multisoliton solutions for the $M$-component sdC integrable system. Additionally, we demonstrate that quasi-Grammian solutions can be simplified to elementary solutions by reducing spectral parameters.