Quasi-Grammian loop dynamics of a multicomponent semidiscrete short pulse equation

A semidiscrete short-pulse equation (sdSPE) is presented via a proposed Lax pair. A multicomponent sdSPE is derived using $2^M\times 2^M$ Lax matrices. The standard binary Darboux transformation (SBDT) is employed by constructing the Darboux matrices from particular eigenvector solutions of the generalized Lax pair not only in the direct space but also in its adjoint space. Explicit expressions of the first- and second-order nontrivial quasi-Grammian loop solutions of the multicomponent sdSPE are computed, by iterating its SBDT. It is also shown that quasi-Grammian loop solutions reduce to the elementary loop solutions by applying reduction of spectral parameters.