Quasi-Grammian Loop Dynamics of a Multi-Component Semi-Discrete Short Pulse Equation

In this article, a semi-discrete short pulse equation (sdSPE) is presented via a proposed Lax pair. A multi-component sdSPE is derived by 2M ×2M Lax matrices. The standard binary Darboux transformation (SBDT) is employed by constructing the Darboux matrices from particular eigen-vector solutions to the generalized Lax pair not only in the direct space but also in its adjoint space. Explicit expressions of first and second order non-trivial quasi-grammian loop solutions of multi-component 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.