Integrable semi-discretization of a multi-component short pulse equation

In the present paper, we mainly study the integrable semi-discretization of a multi-component short pulse equation. Firstly, we briefly review the bilinear equations for a multi-component short pulse equation proposed by Matsuno (J. Math. Phys. $\mathbf{\text{52}}$ 123705) and reaffirm its $N$-soliton solution in terms of pfaffians. Then by using a B"{a}cklund transformation of the bilinear equations and defining a discrete hodograph (reciprocal) transformation, an integrable semi-discrete multi-component short pulse equation is constructed. Meanwhile, its $N$-soliton solution in terms of pfaffians is also proved.