  In this article, a fully discrete short pulse (SP) equation is presented as
an integrability condition of a linear system of difference equations (also
known as discrete Lax pair). Additionally, two semi-discrete versions of the SP
equation have also been obtained from fully discrete SP equation under
continuum limits. Darboux transformation is employed to compute multi-soliton
solutions of fully discrete and semi-discrete SP equations. Explicit
expressions of first and second nontrivial soliton solutions are computed. We
also derived explicit expression of breather solution for fully discrete SP
equation. The dynamics of single loop soliton and interaction mechanism of
loop-loop and loop-antiloop solutions has been explored and illustrated.
