Continuous limit and location-manageable discrete loop rogue wave solutions for the semi-discrete complex short pulse equation

This paper focuses on the semi-discrete complex short pulse equation which may describe the ultra-short pulse propagation in optical fiber. First, we study the continuous limit of the semi-discrete complex short pulse equation and establish a certain connection with a continuous complex short pulse equation. Then the discrete generalized (p,N−p)-fold Darboux transformation is constructed to generate three kinds of location-manageable discrete localized wave solutions including loop rogue wave, loop periodic wave, and their mixed collision loop structures. With the help of the hodograph transformation, we find a class of implicit solutions of semi-discrete complex short pulse equation, and some specific parameters are added to the analytical expressions so that we can theoretically manage these loop structures where we want them to appear. These unique structures may help researchers better grasp the characteristics of location-manageable loop localized wave solutions and explain many physical phenomena in nonlinear optics. •The continuous limit of the semi-discrete system is discussed and investigated.•The discrete generalized (p, N −p)-fold Darboux transformation is first constructed.•Multivalued rogue wave, periodic wave and mixed collision solutions are obtained.•These loop discrete localized wave structures are shown and discussed graphically.