Modulational Instability and Position Controllable Discrete Rogue Waves with Interaction Phenomena in the Semi-Discrete Complex Coupled Dispersionless System

In this paper, the position controllable discrete analytical higher-order rogue wave and periodic wave solutions in the semi-discrete complex coupled dispersionless system, as well as their mixed interaction phenomena, are theoretically provided in this study. Our first priority is to study the modulation instability for this system and deduce the formation mechanism from its plane wave solutions. And then, we use the generalized (n, N−n)-fold Darboux transformation to create three types of position controlled localized wave solutions, including rogue waves, periodic waves and their mixed interaction solutions. In particular, we found three different kinds of rogue wave structures in the same discrete system, including dark-bright structure, bright-bright structure, and the structure with two peaks and two depressions, which is a very rare phenomenon.And these innovative localized wave structures are illustrated graphically. Besides, the principal characteristic and placements of these novel solutions may be controlled by a set of unique parameters, it is shown that we can control the location of different localized waves by changing these parameters so that we can theoretically control them to appear where we need them to occur. It is hoped that the results in the present work can play a potential role in describing the interaction between the current-fed string and the external magnetic field and other physical phenomena