Abstract
In this paper, we explore the topic of rogue waves on periodic backgrounds. Using the travelling wave reduction method, we derive two families of periodic solutions for the coupled integrable dispersionless (CD) equation expressed in Jacobi elliptic dnoidal and conoidal functions, respectively. We use these Jacobi elliptic functions as seed solutions to construct algebraic decaying solitons on the dnoidal background of the CD equation based on the one-fold Darboux transformation (DT). Furthermore, we utilize the two-fold DT to find rogue wave solutions on the conoidal background of the CD equation. To obtain solutions of the short pulse equation, we use the hodograph transformation. We analyse the dynamics of the explicit solutions obtained from this research through plots.
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