Rogue waves of the Kraenkel–Manna–Merle system on a periodic background

In this paper, we summarize the construction of rogue wave solutions for the Kraenkel–Manna–Merle system on the background of Jacobian elliptic dn- and cn-periodic waves. Our approach involved nonlinearizing the Lax pair to derive eigenvalues and eigenfunctions, introducing periodic and non-periodic solutions of the Lax pair, and utilizing the Darboux transformation to establish potential relations. Consequently, we obtain periodic rogue wave solutions and conducted a nonlinear dynamics analysis, revealing significant insights into the behavior of the Kraenkel–Manna–Merle system. A rogue wave on a skewed periodic wave background is obtained which is a novel phenomenon in the nonlinear system.