Localized wave solutions and rogue wave solutions for the complex modified KdV equation under the elliptic function background

Nonlinear wave phenomena frequently evolve on periodically fluctuating backgrounds, yet systematic theoretical frameworks for the complex modified Korteweg–de Vries (cmKdV) equation on elliptic backgrounds remain incomplete. To address this, this study systematically investigates localized and rogue wave solutions of the cmKdV equation on elliptic function backgrounds. The methodology combines the Theta function representation with the Darboux transformation within a genus-1 algebraic geometry framework. By utilizing linear combinations of Lax pair solutions and small-parameter expansions near the branch points of the spectral curve, the study extracts degenerate vector solutions. The results explicitly present five typical localized waves, including elliptic solitons and elliptic breathers, on both cn-type and dn-type backgrounds. Furthermore, general formulas for higher-order rogue waves are derived. The work reveals distinct dynamic behaviors, such as the coexistence of different families of rogue waves on cn-type backgrounds and nonlinear superpositions on modulationally stable dn-type backgrounds.