Rogue waves on the periodic background in the extended mKdV equation

Abstract We construct new exact solutions for the extended mKdV (emKdV) equation, The exact solutions are obtained by nonlinearization of spectral problem associated with the travelling periodic waves and using the one-fold, two-fold Darboux transformation. We consider the dnoidal travelling periodic wave and cnoidal travelling periodic wave of the emKdV equation. Since the dnoidal travelling periodic wave is modulationally stable, the algebraic solutions propagating on dnoidal periodic background. However, since the cnoidal travelling periodic wave is modulationally unstable, the rogue wave generated on the cnoidal periodic background.