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We address the most general periodic travelling wave of the modifiedKorteweg-de Vries (mKdV) equation written as a rational function of Jacobianelliptic functions. By applying an algebraic method which relates the periodictravelling waves and the squared periodic eigenfunctions of the Lax operators,we characterize explicitly the location of eigenvalues in the periodic spectralproblem away from the imaginary axis. We show that Darboux transformations withthe periodic eigenfunctions remain in the class of the same periodic travellingwaves of the mKdV equation. In a general setting, there are exactly threesymmetric pairs of eigenvalues away from the imaginary axis, and we give a newrepresentation of the second non-periodic solution to the Lax equations for thesame eigenvalues. We show that Darboux transformations with the non-periodicsolutions to the Lax equations produce rogue waves on the periodic background,which are either brought from infinity by propagating algebraic solitons orformed in a finite region of the time-space plane.

Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background