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

We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there are exactly three symmetric pairs of eigenvalues away from the imaginary axis, and we give a new representation of the second non-periodic solution to the Lax equations for the same eigenvalues. We show that Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background, which are either brought from infinity by propagating algebraic solitons or formed in a finite region of the time-space plane.