Rogue periodic waves of the mKdV equation

Rogue periodic waves stand for rogue waves on the periodic background. Two families of traveling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and twofold Darboux transformations, we construct explicitly the rogue periodic waves of the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the “rogue” dn-periodic solution is not a proper rogue wave on the periodic background but rather a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the rogue cn-periodic wave is a proper rogue wave on the periodic background, which generalizes the classical rogue wave (the so-called Peregrine’s breather) of the nonlinear Schrödinger (NLS) equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remains for all amplitudes the same as in the small-amplitude NLS limit. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the AKNS spectral problem associated with the dn and cn periodic waves of the mKdV equation.