Rogue periodic waves of the modified KdV equation

Abstract

Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation\nare considered in the focusing case. By using one-fold and two-fold Darboux\ntransformations, we construct explicitly the rogue periodic waves of the mKdV\nequation expressed by the Jacobian elliptic functions dn and cn respectively.\nThe rogue dn-periodic wave describes propagation of an algebraically decaying\nsoliton over the dn-periodic wave, the latter wave is modulationally stable\nwith respect to long-wave perturbations. The rogue cn-periodic wave represents\nthe outcome of the modulation instability of the cn-periodic wave with respect\nto long-wave perturbations and serves for the same purpose as the rogue wave of\nthe nonlinear Schrodinger equation (NLS), where it is expressed by the rational\nfunction. We compute the magnification factor for the cn-periodic wave of the\nmKdV equation and show that it remains the same as in the small-amplitude NLS\nlimit for all amplitudes. As a by-product of our work, we find explicit\nexpressions for the periodic eigenfunctions of the AKNS spectral problem\nassociated with the dn- and cn-periodic waves of the mKdV equation.\n