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J. Chen and D.E. Pelinovsky

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Rogue periodic waves of the modified Korteweg-de Vries equation

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**Abstract:**

Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation are considered
in the focusing case. By using one-fold and two-fold Darboux transformations, we construct
explicitly the rogue periodic waves of the mKdV equation expressed by the Jacobian elliptic
functions dn and cn respectively. The rogue dn-periodic wave describes propagation of an
algebraically decaying soliton over the dn-periodic wave, the latter wave is modulationally stable
with respect to long-wave perturbations. The rogue cn-periodic wave represents the outcome of
the modulation instability of the cn-periodic wave with respect to long-wave perturbations and
serves for the same purpose as the rogue wave of the nonlinear Schrodinger equation (NLS),
where it is expressed by the rational function. We compute the magnification factor for the cn-
periodic wave of the mKdV equation and show that it remains the same as in the small-amplitude
NLS limit for all amplitudes. 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.

**Keywords**:

modified Korteweg-de Vries equation, periodic travelling waves, rogue waves,