##
J. Chen and D.E. Pelinovsky

#
Rogue periodic waves of the modified KdV equation

###
Nonlinearity 31 (2018) 1955-1980

**Abstract:**

Rogue periodic waves stand for rogue waves on a periodic background. Two
families of travelling 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 two-fold Darboux transformations
of the travelling periodic waves, we construct new explicit solutions for the
mKdV equation. Since the dn-periodic wave is modulationally stable with
respect to long-wave perturbations, the new solution constructed from the
dn-periodic wave is 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
new solution constructed from the cn-periodic wave is a rogue wave on the
cn-periodic background, which generalizes the classical rogue wave (the so-called
Peregrine breather) of the nonlinear Schrodinger equation. We compute the
magnification factor for the rogue cn-periodic wave of the mKdV equation and
show that it remains constant for all amplitudes. As a by-product of our work, we
find explicit expressions for the periodic eigenfunctions of the 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,