# Rogue periodic waves of the mKdV equation

## TL;DR

This paper constructs explicit rogue periodic wave solutions of the focusing mKdV equation using Darboux transformations, analyzes their stability, and compares their magnification factors to those of the NLS equation.

## Contribution

It provides explicit formulas for rogue periodic waves of the mKdV equation and analyzes their stability and magnification factors, extending understanding of rogue wave phenomena.

## Key findings

- Rogue dn-periodic wave describes an algebraically decaying soliton over a stable periodic wave.
- Rogue cn-periodic wave results from modulation instability and mimics NLS rogue waves.
- Magnification factor for the cn-periodic wave remains consistent with the small-amplitude NLS limit.

## 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.