Rogue periodic waves of the modified KdV equation

• Published in: Nonlinearity Vol. 31; no. 5; pp. 1955 - 1980
• Main Authors: Chen, Jinbing, Pelinovsky, Dmitry E
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.05.2018
• Subjects: fold Darboux transformation; modified Korteweg-de Vries equation; periodic waves; rogue waves
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/aaa2da

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's breather) of the nonlinear Schrödinger 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.