Rogue periodic waves of the mKdV equation

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.

24 pages, 3 figures

References in corpus (4)

• Baseband Modulation Instability as the Origin of Rogue Waves
• The role of multiple soliton and breather interactions in generation of rogue waves: the mKdV framework
• The Stability Spectrum for Elliptic Solutions to the Focusing NLS Equation
• Rogue waves in multiphase solutions of the focusing NLS equation

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• Localized structures on librational and rotational travelling waves in the sine-Gordon equation
• Formation of rogue waves on the periodic background in a fifth-order nonlinear Schrödinger equation
• The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation
• Rogue waves on an elliptic function background in complex modified Korteweg-de Vries equation
• Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schrödinger equation