Direct dynamical energy cascade in the modified KdV equation

Denys Dutykh (1, 2, 3, 4) , Elena Tobisch (5)

1 EDPs2
2 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 LAMA - Laboratoire de Mathématiques
5 Institut für Analysis

In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg-de Vries equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D-cascade model are outlined and thoroughly discussed. Finally, the obtained theoretical results are validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. A general good agreement is reported in this study. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.

Mots clés: NLS equation; Korteweg-de Vries (KdV) equation; Modulational Instability; energy cascade; modified KdV equation

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Milieux et Changements globaux; Physique Atmosphérique et Océanique [physics.ao-ph]; Dynamique Chaotique [nlin.CD]; Formation de Structures et Solitons [nlin.PS]; Mathématiques générales [math.GM]; Physique Générale [physics.gen-ph]; Systèmes Solubles et Intégrables [nlin.SI]; Dynamique des Fluides [physics.flu-dyn]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Physique Numérique [physics.comp-ph]; Géophysique [physics.geo-ph]; Géophysique [physics.geo-ph]

Denys Dutykh, Elena Tobisch. Direct dynamical energy cascade in the modified KdV equation. Physica D: Nonlinear Phenomena, 2015, 297, pp.76-87. ⟨10.1016/j.physd.2015.01.002⟩. ⟨hal-00990724v3⟩