Observation of the Inverse Energy Cascade in the modified Korteweg-de Vries Equation

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

1 LAMA - Laboratoire de Mathématiques
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 Institut für Analysis

In this Lettre we demonstrate for the first time the formation of the inverse energy cascade in the focusing modified Korteweg-de Vries (mKdV) equation. We study numerically the properties of this cascade such as the dependence of the spectrum shape on the initial excitation parameter (amplitude), perturbation magnitude and the size of the spectral domain. Most importantly we found that the inverse cascade is always accompanied by the direct one and they both form a very stable quasi-stationary structure in the Fourier space in the spirit of the FPU-like reoccurrence phenomenon. The formation of this structure is intrinsically related to the development of the nonlinear stage of the Modulational Instability (MI). These results can be used in several fields such as the internal gravity water waves, ion-acoustic waves in plasmas and others.

Mots clés: modified KdV equation; inverse cascade; Modulational Instability; energy cascade

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

Denys Dutykh, Elena Tobisch. Observation of the Inverse Energy Cascade in the modified Korteweg-de Vries Equation. EPL - Europhysics Letters, 2014, 107, pp.14001. ⟨10.1209/0295-5075/107/14001⟩. ⟨hal-00991944v2⟩