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

Abstract

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.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2014observationinverse,
  author = {Dutykh, D. and Tobisch, E.},
  title = {Observation of the Inverse Energy Cascade in the modified Korteweg—de Vries Equation},
  journal = {Eur. Phys. Lett.},
  year = {2014},
  volume = {107},
  number = {14001},
  pages = {14001},
  doi = {10.1209/0295-5075/107/14001},
  abstract = {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.},
  keywords = {Fourier analysis, energy cascade, water waves, numerical simulation, modified Korteweg-de Vries equation, plasma waves, nonlinear dynamics, Modulational Instability, modified KdV equation, inverse energy cascade, inverse cascade, modulational instability}
}