Direct dynamical energy cascade in the modified KdV equation

D. Dutykh; E. Tobisch

doi:10.1016/j.physd.2015.01.002

2015

Direct dynamical energy cascade in the modified KdV equation

PPhys. D

D. Dutykh and E. Tobisch

Phys. D 297: 76-87 (2015)

Open Access

HAL Version DOI

Abstract

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.

Keywords

NLS equationKorteweg-de Vries (KdV) equationModulational Instabilityenergy cascademodified KdV equation

Bibliographic record

Journal: Phys. D

Volume: 297

Pages: 76-87

DOI: 10.1016/j.physd.2015.01.002

HAL: hal-00990724

BibTeX Citation

@article{Dutykh2015directdynamical,
  author = {Dutykh, D. and Tobisch, E.},
  title = {Direct dynamical energy cascade in the modified KdV equation},
  journal = {Phys. D},
  year = {2015},
  volume = {297},
  pages = {76--87},
  doi = {10.1016/j.physd.2015.01.002},
  abstract = {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.},
  keywords = {NLS equation, Korteweg-de Vries (KdV) equation, Modulational Instability, energy cascade, modified KdV equation}
}