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Open AccessArticle

Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations

by 1,† and 2,*,†
1
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
2
Institute of Analysis, Johannes Kepler University, 4020 Linz, Austria
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2020, 12(8), 1254; https://doi.org/10.3390/sym12081254
Received: 11 June 2020 / Revised: 21 July 2020 / Accepted: 25 July 2020 / Published: 29 July 2020
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
In this study we investigate for the first time the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards we focus on higher order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. The direct cascade structure remains invariant regardless of the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. The results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics. View Full-Text
Keywords: energy cascade; modulational instability; Fourier power spectrum; Korteweg–de Vries equations energy cascade; modulational instability; Fourier power spectrum; Korteweg–de Vries equations
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MDPI and ACS Style

Dutykh, D.; Tobisch, E. Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations. Symmetry 2020, 12, 1254. https://doi.org/10.3390/sym12081254

AMA Style

Dutykh D, Tobisch E. Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations. Symmetry. 2020; 12(8):1254. https://doi.org/10.3390/sym12081254

Chicago/Turabian Style

Dutykh, Denys; Tobisch, Elena. 2020. "Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations" Symmetry 12, no. 8: 1254. https://doi.org/10.3390/sym12081254

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