@article{b267856324e34cdaae7b65e7db88e589,
title = "Formation of the dynamic energy cascades in quartic and quintic generalized KdV equations",
abstract = "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.",
keywords = "Energy cascade, Fourier power spectrum, Korteweg-de vries equations, Modulational instability",
author = "Denys Dutykh and Elena Tobisch",
note = "Funding Information: Funding: This work has been supported by the Austrian Science Foundation (FWF) under projects P30887 and P31163. The work of DD has been supported by the French National Research Agency, through Investments for Future Program (ref. ANR−18−EURE−0016 — Solar Academy). Funding Information: Acknowledgments: The first Author would like to acknowledge the hospitality of the Johannes Kepler University in Linz during the work on this paper. Both Authors would like to thank late Professor Walter Craig (McMaster University, Canada), and Professors Roger Grimshaw (University College London, United Kingdom) and Efim Pelinovsky (Institute of Applied Physics, Russia) for helpful discussions on the topics of the present study. We would also like to thank our Referees for their remarks which allowed to improve the presentation in our manuscript. Open Access Funding by the Austrian Science Fund (FWF). Funding Information: This work has been supported by the Austrian Science Foundation (FWF) under projects P30887 and P31163. The work of DD has been supported by the French National Research Agency, through Investments for Future Program (ref. ANR-18-EURE-0016-Solar Academy). The first Author would like to acknowledge the hospitality of the Johannes Kepler University in Linz during the work on this paper. Both Authors would like to thank late Professor Walter Craig (McMaster University, Canada), and Professors Roger Grimshaw (University College London, United Kingdom) and Efim Pelinovsky (Institute of Applied Physics, Russia) for helpful discussions on the topics of the present study. We would also like to thank our Referees for their remarks which allowed to improve the presentation in our manuscript. Open Access Funding by the Austrian Science Fund (FWF). Publisher Copyright: {\textcopyright} 2020 by the authors.",
year = "2020",
month = aug,
doi = "10.3390/SYM12081254",
language = "British English",
volume = "12",
journal = "Symmetry",
issn = "2073-8994",
publisher = "Multidisciplinary Digital Publishing Institute (MDPI)",
number = "8",
}