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

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## Abstract

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## 1. Introduction

- Big whirls have little whirls;
- Which feed on their velocity;
- And little whirls have lesser whirls;
- And so on to viscosity.

## 2. mKdV Equation With Broad Spectrum Initial Perturbation

#### 2.1. Direct cascade

#### 2.2. Double Cascade

## 3. Quartic and Quintic gKdV Equations

- Investigate the existence of the direct and inverse energy cascades
- If they exist:
- -
- To compare their formation time scales;
- -
- To compare their structure, i.e., the number of cascading modes, their location, spacing, etc.

#### 3.1. Quartic gKdV Equation

#### 3.1.1. Direct Cascade

#### 3.1.2. Double Cascade

#### 3.2. Quintic gKdV Equation

## 4. Discussion

- First of all, we extended the study of the mKdV equation presented in [16,17] for the case of a broader initial perturbation of the base wave. It was found that previous findings can be transposed to this case as well: formation of the direct and double cascades, their structure is preserved and appears to be quite stable. However, in the broad perturbation case all nonlinear processes are accelerated (i.e., they develop in shorter times).
- Not only the general structure of the energy cascade is preserved, but also the number of the cascading modes (for the fixed spectral domain and the base wave amplitude) and their location in the Fourier space.
- The main difference between the direct and double cascades consists in the fact that the change of the size of the spectral domain does not induce the energy redistribution in the direct cascade, while it changes completely the number of entities, and thus the energy distribution, in the double cascade(s).
- In all the mKdV cases, the formation of dynamical cascades is accompanied by the nonlinear stage of the MI development and the substantial amplification of the base wave in the physical space.
- In the case of the quartic gKdV equation, all the results about the dynamical cascades formation in the spectral domain remain the same. However, no amplification of the base wave amplitude was observed in the physical space.
- The quintic gKdV case is mostly similar to the mKdV equation (with positive nonlinearity) except for the presence of the amplitude threshold in the physical space, below which the amplification does not occur on the time horizons investigated in this study. However, if the amplitude is taken above this threshold, the base wave amplification occurs on longer time scales (virtually three times slower than in mKdV for the parameters considered in this study).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

List of acronyms | |

BFI | Benjamin–Feir Index |

CNRS | Centre National de la Recherche Scientifique |

CPU | Central Processor Unit |

FFT | Fast Fourier Transform |

FWF | Austrian Science Foundation |

gKdV | generalized Korteweg–de Vries equation |

ICE | Incremental Chain Equations |

LAMA | LAboratory of MAthematics |

KdV | Korteweg–de Vries equation |

MI | Modulational Instability |

mKdV | modified Korteweg–de Vries equation |

NLS | Nonlinear Schrödinger equation |

PDE | Partial Differential Equation |

WTT | Wave Turbulence Theory |

Nomenclature | |

a | base wave amplitude |

$\alpha ,\phantom{\rule{0.166667em}{0ex}}\beta ,\phantom{\rule{0.166667em}{0ex}}\gamma ,\phantom{\rule{0.166667em}{0ex}}\delta $ | coefficients in the gKdV equation |

${c}_{k}$ | Fourier coefficient corresponding to the wave number k |

$\delta $ | perturbation magnitude; also appears as a coefficient in the quartic KdV equation |

$\epsilon $ | dimensionless nonlinearity parameter |

k | wave number (dual variable to x) |

${k}_{0}$ | base wave number |

${K}_{0}$ | modulation wave number |

ℓ | characteristic size of the vortex |

$\mu $ | coefficient in the quintic KdV equation |

N | the number of Fourier modes used in our numerical simulations |

p | power of the nonlinearity in the KdV family (7) |

t | time variable |

T | time horizon or final simulation time |

$u(x,t)$ | solution to the corresponding (m,g)KdV equation (should be clear from the context) |

${u}_{t},\phantom{\rule{0.166667em}{0ex}}{u}_{x}$ | partial derivatives of the function u with respect to its independent variables t and x correspondingly |

x | horizontal space coordinate |

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**Figure 1.**Left panels (

**a**,

**c**,

**e**): the same initial condition subject to the different number of perturbation modes. Right panels (

**b**,

**d**,

**f**): evolution of the corresponding initial conditions up the same time instance $t\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}160.0$. Each panel contains upper and lower sub-panels. The upper sub-panel shows the snapshot of the solution $x\phantom{\rule{4pt}{0ex}}\u27fc\phantom{\rule{4pt}{0ex}}u\phantom{\rule{0.166667em}{0ex}}(x,\phantom{\rule{0.166667em}{0ex}}t)$ at some fixed instance of time t. The lower sub-panel shows its Fourier power spectrum. The same convention applies to all subsequent figures.

**Figure 2.**(

**a**) Development of the nonlinear stage of the MI from one unstable mode. (

**b**) Development of the nonlinear stage of the MI from seven unstable modes and with double initial amplitude $a\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}0.16$.

**Figure 3.**Formation of a double energy cascade in the spectral domain due to the perturbation with seven modes shown on panel (

**a**) $t=0.0$; (

**b**) $t=5.0$; (

**c**) $t=160.0$.

**Figure 4.**Direct cascade in the quartic KdV Equation (5) with seven perturbative modes for the base wave amplitude $a=0.08$.

**Figure 5.**Upper panel (

**a**): Direct cascade in the quartic KdV Equation (5) with seven perturbative modes for the base wave amplitude $a\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}0.16$ Lower panel (

**b**): Comparison of Fourier spectra obtained for the quartic KdV Equation (5) with seven perturbation modes at $t\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}3600.0$. Base wave amplitude is $a\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}0.08$ (black dots) and $a\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}0.16$ (red dots).

**Figure 6.**Double energy cascade in the quartic gKdV equation with base wave amplitude$a\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}0.08$ at $t\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}3600.0$. The spectral domain is increasing from up to down.

**Figure 9.**Direct cascade in the quartic KdV Equation (5) with seven perturbative modes for the base wave amplitude a = 10 × 0.08 = 0.8. Together with perturbation it produces the initial condition of the order of O (1).

**Table 1.**Various parameters used in simulations of the direct energy cascade subject to a broad excitation.

Base wave amplitude, a | $0.08$ |

Perturbation magnitude, $\delta $ | $0.05$ |

Base wavenumber, ${k}_{0}$ | $1.884$ |

Perturbation wavenumber, ${K}_{0}$ | $0.00785$ |

Ratio of wavelengths, ${k}_{0}/{K}_{0}$ | 240 |

Number of Fourier modes, N | 16 384 |

**Table 2.**Various parameters used in simulations of the double energy cascade subject to a broad excitation.

Base wave amplitude, a | $0.08$ |

Perturbation magnitude, $\delta $ | $0.05$ |

Base wavenumber, ${k}_{0}$ | $35\times 1.884$ |

Perturbation wavenumber, ${K}_{0}$ | $0.00785$ |

Ratio of wavelengths, ${k}_{0}/{K}_{0}$ | 240 |

Number of Fourier modes, N | $2\times 16$ 384 |

**Table 3.**The main conclusions of this study summarized in this table. The sign ‘−’ designates the absence of the corresponding property, while ‘+’ means its presence.

Property/Model | KdV (1) | mKdV (2) | Quartic gKdV (5) | Quintic gKdV (6) |
---|---|---|---|---|

Nonlinear Term | ${\mathit{uu}}_{\mathit{x}}$ | ${\mathit{u}}^{\mathbf{2}}{\mathit{u}}_{\mathit{x}}$ | ${\mathit{u}}^{\mathbf{3}}{\mathit{u}}_{\mathit{x}}$ | ${\mathit{u}}^{\mathbf{4}}{\mathit{u}}_{\mathit{x}}$ |

MI | − | + | − | + |

Direct cascade | − | + | + | + |

Inverse cascade | − | − | − | − |

Double cascade | − | + | + | + |

Energy redistrib. | − | + | + | + |

MI nonlin. stage | − | + | − | + |

(amplif. $>2a$) | ||||

Threshold | − | − | − | + |

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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, and Elena Tobisch.
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