Direct dynamical energy cascade in the modified KdV equation

TL;DR

This paper investigates the energy transfer during the nonlinear stage of modulational instability in the modified KdV equation, using a Fourier space approach and validating results through numerical simulations.

Contribution

It adapts a dynamical energy cascade model from NLS equations to the mKdV setting, providing new insights into energy transfer mechanisms.

Findings

Good agreement between theoretical predictions and numerical simulations

Detailed analysis of nonlinear stages of modulational instability in mKdV

Extension of D-cascade model to KdV-type equations

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.