Observation of the Inverse Energy Cascade in the modified Korteweg-de Vries Equation

TL;DR

This paper demonstrates the first numerical observation of the inverse energy cascade in the focusing modified Korteweg-de Vries equation, revealing stable quasi-stationary structures linked to modulational instability.

Contribution

It provides the first numerical evidence of inverse energy cascade formation in the focusing mKdV equation and analyzes its properties and stability.

Findings

Inverse cascade always accompanies direct cascade.

Stable quasi-stationary Fourier space structures form.

Cascade development relates to modulational instability.

Abstract

In this Letter we demonstrate for the first time the formation of the inverse energy cascade in the focusing modified Korteweg-de Vries (mKdV) equation. We study numerically the properties of this cascade such as the dependence of the spectrum shape on the initial excitation parameter (amplitude), perturbation magnitude and the size of the spectral domain. Most importantly we found that the inverse cascade is always accompanied by the direct one and they both form a very stable quasi-stationary structure in the Fourier space in the spirit of the FPU-like reoccurrence phenomenon. The formation of this structure is intrinsically related to the development of the nonlinear stage of the Modulational Instability (MI). These results can be used in several fields such as the internal gravity water waves, ion-acoustic waves in plasmas and others.