Direct dynamical energy cascade in the modified KdV equation

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 (mKdV) 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 validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.