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 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.
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Denys Dutykh, Elena Tobisch. Direct dynamical energy cascade in the modified KdV equation. Physica D: Nonlinear Phenomena, Elsevier, 2015, 297, pp.76-87. ⟨http://www.sciencedirect.com/science/article/pii/S0167278915000032⟩. ⟨10.1016/j.physd.2015.01.002⟩. ⟨hal-00990724v3⟩

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