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.

21 pages, 8 figures, 3 tables, 37 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/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

Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations u t + s u p u x + u x x x (here s = ± 1 and p > 0 is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper.

One physical mechanism that is responsible for the focusing of uni-directional water waves is the modulation instability (MI). This occurs when side-bands around the main frequency are excited either deterministically or randomly and subsequently grow exponentially. In physical space the periodically-perturbed wave group can reach significant wave amplifications and in the case of infinite modulation period even three times the initial amplitude of the regular Stokes wave train. These periodic wave groups propagate in deep-water with a group velocity half the wave's phase speed. In this experimental study, we investigate the dynamics of modulationally unstable wave groups that propagate with a velocity that is higher than the packet's group velocity in deep-water. It is shown that when this additional velocity to the wave group is marginal, a very good agreement with nonlinear Schrödinger (NLS) hydrodynamics is reached at all stage of propagation and the characteristic wave energy cascade remains symmetric and stationary. Otherwise, a significant deviation is observed only at the stage of large wave focusing of the isolated wave packet. In this case, the wave field experiences an almost perfect return to the initial conditions after the compression, a particular dynamics also known as the Fermi-Pasta-Ulam paradox.

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