Direct dynamical energy cascade in the modified KdV equation

Abstract

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

Reduction to modified KdV and its KP-like generalization via phase modulation

“…For example the focusing mKdV equation has periodic travelling wave solutions which are Benjamin-Feir unstable [8]. It is shown in [4,5] that this Benjamin-Feir instability can be the catalyst for an energy cascade leading to a continuous spectrum and a highly-complex wave field. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 When B kk is not zero but small, for example B kk = εµ, the mKP-KD + has an unfolding into the 2 + 1 Gardner system which includes both cubic and quadratic nonlinearities.…”

Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations

“…In general, the MI is an effect which describes a special type of the instability of a single spectral component subject to narrow band excitation in an experiment (regardless whether it is numerical or laboratory). In our previous investigations [14,15] we followed this ideology and all the numerical experiments toward the observation of the direct and inverse cascades had only two spectral harmonics in the initial condition (i.e. the base wave k 0 along with its modulation K 0 k 0 ).…”

“…Consequently, the priveleged method of studying the non-integrable and even integrable, but not solvable equations is the numerical simulation. The formation of the dynamical energy cascade in the integrable mKdV equation (2) was studied numerically in our previous papers [14,15]. However, it was done for narrow initial perturbations in the FOURIER space, since originally it was discovered for the NLS equation which is valid only in this narrow band approximation [4].…”

Constructive Study of Modulational Instability in Higher Order Korteweg-de Vries Equations

“…In the case of the modified KdV with s = +1 a wave train is modulationally unstable and this leads to the generation of so-called rogue waves [13][14][15][16][17]. Dynamical energy cascades (triggered by the modulational instability) were introduced and studied analytically in the frame of the modified NLS equations [18][19][20][21], while the cascade formation governed by the focusing modified KdV equations was studied numerically [22,23].…”

“…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 (m)NLS equations obtained, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. These results can be used for planning numerical and laboratory experiments similar to [15,16], and for explaining the available data, e.g., [22,23]. An important issue here would be to choose small parameters and initial amplitudes facilitating these results which is not a simple problem.…”