Abstract
In this study we investigate for the first time the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards we focus on higher order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. The direct cascade structure remains invariant regardless of the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. The results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics.
Highlights
When we want to describe an open or a closed physical system, the main question is to describe the energy flux inside this system and with surrounding environment
In the present study we investigated several Korteweg–de Vries (KdV)-type equations involving higher order nonlinearities
The main focus was on the formation and properties of the dynamical energy cascade in the Fourier space
Summary
When we want to describe an open or a closed physical system, the main question is to describe the energy flux inside this system and with surrounding environment. It should be noted that the BFI was introduced, strictly speaking, for surface waves in deep water which are described by the NLS-type equations This notion might be transposed to the mKdV case as well, since the analytical bridges between NLS and (m)KdV are well studied [21] (and the references therein). The formation of the dynamical energy cascade in the integrable mKdV Equation (2) was studied numerically in our previous papers [16,17] 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]. In all the experiments this norm was conserved up to the 8th digit with smaller oscillations around the mean value
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
