Nonlinear wave dynamics in Faraday instabilities

Abstract

Nonlinear wave dynamics in parametrically driven surface waves are studied in numerical simulations of the two-dimensional Navier-Stokes equation, with an emphasis on the evolution and interaction between different wave number modes. The dynamics are found to be closely correlated with the single-mode nonlinear saturated wave amplitudes. Modulating behavior of primary wave modes in a particular parameter range and in time scales much longer than the underlining wave periods is observed.

Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators

“…The form of Eq. (1.112) is also applicable to the onset of parametrically driven standing waves in continuum systems with weak nonlinear damping, and combines in a single equation a number of effects studied previously [64][65][66][67][68][69].…”

Nonlinear Dynamics of Nanomechanical Resonators

“…The reader is encouraged to consult [7] for a more detailed account of the derivation of the BCL equation. The form of (8.112) is also applicable to the onset of parametrically driven standing waves in continuum systems with weak nonlinear damping, and combines in a single equation a number of effects studied previously [13,14,23,29,46,53].…”

Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

“…[20][21][22]). For the DNLS solitons with VBC, a direct perturbation theory was recently developed [23], in which the eigenfunctions of the linearized equation around soliton solution were constructed with the squared Jost solutions obtained from the IST [13]. The linearization operator and the way to construct its eigenfunctions with the squared Jost solutions are the same for both VBC and NVBC.…”

“…The linearization operator and the way to construct its eigenfunctions with the squared Jost solutions are the same for both VBC and NVBC. Therefore, with results of IST for the DNLS equation with NVBC, in principle, the direct perturbation theory for the DNLS soliton with VBC [23] can be extended to that with NVBC. However, the IST and the soliton solution for the DNLS equation with NVBC in present literature [4,17] seem too complicated to be applied in developing a perturbation theory.…”