The reductive perturbation method and some of its applications

Abstract

The reductive perturbation method is a very powerful way of deriving simplified models describing nonlinear wave propagation and interaction. In abstract frames chosen for the sake of clarity, we describe the fundamentals of the method: envelope equations, long-wave approximation, three-wave resonant interaction. We give an insight into the mathematical properties of the perturbative schemes. Then some applications are given, which either illustrate the typical situation or introduce additional features of perturbative expansions, and have their own physical interest. The applications concern either nonlinear optics, especially ultrafast, or wave propagation in ferromagnetic media, in the so-called electromagnetic or polariton range.

Nonlinear waves in second order conformal hydrodynamics

“…(2), (3), and (4) from (x, t) to the (X, Y ) space defined by the "stretched coordinates" [22][23][24][25] …”

“…In [21] the authors went beyond linearization and considered the effects of shear viscosity on the propagation of nonlinear waves. This study was performed with the help of the well established reductive perturbation method [22][23][24][25].…”

Solitary waves propagation described by Korteweg-de Vries equation in the granular chain with initial prestress

“…Yang et al AIP Advances 6, 075317 (2016) In order to study the small amplitude nonlinear waves, we use the traditional reductive perturbation technique [55][56][57] and the stretched coordinates: ξ = ϵ[li − λt], l = ±1, τ = ϵ 3 i, where ϵ is a small parameter and λ is the velocity of the solitary wave. The variable s is expanded as:…”

Few-cycle solitons in supercontinuum generation dynamics

“…We first consider the UV transition only, with the assumption that (1/τ p ) ≪ ω 2 . This assumption corresponds to a long-wave approximation, which is performed according to the standard procedure of the reductive perturbation method, or multiscale expansion method [12]. It results in an asymptotic model, which is the modified Korteweg-de Vries (mKdV) equation, as [8] ∂E ∂ζ = 1 6…”