Perturbation Method for a Nonlinear Wave Modulation. I

Abstract

In this paper we consider a system of nonlinear wave equations which admits, in a linear approximation, a planewave solution with high-frequency oscillation. Then, for the wave of small but finite amplitude, we investigate how slowly varying parts of the wave such as the amplitude are modulated by nonlinear self-interactions. A stretching transformation shows that, in the lowest order of an asymptotic expansion, the original system of equations can be reduced to a tractable, single, nonlinear equation to determine the amplitude modulation.

Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser

“…The small correction of order ε in equations (22) or (25) gives account for the variations of f on 'propagation distances', which are here number of round trips, very large, of order 1/ε. This can be shown in a rigorous way using the multiscale formalism, commonly used for the derivation of the model equations in the soliton theory [15]. We introduce a slow variable ζ = εz, in such a way that…”

Acoustic Nonlinearity in Dispersive Solids

“…However, it is in general necessary to account for deviations from such "adiabatic" assumptions by adding to Eq. (8) a term, analogous to that obtained by Taniuti and Yajima [10] and by Karpman [11], that properly includes the effects oflarge amplitude and wave number gradients. Adding this term to Eq.…”

“…An appropriate expression is the energy conservation equation [9] d(A2) ~(A2) = 0 at + da Cg (9) where Cg is the group velocity in the linear approximation. We may write the group velocity in tenns of (o<l>/oa) as (10) where we have used Eqs. (6) and (7).…”

Amplitude modulation of three-dimensional low-frequency solitary waves in a magnetized dusty superthermal plasma

“…In order to derive the evolution equation for weakly nonlinear DIA wave envelopes, we employ the standard multiple-scale technique [32,33] in which the coordinates are stretched as ðn; g; f; sÞe ilðkzÀxtÞ , where x and k, respectively, represent the carrier wave frequency and the wavenumber. In order that n, U, /; etc.…”