The theory of nonlinear ion-acoustic waves revisited

Abstract

The basic set of equations describing nonlinear ion-acoustic waves in a cold collisionless plasma, in the limit of long wavelengths, is reconsidered. First, a travelling-wave solution is found up to third order by means of a straightforward perturbation approach based on the smallness of the wavenumber. As a result, a positive dressed solitary wave shows up, which is larger, taller and faster than the KdV soliton, the first-order result. Furthermore, the accuracy of this approach is tested and compared with previous result. Secondly, the reductive perturbation techique to study higher-order corrections is revised and adapted to the present problem.

Multiple - scale Expansion for Nonlinear Ion-acoustic Waves

“…It is shown that the wave speed corresponding to the lowest order time parameter r gives the wave speed of the classical perturbation expansion, whereas the wave speeds corresponding to higher order time parameters give the speed corrections terms. The results obtained here are exactly the same as those given in Malfliet and Wieers [6] and Demiray [7],…”

“…This requirement made it possible to remove the secularities that might occur and to determine the higher order correction terms to the wave speed. The result obtained in [7] was found to be in complete agreement with that of Malfliet and Wieers [6].…”

“…A particular solution to equations (33) and (34) takes the form: This result is exactly the same as those obtained by Malfliet and Wieers [6] and Demiray [7], The present derivation can be reduced to the modified reductive perturbation method presented in [7] by introducing the …”

“…This causes some ambiguity over the correction terms. In order to remove this uncertainty, Malfliet and Wieers [6] presented a dressed solitary wave approach, based on the assumption that the field variables admit a localized travelling wave solution. Then, for the long-wave limit, they expanded the field quantities into a power series of the wave number, which is assumed to be the only smallness variable, and obtained the explicit solution for various order terms in the expansion.…”

Weakly Nonlinear Waves in Elastic Tubes Filled with a Layered Fluid

“…For this type of problems it is a common practice to use the hyperbolic tangent method [16]. For this we introduce a new variable η as η = tanh ζ and propose a solution of the form…”