The Gardner equation and acoustic solitary waves in plasmas

TL;DR

This paper investigates ion-acoustic solitary waves in dusty plasmas with Cairns ion distribution and Boltzmann electrons, comparing numerical SPA and analytical RPT methods to derive and analyze soliton profiles.

Contribution

It introduces the use of the Gardner equation derived via RPT for analytical soliton solutions in dusty plasmas, comparing it with numerical SPA results.

Findings

Soliton profiles from SPA and RPT are consistent under certain parameters.

The Gardner equation captures both quadratic and cubic nonlinearities in plasma waves.

Analytical solutions provide insight into the structure of solitary waves in dusty plasmas.

Abstract

Ion-acoustic waves in a dusty plasma are investigated where it is assumed that the ions follow a Cairns distribution and the electrons are Boltzmann distributed. Two theoretical methods are applied: Sagdeev pseudopotential analysis (SPA) and reductive perturbation theory (RPT). Since SPA incorporates all nonlinearities of the model it is the most accurate but deriving soliton profiles requires numerical integration of Poisson's equation. By contrast, RPT is a perturbation method which at second order yields the Gardner equation incorporating both the quadratic nonlinearity of the KdV equation and the cubic nonlinearity of the modified KdV equation. For consistency with the perturbation scheme the coefficient of the quadratic term needs to be at least an order of magnitude smaller than the coefficient of the cubic term. Solving the Gardner equation yields an analytic expression of the soliton profile. Selecting an appropriate set of compositional parameters, the soliton solutions obtained from SPA and RPT are analyzed and compared.