Time-Fractional KdV Equation for the plasma in auroral zone using Variational Methods

TL;DR

This paper derives and solves a time fractional KdV equation for plasma waves in the auroral zone, revealing how fractional derivatives influence solitary wave structures using variational methods.

Contribution

It introduces a novel application of variational methods to derive and analyze a time fractional KdV equation for auroral zone plasma.

Findings

Time fractional parameter affects solitary wave profiles.

Numerical solutions show modified wave structures.

Plasma parameters are close to auroral zone conditions.

Abstract

The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small but finite amplitude electrostatic waves. The Lagrangian of the time fractional KdV equation is used in similar form to the Lagrangian of the regular KdV equation. The variation of the functional of this Lagrangian leads to the Euler-Lagrange equation that leads to the time fractional KdV equation. The Riemann-Liouvulle definition of the fractional derivative is used to describe the time fractional operator in the fractional KdV equation. The variational-iteration method given by He is used to solve the derived time fractional KdV equation. The calculations of the solution with initial condition A0*sech(cx)^2 are carried out. Numerical studies have been made using plasma parameters close to those values corresponding to the dayside auroral zone. The effects of the time fractional parameter on the electrostatic solitary structures are presented.