NLS approximation and dark solitons for the Adlam-Allen model of cold collisionless plasmas

TL;DR

This paper derives a defocusing nonlinear Schr{"o}dinger equation from the Adlam-Allen plasma model to construct and analyze long-lived dark soliton solutions, expanding understanding of plasma wave dynamics.

Contribution

It introduces a novel multiple-scale approach to derive dark solitons in the Adlam-Allen model, which had not been achieved before.

Findings

Dark soliton waveforms are physically realistic and long-lived.

Numerical simulations confirm the stability of the dark solitons.

The approach extends previous KdV-based analyses to a NLS framework.

Abstract

The present work extends earlier considerations on a quintessential model of cold, collisionless plasmas, namely the Adlam-Allen model. Previously, an analysis of homoclinic solutions around a non-vanishing background (associated with a saddle equilibrium) led to a Korteweg-de Vries reduction. Here, we consider a different equilibrium of the co-traveling frame model, namely a center, and expanding around it, by means of a multiple-scale methodology and suitable scalings, leads to an effective defocusing nonlinear Schr{\"o}dinger equation. Leveraging the latter, we construct, for the first time to our knowledge, physically realistic dark soliton waveforms of the Adlam-Allen model. We subsequently test the numerical evolution of such coherent structures, identifying them as long-lived waveforms of the full model.