On the existence of traveling wave solutions for cold plasmas

TL;DR

This paper proves the existence of various traveling wave solutions in a plasma model using bifurcation theory and dynamical systems analysis, including periodic, monotone, and non-monotone waves.

Contribution

It introduces a rigorous bifurcation-based approach and a dynamical systems analysis to establish the existence of diverse traveling wave solutions in plasma models.

Findings

Existence of small amplitude periodic traveling waves.

Identification of different wave types, including monotone and non-monotone.

Numerical evidence supporting the theoretical results.

Abstract

The present paper is concerned with the existence of traveling wave solutions of the asymptotic model, derived by the authors in a previous work, to approximate the unidirectional evolution of a collision-free plasma in a magnetic field. First, using bifurcation theory, we can rigorously prove the existence of periodic traveling waves of small amplitude. Furthermore, our analysis also evidences the existence of different type of traveling waves. To this end, we present a second approach based on the analysis of the differential system satisfied by the traveling-wave profiles, the existence of equilibria, and the identification of associated homo-clinic and periodic orbits around them. The study makes use of linearization techniques and numerical computations to show the existence of different types of traveling-wave solutions, with monotone and non-monotone behaviour and different regularity, as well as periodic traveling waves.