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On the existence of traveling wave solutions for cold plasmas

Diego Alonso-Orán, Angel Durán, Rafael Granero-Belinchón

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.

On the existence of traveling wave solutions for cold plasmas

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.

Paper Structure

This paper contains 14 sections, 7 theorems, 70 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and auxiliary results
  3. Existence of periodic traveling wave solutions
  4. Formulation and functional spaces
  5. Spectral and tranversality properties of the linearized operator
  6. Proof of Theorem \ref{['main:theorem']}
  7. Traveling wave solutions of the nonlocal wave equation
  8. The case $g\leq g_{1}(c_{s})$
  9. Some previous lemmas
  10. Existence of non-periodic traveling wave solutions
  11. Dynamics around the equilibrium $y_{+}$
  12. Dynamics around the equilibrium $y_{-}$
  13. Concluding remarks
  14. A procedure for the numerical generation of traveling waves

Key Result

Lemma 2.1

The operator $\mathscr{Q}$ maps $C^{k,\alpha}(\mathbb{T})$ isomorphically onto $C^{k+2,\alpha}(\mathbb{T})$. More precisely, there exists a constant $C>0$ such that

Figures (5)

  • Figure 1: Linearization at the origin of (\ref{['adg19']}) (cf. Champ): Regions $1$ to $4$ in the $(b,a)$-plane, delimited by the bifurcation curves $\mathbb{C}_{0}$ to $\mathbb{C}_{3}$ given by (\ref{['bifurcurv']}), and schematic representation of the position in the complex plane of the eigenvalues of $L$ for each curve and region. (Dot: simple root, larger dot: double root.)
  • Figure 2: Illustration of Lemma \ref{['adg_lemma2']}. (a), (b) Case (1); (c) Case (2), where the remarked dot corresponds to $c_{s}= 3(1+\sqrt{2})$; (d) Case (c), where the remarked dot corresponds to $c_{s}= 3$
  • Figure 11: Approximations to traveling-wave solutions of (\ref{['adg9']}) for two values of $c_{s}$. Case $g=g_{+}(c_{s})-0.1$. (a) $\widetilde{h}+y_{-}$ profile; (b) $\widetilde{u}+y_{-}$ profile.
  • Figure 12: Approximations to traveling-wave solutions of (\ref{['adg9']}) for two values of $c_{s}$. Case $g=0$. (a) $\widetilde{h}+y_{-}$ profile; (b) $\widetilde{u}+y_{-}$ profile.
  • Figure 13: Approximations to traveling-wave solutions of (\ref{['adg9']}) for two values of $c_{s}$. Case $g=g_{1}(c_{s})-0.001$. (a) $\widetilde{h}+y_{-}$ profile; (b) $\widetilde{u}+y_{-}$ profile.

Theorems & Definitions (8)

  • Lemma 2.1
  • Definition 1: Fredholm operator
  • Theorem 1: Crandall-Rabinowitz Theorem
  • Theorem 2
  • Proposition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4