Laplacian Instability of Planar Streamer Ionization Fronts—An Example of Pulled Front Analysis

  • Gianne DerksEmail author
  • Ute Ebert
  • Bernard Meulenbroek
Research Article


Streamer ionization fronts are pulled fronts that propagate into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific long-time attractor out of a continuous family. A stability analysis for perturbations in the transverse direction has to take these features into account. In this paper we show how to apply the Evans function in a weighted space for this stability analysis. Zeros of the Evans function indicate the intersection of the stable and unstable manifolds; they are used to determine the eigenvalues. Within this Evans function framework, we define a numerical dynamical systems method for the calculation of the dispersion relation as an eigenvalue problem. We also derive dispersion curves for different values of the electron diffusion constant and of the electric field ahead of the front. Numerical solutions of the initial value problem confirm the eigenvalue calculations. The numerical work is complemented with an analysis of the Evans function leading to analytical expressions for the dispersion relation in the limit of small and large wave numbers. The paper concludes with a fit formula for intermediate wave numbers. This empirical fit supports the conjecture that the smallest unstable wave length of the Laplacian instability is proportional to the diffusion length that characterizes the leading edge of the pulled ionization front.


Pulled front Stability analysis Streamer ionization front Dispersion relation Wave selection of Laplacian instability 

Mathematics Subject Classification (2000)

37L15 34L16 35Q99 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Gianne Derks
    • 1
    Email author
  • Ute Ebert
    • 2
    • 3
  • Bernard Meulenbroek
    • 2
    • 4
  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK
  2. 2.Center for Mathematics and Computer Science (CWI)AmsterdamThe Netherlands
  3. 3.Department of Applied PhysicsEindhoven University of TechnologyEindhovenThe Netherlands
  4. 4.Faculty of Electrical Engineering, Mathematics and Computer ScienceDelft Univ. Techn.DelftThe Netherlands

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