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Stability analysis of fronts in a tristable reaction-diffusion system

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Abstract.

A stability analysis is performed analytically for the tristable reaction-diffusion equation, in which a quintic reaction term is approximated by a piecewise linear function. We obtain growth rate equations for two basic types of propagating fronts, monotonous and nonmonotonous ones. Their solutions show that the monotonous front is stable whereas the nonmonotonous one is unstable. It is found that there are two values of the growth rate for the most dangerous modes (corresponding to the longest possible wavelengths), \(\omega = 0\) and \(\omega < 0\), for the monotonous front, so that at \(\omega = 0\) the perturbation eigenfunction is positive whereas when \(\omega < 0\) it changes sign. It is also noted that the eigenvalue \(\omega = 0\) becomes negative in an inhomogeneous system with a particular (stabilizing) inhomogeneity. Counting arguments for the number of eigenmodes of the linear stability operator are presented.

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Authors and Affiliations

  1. Institut für Theoretische Physik, Otto-von-Guericke-Universität, Universitätsplatz 2, 39106, Magdeburg, Germany

    E. P. Zemskov & K. Kassner

Authors
  1. E. P. Zemskov
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  2. K. Kassner
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Correspondence to K. Kassner.

Additional information

Received: 9 August 2004, Published online: 23 December 2004

PACS:

05.45.-a Nonlinear dynamics and nonlinear dynamical systems - 47.20.Ma Interfacial instability - 47.54. + r Pattern selection; pattern formation

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Zemskov, E.P., Kassner, K. Stability analysis of fronts in a tristable reaction-diffusion system. Eur. Phys. J. B 42, 423–429 (2004). https://doi.org/10.1140/epjb/e2004-00399-x

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  • DOI: https://doi.org/10.1140/epjb/e2004-00399-x

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