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Transition to Longitudinal Instability of Detonation Waves is Generically Associated with Hopf Bifurcation to Time-Periodic Galloping Solutions

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Abstract

We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier–Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic “galloping”, or “pulsating”, solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier–Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works.

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Authors and Affiliations

  1. Université Paris Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR CNRS 7586, 75205, Paris Cedex 13, France

    Benjamin Texier

  2. Indiana University, Bloomington, IN, 47405, USA

    Kevin Zumbrun

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  1. Benjamin Texier
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  2. Kevin Zumbrun
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Correspondence to Kevin Zumbrun.

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Communicated by P. Constantin

Research of B.T. was partially supported under NSF grant number DMS-0505780.

Research of K.Z. was partially supported under NSF grants no. DMS-0300487 and DMS-0801745.

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Texier, B., Zumbrun, K. Transition to Longitudinal Instability of Detonation Waves is Generically Associated with Hopf Bifurcation to Time-Periodic Galloping Solutions. Commun. Math. Phys. 302, 1–51 (2011). https://doi.org/10.1007/s00220-010-1175-8

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