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Review
. 2018 Apr 13;376(2117):20170184.
doi: 10.1098/rsta.2017.0184.

Evans function computation for the stability of travelling waves

B Barker  1 , J Humpherys  2 , G Lyng  3 , J Lytle  1
Affiliations
Review

Evans function computation for the stability of travelling waves

B Barker et al. Philos Trans A Math Phys Eng Sci. .

Abstract

In recent years, the Evans function has become an important tool for the determination of stability of travelling waves. This function, a Wronskian of decaying solutions of the eigenvalue equation, is useful both analytically and computationally for the spectral analysis of the linearized operator about the wave. In particular, Evans-function computation allows one to locate any unstable eigenvalues of the linear operator (if they exist); this allows one to establish spectral stability of a given wave and identify bifurcation points (loss of stability) as model parameters vary. In this paper, we review computational aspects of the Evans function and apply it to multidimensional detonation waves.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Keywords: Evans function; stability; travelling waves.

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Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.
Figure 1.
The profile (a) and Evans function output (b) for the p-system as v+→0.
Figure 2.
Figure 2.
A left-moving travelling wave solution (strong detonation) of (5.1) in Eulerian coordinates, with activation energy EA=2.7, (2μ+η)=0.1, e−=0.0623, Γ=0.2 and q=0.623. This is the same profile computed in [18], although computed in Eulerian coordinates and oriented towards the left.
Figure 3.
Figure 3.
In (a), we show the Evans function output for EA=2.7 on a semicircular contour of radius R=0.1. The inset in (a) is magnified in (b), where it is easy to see that the winding number changes between EA=2.7 (solid) and EA=2.8 (dotted) as a Hopf bifurcation occurs. The other independent parameters are ξ=0, e−=0.0623, Γ=0.2 and q=0.623. (Online version in colour.)
Figure 4.
Figure 4.
Unstable eigenvalues as heat conductivity ν varies. Each line is parametrized by Fourier frequency ξ. Eigenvalues return to the left half-plane and restabilize as ν increases.
Figure 5.
Figure 5.
Evans function output for several Fourier frequencies ξ∈[0,0.8] on a semicircular contour with radius 0.4. (a) ν=1/10, (b) ν=2/5 and (c) ν=8/5. (Online version in colour.)
See this image and copyright information in PMC

References

    1. Meshkov EE. 1969. Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4, 101–104. (10.1007/BF01015969) - DOI
    1. Urtiew PA, Oppenheim AK. 1966. Experimental observations of the transition to detonation in an explosive gas. Proc. R. Soc. Lond. A 295, 13–28. (10.1098/rspa.1966.0223) - DOI
    1. Torruellas WE, Wang Z, Hagan DJ, VanStryland EW, Stegeman GI, Torner L, Menyuk CR. 1995. Observation of two-dimensional spatial solitary waves in a quadratic medium. Phys. Rev. Lett. 74, 5036–5039. (10.1103/PhysRevLett.74.5036) - DOI - PubMed
    1. Davis RE, Acrivos A. 1967. Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607. (10.1017/S0022112067001041) - DOI
    1. Rauprich O, Matsushita M, Weijer CJ, Siegert F, Esipov SE, Shapiro JA. 1996. Periodic phenomena in proteus mirabilis swarm colony development. J. Bacteriol. 178, 6525–6538. (10.1128/jb.178.22.6525-6538.1996) - DOI - PMC - PubMed

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