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Journal of Engineering Mathematics
Erschienen in: Journal of Engineering Mathematics 1/2017

07.11.2015

Pushed and pulled fronts in a discrete reaction–diffusion equation

verfasst von: J. R. King, R. D. O’Dea

Erschienen in: Journal of Engineering Mathematics | Ausgabe 1/2017

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Abstract

We consider the propagation of wave fronts connecting unstable and stable uniform solutions to a discrete reaction–diffusion equation on a one-dimensional integer lattice. The dependence of the wavespeed on the coupling strength \(\mu \) between lattice points and on a detuning parameter (a) appearing in a nonlinear forcing is investigated thoroughly. Via asymptotic and numerical studies, the speed both of ‘pulled’ fronts (whereby the wavespeed can be characterised by the linear behaviour at the leading edge of the wave) and of ‘pushed’ fronts (for which the nonlinear dynamics of the entire front determine the wavespeed) is investigated in detail. The asymptotic and numerical techniques employed complement each other in highlighting the transition between pushed and pulled fronts under variations of \(\mu \) and a.
Vorheriger Artikel Gas-cushioned droplet impacts with a thin layer of porous media
Nächster Artikel Initial stage of plate lifting from a water surface

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Fußnoten
1
An early description of the distinct classes of wave in reaction–diffusion equations of the form (7) was given by Hadeler and Rothe [20]; detailed analysis of the different front types in such a PDE, as well as the pushed/pulled terminology, is due to Stokes [21].
 
2
Due to the rescaling of j, the constant c in the remainder of this section corresponds to dividing that elsewhere in the paper by \(\sqrt{\mu }\).
 
3
We stress that, because \(x=j/\sqrt{\mu }\), the quantities z, S, c and \(\lambda \) henceforth are scaled differently from those in the previous section.
 
4
Provided—as in the case of the cubic nonlinearity on which we mainly focus—only one such transition occurs; a characterisation of the nonlinearities f(u; a) in such regards would be valuable.
 
5
We suppress the dependence of U on \(\mu \) in our far-field expressions.
 
6
Note that \(c^\dag (a_T,\mu )=c^*(\mu )\).
 
7
The far-field expansion (39) will also in general contain a contribution of the form (40) with \(A_+^\dag \) replaced by \(A_+(a,c)\), with \(A_+^\dag (a)=A_+(a,c^\dag )\).
 
8
Significantly, the coefficient of the pre-exponential term linear in z is positive in (41) and negative in (42).
 
9
Here we assume \(a={O}(1)\); the behaviour differs significantly for \(a={O}(\varepsilon )\)—see Sect. 5.3 below.
 
10
Setting the coefficient of the exponential to unity in the initial condition requires appropriate choice of the \({O}\left( 1/\ln (1/\varepsilon ) \right) \) contribution to \(T_j(\varepsilon )\).
 
11
The ‘\(-1\)’ in (67), which arises from the \(U(z-1)\) term in (26), is the only leading-order contribution of the difference operator in (26) to enter either of these regions.
 
12
That the other terms from the central difference operator are negligible in (85) follows from the factors \(\mathrm{e}^{-s}\) arising from the transformation from U to V.
 
13
Because \(\zeta \ll s\) in (78), this fully nonlinear region makes no leading-order contribution to the integral in (109), though the contribution it does make is only logarithmically smaller: self-consistency checks incorporating such correction terms have been undertaken to confirm that they do not lead to the expansions derived in this section becoming invalid in the regimes considered.
 
14
The notation \(\lambda _-\) is that of Appendix 2, not that above; the two usages of \(\lambda _\pm \) are in correspondence, however.
 
15
It will be clear by now that we are, in the interests of brevity, including in a number of such expressions terms that may not be of the same order. We affirm, in line with the footnote before last, that the analysis is nevertheless not ad hoc: that the various terms that contribute to the final conclusions (and only such terms) have each been retained has been subject to post hoc analysis; the linearity of (87) plays an important part in such considerations.
 
16
Again, it can be confirmed that the above expressions remain valid at leading order under this scaling, notwithstanding the title of this subsection.
 
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Metadaten
Titel
Pushed and pulled fronts in a discrete reaction–diffusion equation
verfasst von
J. R. King
R. D. O’Dea
Publikationsdatum
07.11.2015
Verlag
Springer Netherlands
Erschienen in
Journal of Engineering Mathematics / Ausgabe 1/2017
Print ISSN: 0022-0833
Elektronische ISSN: 1573-2703
DOI
https://doi.org/10.1007/s10665-015-9829-3

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