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 µ between lattice points and on a detuning parameter (α) 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 µ and α.
Highlights
Mathematical analyses of a wide variety of physical and biological systems lead to investigation of spatially-discrete nonlinear reaction-diffusion equations of the form duj dt = μ + f, (1)on a discrete integer lattice with lattice points j ∈ Z at which uj = uj(t), the parameter μ > 0 dictating the coupling strength between lattice points while the constant a parameterises the nonlinearity
That the formulations investigated in the above studies, and the equation analysed are to be viewed as truly spatially discrete, and not as discretised versions of PDEs; the results that we present serve to illustrate how the behaviour of the continuous analogue of (1) relates to that of the discrete system, representing one of the simplest systems in which such a discrete-to-continuous transition can be explored
Discrete diffusion equations with a bistable nonlinearity have been widely studied: see Keener [11], Elmer and van Vleck [12], Cahn et al [13], Chow et al [14], Fath [15] and King and Chapman [16], and references therein, in which propagation failure was considered in detail, highlighting some key differences between discrete models and their continuous counterparts
Summary
As a decreases (in the case of (2), and more generally in (1), (3) if a is suitably defined), A reaches zero at a = aT , say, and for smaller a with 2 < c < c† for some c†(a), U (z) in (10) becomes negative at some z, approaching zero from below as z → +∞ (as can readily be demonstrated by a phase-plane analysis, a method that is not available in the discrete the2pDaupeertobyth√e μre.scaling of j the constant c in the remainder of this section corresponds to dividing that elsewhere in case); such non-monotonic waves are unstable and are in any case again precluded for non-negative initial data by the comparison theorem.
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