Pushed and pulled fronts in a discrete reaction–diffusion equation

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.

Appendices

Appendix 1: The limit $a\rightarrow 0^+$ in (2)

The system (1), (2) contains two parameters, $\mu$ and a; the bulk of this paper focusses on the dependence on the former, while here we briefly address the latter. The limit $a\rightarrow +\infty$ is a regular one in which the ‘u / a’ term in f(u; a) is simply disregarded at leading order and the front is a pulled one (that the front is pulled when $a\rightarrow +\infty$ for any $\mu$ is a conjecture based on the results above; a stronger conjecture also suggested by our results is that $a_T(\mu )$ is an increasing function of $\mu$ with $a_T(\infty ) = 1/2$, and $a_T(\mu )\sim 1/\ln (1/\mu )$ as $\mu \rightarrow 0^+$). We are therefore concerned here with the limit $a\rightarrow 0^+$ when pushed fronts are to be expected.

If we set $\mu =\hat{\mu }/a$, $c=\hat{c}/a$ and take the limit $a\rightarrow 0$, (26) becomes

$$\begin{aligned} \hat{\mu } \left( U(z+1) - 2U(z) +U(z-1) \right) + \hat{c}\frac{\mathrm {d}U(z)}{\mathrm {d}z} + U^2(z)(1-U(z))=0, \end{aligned}$$

(104)

and it is clear a priori that the wave must then be pushed (the linearisation of f(u) being trivial); moreover, absorbing a by the above rescaling implies

$$\begin{aligned} c_{{\text {min}}}(a,\mu ) \sim \hat{c}_{{\text {min}}}\frac{(a\mu )}{a} \quad \text{ as } \quad a\rightarrow 0^+,\ \mu =O(1), \end{aligned}$$

(105)

and the preceding analyses of pushed fronts imply with minor modifications that

$$\begin{aligned} \hat{c}_{{\text {min}}}(\hat{\mu }) \sim \sqrt{\frac{\hat{\mu }}{2}} \quad \text{ as } \quad \hat{\mu }\rightarrow +\infty , \quad \hat{c}_{{\text {min}}}(\hat{\mu }) \sim 2\sqrt{\frac{\hat{\mu }}{\pi }} \quad \text{ as } \quad \hat{\mu }\rightarrow 0^+ \end{aligned}$$

(106)

(cf. (22) and (74) as $\alpha \rightarrow +\infty$). It is noteworthy in (106) that $\hat{c}_{{\text {min}}}$ scales with $\hat{\mu }$ in the same fashion in both limits.

The scaling result (105) does not, however, apply when $\mu$ is small with respect to a. For $\mu =O(a)$, the distinguished limit in Sect. 5.3 applies, and a further regime with $\mu$ exponentially small in a gives the transition from pushed to pulled fronts, as in Sect. 5.4. Thus, while for $a>1/2$ fronts are pulled for any $\mu$, for small a, pushed fronts occur except in a very small region of $\mu$ parameter space.

Appendix 2: A linear differential–difference equation

The differential–difference equation

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}\xi }v(\xi +1) + \sigma v(\xi )=0 \end{aligned}$$

(107)

plays a central role in Sect. 5.4 and warrants brief separate discussion, not least because it constitutes a rare instance in which the appropriate treatment of a linearised problem in such a wavespeed-selection analysis is not simply the Liouville–Green (or JWKB) approximation. The notation in this appendix differs from elsewhere.

Introducing the Laplace transform

$$\begin{aligned} \hat{v}(\rho ) = \int _0^\infty v(\xi )\mathrm{e}^{-\rho \xi }\, \mathrm {d}\xi \quad {v}(\xi ) = \frac{1}{2\pi \mathrm{i}}\int _{-\mathrm{i}\infty }^{\mathrm{i}\infty } \hat{v}(\rho )\mathrm{e}^{\rho \xi }\, \mathrm {d}\rho \end{aligned}$$

(108)

(the poles of $\hat{v}(\rho )$ have ${\mathbb {R}}(\rho )<0$) gives

$$\begin{aligned} (\rho \mathrm{e}^\rho +\sigma )\hat{v}(\rho ) = v(1) + \rho \mathrm{e}^\rho \int _0^1 v(\xi ) \mathrm{e}^{-\rho \xi }\, \mathrm {d}\xi . \end{aligned}$$

(109)

The poles of $\hat{v}$ thus occur at $\rho =-\lambda$ with

$$\begin{aligned} \lambda \mathrm{e}^{-\lambda } = \sigma \end{aligned}$$

(110)

(cf. (63)), so for $\sigma <1/\mathrm{e}$ there are two (real) roots for $\lambda$ that we denote here by $\lambda _-$ and $\lambda _+$, with $0<\lambda _-<\lambda _+$, and it is easy to show that the complex roots all have real part larger than $\lambda _+$.

Equation (107) requires initial data for $0\le \xi \le 1$ and for our purposes in Sect. 5 it suffices to consider the case

$$\begin{aligned} v(\xi ) = \mathrm{e}^{\nu \xi } \quad \text{ for } \quad 0\le \xi \le 1, \end{aligned}$$

(111)

for constant $\nu$, and it then follows from (109) that

$$\begin{aligned} (\rho \mathrm{e}^\rho + \sigma )\hat{v}(\rho ) = \frac{\rho \mathrm{e}^\rho - \nu \mathrm{e}^\nu }{\rho - \nu }. \end{aligned}$$

(112)

If $\nu =-\lambda$, with $\lambda$ satisfying (110), we have

$$\begin{aligned} \hat{v}(\rho ) = \frac{1}{\rho + \lambda }, \quad v(\xi ) = \mathrm{e}^{-\lambda \xi }, \end{aligned}$$

(113)

as is clear beforehand. More significantly for our purposes, the far-field behaviour

$$\begin{aligned} v(\xi ) \sim \frac{\lambda _- + \nu \mathrm{e}^{\lambda _-+\nu }}{(1-\lambda _-)(\lambda _-+\nu )}\mathrm{e}^{-\lambda _-\xi } \quad \text{ as }\ \xi \rightarrow \infty \end{aligned}$$

(114)

follows from (112) as a residue contribution; suppressing such a slowly decaying term is a central ingredient in the selection mechanism for a pushed front in §5.4.

Appendix 3: Pulled and pushed waves in a discrete diffusion equation with nonlinear coupling

In this appendix, we show that travelling-wave speeds for (1), with nonlinearity given by (2), may be constructed explicitly in both the pushed and pulled regimes in the case for which constant coupling strength is replaced by the nonlinear function

$$\begin{aligned} \overline{\mu } = \frac{\mu u_j^2}{u_{j+1}u_{j-1}}, \end{aligned}$$

(115)

in which ${\mu }$ is constant. The nonlinearity (115) converges to the constant-coupling case in the continuum (slowly varying) limit and is otherwise mathematically convenient, as we shall highlight below.

Travelling waves U(z), propagating at speed c obey

$$\begin{aligned} \frac{\mu U^2(z)}{U({z+1})U({z-1})}\left( U({z+1})-2U(z)+U({z-1})\right) + c\frac{\mathrm {d}{U}(z)}{\mathrm {d}z} + f(U(z);a)=0. \end{aligned}$$

(116)

1.1 Stable–unstable connections

I Pulled waves   Because

$$\begin{aligned} U(z) \sim \mathrm{e}^{-\lambda z}, \quad z \rightarrow +\infty \end{aligned}$$

(117)

has $\mu \sim \overline{\mu }$ in (115), the pulled wavespeed $c^*(\mu )$ is again given by (4), (5).

II Pushed waves   Wave propagation speeds determined by the whole nonlinear wavefront may be obtained by the ansatz (which motivated (115) in the first place):

$$\begin{aligned} U(z) = \frac{1}{V(z)}, \quad V(z)=1+\mathrm{e}^{\lambda z}. \end{aligned}$$

(118)

We thereby obtain

$$\begin{aligned} cV(z)\frac{\mathrm {d}{V(z)}}{\mathrm {d}z} - {\mu }\left( V(z)V({z+1})-2V(z+1)V(z-1)+V(z)V({z-1})\right) - (V(z)-1)\left( V(z)+\frac{1}{a}\right) =0,\quad \end{aligned}$$

(119)

and hence

$$\begin{aligned} c^\dag \lambda ^\dag = 2\mu \left( \cosh (\lambda ^\dag ) -1 \right) +1, \quad c^\dag \lambda ^\dag = 1+\frac{1}{2a}, \end{aligned}$$

(120)

with a pushed front occurring when $\lambda ^\dag >\lambda ^*$; $\lambda ^\dag (a,\mu )$, and $c^\dag (a,\mu )$ can be determined explicitly in the form:

$$\begin{aligned} {{\lambda ^\dag = \ln \left( 1+\frac{1+\sqrt{1+8a\mu }}{4a\mu } \right) }}, \quad c^\dag = \left( \frac{{1 +1/2a}}{\lambda ^\dag }\right) , \end{aligned}$$

(121)

and the transition relationships $\lambda ^\dag =\lambda ^*$, $c^\dag = c^*$ imply that $a_T(\mu )$ is given by

$$\begin{aligned} {{2a_T+1 = \sqrt{1+8a_T\mu } \ln \left( 1+\frac{1+\sqrt{1+8a_T\mu }}{4a_T\mu } \right) .}} \end{aligned}$$

(122)

Equation (116) is evidently not of the class discussed in Appendix 3; moreover, setting

$$\begin{aligned} U(z) \sim {\mathrm {e}}^{-\lambda ^* z} W(z) \end{aligned}$$

(123)

in (116) with $c=c^*$ and W(z) slowly varying yields the dominant balance

$$\begin{aligned} \frac{{2(\cosh (\lambda ^*) - 1)}}{W}\left( \left( \frac{\mathrm {d}W}{\mathrm {d}z} \right) ^2 - W\frac{\mathrm {d}^2 W}{\mathrm {d}z^2} \right) + \cosh ({{\lambda ^*}})\frac{\mathrm {d}^2 W}{\mathrm {d}z^2}=0, \end{aligned}$$

(124)

so the far-field behaviour differs from (37), and the behaviour outlined in Appendix 3 might not be expected to pertain. Nevertheless, it is easy to see that $\partial c^\dag (a_T,\mu )/\partial a=0$ also holds in this case.

Equation (121) implies

$$\begin{aligned} c^\dag \sim {1\over 2a \ln \left( {1\over 2a\mu }\right) } \quad \text{ as } \quad a \rightarrow 0^+, \quad c^\dag \sim \sqrt{2a\mu } \quad \text{ as } \quad a\rightarrow +\infty , \end{aligned}$$

(125)

the former being consistent with the scaling argument embodied by (105) and the latter corresponding to the continuum limit. More importantly,

$$\begin{aligned} c^\dag \sim \left( 1+\frac{1}{2a}\right) /\ln \left( \frac{1}{2a\mu }\right) \quad \text{ as } \quad \mu \rightarrow 0^+,\ c^\dag \sim \left( \sqrt{2a}+\frac{1}{\sqrt{2a}} \right) \sqrt{\mu } \quad \text{ as } \quad \mu \rightarrow +\infty , \end{aligned}$$

(126)

and it follows that

$$\begin{aligned} a_T \sim \ln \left( {1\over 2\mu }\right) \quad \text{ as }\ \mu \rightarrow 0^+,\qquad a_T \sim \frac{1}{2} \quad \text{ as } \ \mu \rightarrow +\infty ; \end{aligned}$$

(127)

the latter is as expected from the continuum limit, but the former implies $a_T\rightarrow +\infty$ as $\mu \rightarrow 0^+$, i.e. the opposite behaviour from that in Sect. 5.4. Thus the analysis of (119) is counterproductive in terms of gaining insight into (26)—it does, however, reinforce the point that intuition into whether pushed or pulled behaviour is to be expected is hard to come by in the weakly coupled case. Figure 10 shows the curve (122) separating pushed and pulled waves, together with the weak- and strong-coupling limits (127). The offset of the latter for $\mu \ll 1$ illustrates further the implications of logarithmic terms in the associated asymptotic expansions (here through closed-form expressions, in contrast to §5.4).

1.2 Stable–stable connections

For completeness, we exploit the analytic tractability of (115) to address this case also. For a connection between the stable states $U=1,-a$ (stable for $a>0$), representing the propagation of the state $U=-a$ overrunning $U=1$ (and vice versa, denoted $\tilde{U}$), we set

$$\begin{aligned} U(z)=1-\frac{1+a}{V(z)},\quad \tilde{U}(z)=-a+\frac{1+a}{V(z)}, \end{aligned}$$

(128)

where V(z) is defined in (118). In each case, the wavespeed and decay rate may be obtained as in Sect. 3.1. The resulting wavespeeds are

$$\begin{aligned} c^\pm =\pm \frac{1}{2\lambda }\left( a-\frac{1}{a} \right) , \end{aligned}$$

(129)

where $c^+$ corresponds to the wave U(z) and $c^-$ to $\tilde{U}$, and the decay rate $\lambda (\mu ,a)$ is given in each case by

$$\begin{aligned} \lambda (\mu ,a)=\cosh ^{-1}\left( 1+ \frac{(a+1)^2}{4a\mu }\right) ; \end{aligned}$$

(130)

indeed the ostensibly distinct solution ansätze (128) in fact represent the same propagating front, moving in opposite directions.

Appendix 4: The generic pushed/pulled transition

Here we revisit briefly the analysis of Sect. 4 in a much more general setting. We consider the travelling-wave problem

$$\begin{aligned} P\left( \frac{\mathrm {d}}{\mathrm {d}z} \right) U + c\frac{\mathrm {d}U}{\mathrm {d}z} + f(U;a) = 0 \end{aligned}$$

(131)

for a pseudo-differential operator with symbol P, whereby

$$\begin{aligned} P\left( \frac{\mathrm {d}}{\mathrm {d}z} \right) {\mathrm {e}}^{-\lambda z} = P(-\lambda ){\mathrm {e}}^{-\lambda z} \end{aligned}$$

(132)

and the analysis henceforth will for the most part depend on P only in the form of the function $P(-\lambda )$; for the discrete problem (1), $P(-\lambda )$ is

$$\begin{aligned} P(-\lambda )=2\mu (\cosh (\lambda )-1). \end{aligned}$$

(133)

We again let

$$\begin{aligned} f(u;a) = u+o(u) \quad \text{ as }\ u\rightarrow 0 \end{aligned}$$

(134)

and assume P and f are such that for any given $c>0$ a unique connection satisfying

$$\begin{aligned} U\rightarrow 1 \quad \text{ as }\ z\rightarrow -\infty , \quad U\rightarrow 0 \quad \text{ as } \quad z\rightarrow +\infty ,\ U(0)=\frac{1}{2} \end{aligned}$$

(135)

exists. As $z\rightarrow +\infty$ we will in general have (39), where $\lambda$ is a root of

$$\begin{aligned} P(-\lambda ) - \lambda c +1 =0 \end{aligned}$$

(136)

with smallest real part. The repeated-root case has $c^*$ and $\lambda ^*$ determined by

$$\begin{aligned} P(-\lambda ^*)-\lambda ^*c^*+1=0, \quad -P^\prime (-\lambda ^*)-c^*=0, \end{aligned}$$

(137)

in which case U satisfies (37). The transition value $a=a_T$ is given by $A(a_T)=0$ with, generically, $B(a_T)>0$, and we again take the dependence of f upon a to be such that a pulled front arises for $a<a_T$ and a pushed one for $a>a_T$.

In the Liouville–Green approach (14), we have

$$\begin{aligned} \frac{\partial \phi }{\partial t} + P\left( -\frac{\partial \phi }{\partial x} \right) + 1 =0,\quad F -\eta \frac{\mathrm {d}F}{\mathrm {d}\eta } + P\left( -\frac{\mathrm {d}F}{\mathrm {d}\eta } \right) + 1 =0; \end{aligned}$$

(138)

on the envelope solution to the second of these (i.e. the Clairaut equation)

$$\begin{aligned} 0 = -P^\prime \left( -\frac{\mathrm {d}F}{\mathrm {d}\eta } \right) -\eta \end{aligned}$$

(139)

holds, so by (137) it follows that $F=0$ on $\eta =c^*$, $\mathrm {d}F/\mathrm {d}\eta = \lambda ^*$, as is to be anticipated. Correspondingly, the expansion-fan solution of the first of (138), parameterised by $q=\partial \phi /\partial x$ (which is constant on rays) is

$$\begin{aligned} x= -P^\prime (-q)t, \quad \phi = - \left( q P^\prime (-q) + P(-q)+1 \right) t, \end{aligned}$$

(140)

and determining where $\phi =0$ gives a further derivation of the same result.

Again setting $a=a_T+\delta$, $\lambda \sim \lambda ^*-\delta \varLambda$ and adopting the expansion (43) implies

$$\begin{aligned} P\left( -\frac{\mathrm {d}}{\mathrm {d}x}\right) U_0 + c^*\frac{\mathrm {d}U_0}{\mathrm {d}z} + f(U_0;a_T)=0, \quad P\left( -\frac{\mathrm {d}}{\mathrm {d}x}\right) U_1 + c^*\frac{\mathrm {d}U_1}{\mathrm {d}z} + \frac{\partial f}{\partial u}(U_0;a_T)U_1 = -\frac{\partial f}{\partial a}(U_0;a_T), \end{aligned}$$

(141)

from which we recover (44). Equations (136), (137) imply

$$\begin{aligned} \frac{1}{2}P^{\prime \prime }(-\lambda ^*)\varLambda ^2 = \lambda ^*C; \end{aligned}$$

(142)

a pushed front (whereby $\delta <0$) has $C>0$, $\varLambda >0$ (and hence $P^{\prime \prime }(-\lambda ^*)>0$) so by (40)

$$\begin{aligned} U^\dag \sim \left( A_+^\dag (a_T) + \delta \left( A_+^{\dag \prime }(a_T)+\varLambda A_+^\dag (a_T)z \right) \right) \mathrm{e}^{-\lambda ^* z} \quad \text{ as } \quad \delta \rightarrow 0^- , \ z\rightarrow +\infty , \end{aligned}$$

(143)

and consistency with (44) demands

$$\begin{aligned} A_+^\dag (a_T)=B(a_T), \quad A_+^{\dag \prime }(a_T)=B^\prime (a_T), \quad \varLambda = \frac{A^\prime (a_T)}{B(a_T)} \end{aligned}$$

(144)

and hence

$$\begin{aligned} C = \frac{1}{2\lambda ^*}P^{\prime \prime }(-\lambda ^*)\left( \frac{A^\prime (a_T)}{B(a_T)} \right) ^2. \end{aligned}$$

(145)

Finally,

$$\begin{aligned} P\left( -\frac{\mathrm {d}}{\mathrm {d}x}\right) U_2 + c^*\frac{\mathrm {d}U_2}{\mathrm {d}z} + \frac{\partial f}{\partial u}(U_0;a_T)U_2 = -\frac{\partial ^2 f}{\partial a^2}(U_0;a_T) -2\frac{\partial ^2 f}{\partial a\partial u}(U_0;a_T)u_1 - \frac{\partial ^2 f}{\partial u^2}(U_0;a_T)u_1^2 - C\frac{\mathrm {d}U_0}{\mathrm {d}z},\quad \quad \end{aligned}$$

(146)

the last term in which dominates the right-hand side as $z\rightarrow +\infty$ and implies

$$\begin{aligned} U_2\sim \frac{\lambda ^* B(a_T)C}{2P^{\prime \prime }(-\lambda ^*)}z^2 \mathrm{e}^{-\lambda ^* z}\quad \text{ as }\ z\rightarrow +\infty ; \end{aligned}$$

(147)

matching this to the corresponding term from (40) again implies (142), representing a useful consistency check.

Salient features arising from the above (providing one of the motivations for this more general analysis) include the following:

• 1. The wavespeed $c^*$ in the pulled-front case, given by (137), is independent of a; this is self-evident from the nature (134) of the linearisation, but is important to stress. Correspondingly, in the current limit, that the pre-exponential in (37) has no term quadratic in z has the consequence that (147) implies $C\equiv 0$ for a pulled front.

• 2. In the pulled-front case, the wavespeed is accordingly known a priori from (137); solving (131) for this wavespeed determines A(a) and B(a) in (37) and hence, through (144), (145), fully determines the pushed-front behaviour (wherein c has in general instead to be treated as an eigenvalue, being found as part of the solution) local to the transition.

• 3. The result $c^\dag - c^* = O( (a_T-a)^2 )$    as $a\rightarrow a_T^-$ seems to be generally applicable.