## Abstract

By replacing Fick's diffusion law with Green and Nagdhi's type-II flux law, a hyperbolic counterpart to the classical Fisher–KPP equation is obtained. In this article, an analytical study of this partial differential equation is presented with an emphasis on shock and related kinematic wave phenomena. First, an exact travelling wave solution (TWS) is derived and examined. Then, using singular surface theory, exact amplitude expressions for both shock and acceleration waves are obtained. In addition, the issue of shock stability is addressed and the limitations of the model are noted.

It is shown that discontinuity (i.e. shock) formation in the TWS occurs only when the propagation speed, which must exceed the characteristic speed, tends to the latter. It is also shown that the shock amplitude equation undergoes a transcritical bifurcation. Lastly, numerical simulations of acceleration waves in a simple model problem are presented.

## 1. Introduction

Named in honour of Fisher (1937) and Kolmogoroff *et al*. (1937), for their (independent) theoretical work on the wave-like spread of an advantageous gene in a population through random mating, the (one-dimensional) Fisher–KPP equation(1.1)is perhaps the best known nonlinear, scalar reaction–diffusion (RD) equation (see also Aronson & Weinberger 1975; Murray 1993; Ebert & van Saarloos 2000). Here, is a density (or concentration) per unit length; the positive constants *ν*, *γ* and *ρ*_{s} are the diffusivity, rate coefficient and saturation density, respectively; and *x* and *t* subscripts denote partial differentiation. Equation (1.1), whose source term consists of the Pearl–Verhulst (or logistic) growth law, arises not only in biology but also in areas of the physical and social sciences (e.g. Aronson & Weinberger 1975; Murray 1993; Jordan & Puri 2002; Debnath 2004 and references therein). Over the years, several variations1 of this partial differential equation (PDE) have been derived and studied (e.g. Newman 1980; Murray 1993; Sherratt & Marchant 1996; Xin 2000; Kay *et al*. 2001; Harris 2005 and references therein), and while they continue to elicit the interest of researchers from diverse fields, in recent years, however, a number of authors have begun to re-examine the basic assumption upon which PDEs of this class are based.

To understand what motivates this inquiry, we must begin with the fact that the Fisher–KPP equation, which like all RD equations describes *kinematic* wave phenomena (Whitham 1974), is equivalent to the first-order system(1.2)where *q* denotes the flux. Equation (1.2)_{1} is the balance law for the species (or mass) while equation (1.2)_{2} is the constitutive relation for *q* known as Fick's (diffusion) law. Hence, like the well-known diffusion (or heat) equation, which corresponds to the *γ*→0 limiting case, equation (1.1) is a PDE of the parabolic type. Consequently, like all other PDEs of this type the Fisher–KPP equation exhibits the ‘paradox of diffusion’. More precisely, the prediction that a disturbance in the density at any point in a body is felt instantly, but unequally, at all others points in the body, however distant. Such physically unrealistic behaviour is a clear indication that Fick's law does not provide a complete description of the diffusion process.

As was first realized by researchers in heat conduction theory, where equation (1.2)_{2} is known as Fourier's law, the root-cause of this anomaly lies in the fact that, according to Fourier's law, changes in the temperature gradient are instantaneously reflected in the thermal flux (e.g. Chandrasekharaiah 1998). In an effort to correct this objectionable feature, various modifications of Fourier's (or Fick's) law have been proposed. Of these, the best known is the Maxwell–Cattaneo (MC) law (e.g. Gurtin & Pipkin 1968; Müller & Ruggeri 1993; Christov & Jordan 2005 and references therein)(1.3)where *τ*_{0}(>0) denotes a lag (or relaxation) time.2 Since it gives rise to a dissipative, hyperbolic transport equation, the MC law predicts that mass/thermal diffusion actually occurs by means of damped density/temperature waves propagating with finite speed, a phenomenon known as ‘second sound’ in the heat conduction literature (e.g. Müller & Ruggeri 1993; Christov & Jordan 2005 and references therein). Over the last decade, a number of classical RD models have been reformulated in terms of the MC law (e.g. Méndez & Camacho 1997; Niwa 1998; Needham & King 2002; Dolak & Hillen 2003; Mickens & Jordan 2004).

In the early 1990s, Green & Nagdhi (1991, 1993) developed a thermodynamically consistent theory of thermoelasticity which, like the MC law, is capable of admitting second sound. The novelty of their formulation consists in replacing the usual entropy production inequality with an entropy balance law. These authors presented three versions of their theory, which they termed types I, II and III. The second version, which we will refer to as GNII, is the most controversial of the three since it predicts finite speed heat conduction with *no* dissipation, i.e. *undamped* waves of second sound (Quintanilla & Straughan 2004; Jaisaardsuetrong & Straughan 2007). In the context of RD theory, the linearized GNII flux law assumes the form(1.4)where *c*_{∞} is a positive constant with (SI) units of m s^{−1}. The thermal version of this constitutive relation, wherein the empirical temperature *T* and the constant *κ*^{*} play the roles of *ρ* and , can be derived from eqns (2.17)_{3} and (3.13)_{3} of Green & Nagdhi (1993) after linearizing the former and expressing the r.h.s. of the latter in terms of *T*_{x}. Chandrasekharaiah (1998, p. 707) observes that equation (1.4) can also be derived from equation (1.3) by dividing the latter through by *τ*_{0} and then letting *τ*_{0}→∞, under the assumption that the ratio *ν*/*τ*_{0} tends to ; see also Dreyer & Struchtrup (1993, pp. 4–5) and references therein.

In the present article, we replace Fick's law with the GNII flux law, i.e. we consider the system(1.5)where(1.6)and a superscript T denotes transpose. Since and *λ*_{1}≠*λ*_{2}, where are the eigenvalues of the matrix *A*, it follows that this system is *strictly hyperbolic* (Logan 1994), with characteristics defined by . (Clearly, these are also the characteristics for the linear wave equation .)

If we were to eliminate *q* between the two equations in system (1.5), we would find that this system is equivalent to the nonlinear PDE(1.7)

Here, we observe that *Δ* is a positive constant, where is the discriminant of equation (1.7), which is consistent with the fact that system (1.5) is strictly hyperbolic. Thus, as one might expect, the transport equation under the GNII flux law is hyperbolic, for all *x* and *t*, with (finite) characteristic speed .

To the best of the author's knowledge, G. Rosen was the first to study this PDE in the context of physical applications; specifically, he noted that equation (1.7) arises both in one-dimensional gas dynamics involving combustion and in a specialized version of Weyl's unified field theory (see Rosen 1966 and references therein). However, except for the paper by Ockendon *et al*. (2001), wherein equation (1.7) was derived and examined under the Fanno model for turbulent compressible flow, this PDE appears to be absent from the post-1966 literature. With this in mind, the primary aim of the present work is to carry out an analytical study of equation (1.7), which we will refer to here as *Rosen's equation*, with a focus on shock and related phenomena. We also seek to understand the interplay between the nonlinearity and the positive/negative attenuation this PDE can exhibit.

To this end, the present article is arranged as follows. In §2, an exact travelling wave solution (TWS) is derived and its shock formation condition determined. In §§3 and 4, the growth and decay of shock and acceleration waves, respectively, are investigated. And finally, in §5, results are summarized and conclusions stated.

## 2. Travelling wave solution

We begin our search for a TWS with the following observation: since equation (1.7) is invariant under the transformation *x*→−*x*, it is clear that we need only consider, without loss of generality, right-travelling waves. Thus, we seek solutions of the specific form(2.1)where , the propagation (or wave) speed *v*_{0} is a positive constant, and we note that . Let us now substitute equation (2.1) into (1.7) and then integrate once with respect to *ξ*. This yields, after simplifying, the constant-coefficient Riccati equation(2.2)where is the constant of integration and a prime denotes differentiation with respect to *ξ*.

Let us assume that a wavefront is advancing from one steady state to another as *ξ*→∞, where the value of *ρ* behind the front is greater than that ahead, i.e. we seek a TWS in the form of a *Taylor shock* (Dodd *et al*. 1982). Mathematically, this means requiring that , where are constants. Consequently, we find that *ρ*_{1,2} must satisfy the condition(2.3)where , if we are to have a Taylor shock.

Solving our Riccati equation subject to the (usual) ‘initial’ condition yields, after simplifying, the exact solution(2.4)which is very similar to the TWS given by Rosen (1966). Here, the shock thickness *l*, which is defined as (Pierce 1989), is given by(2.5)From equation (2.4), it is clear that if *f* is to be a continuous function that satisfies the conditions at , then *l* must be strictly positive. Clearly, this means that *v*_{0} must satisfy the inequality(2.6)from which we see that the propagation speed *v*_{0} is *supersonic*. We also see from equations (2.5) and (2.6) that a shock wave, i.e. a propagating jump discontinuity in *ρ* (see §3*a*), can only form in the limit .

Using equations (2.3) and (2.5), along with the fact that *f* is in the form of a Taylor shock, it is not difficult to show that(2.7)from which we see that are both precluded.

As originally envisioned by Fisher (1937), the density ahead of the front is zero. In the present study, this corresponds to setting *ρ*_{2}=0. Consequently, the Taylor shock condition, i.e. equation (2.3), becomes simply *ρ*_{1}=*ρ*_{s} and equation (2.4) reduces to(2.8)

Thus, from equations (2.5), (2.7)_{1} and (2.8)_{2}, we can establish that(2.9)

## 3. Growth and decay of shock waves

In this section, we employ the theory of propagating singular surfaces (Chen 1973) to determine how an initial jump discontinuity in *ρ* propagates and evolves over time under system (1.5). The analysis closely follows that of Jordan (2005*a*), who considered an MC-based generalization of Burgers' equation (see equation (5.1)); and as in this earlier work by the author, the singular surface analysis is performed here without approximation. Also, it should be noted that, while it is possible to do so in the present study, due to the fact that *A* is a constant matrix (see equation (1.6)_{1}), singular surface theory generally *cannot* be applied in the context of shock waves (e.g. §1 of Fu & Scott 1990).

### (a) Hadamard's lemma and shock amplitude equation

Consider now a smooth planar wavefront *x*=*Σ*(*t*) propagating along the *x*-axis of a Cartesian coordinate system in, say, the positive *x*-direction. If *ρ* suffers a jump discontinuity across *Σ*, i.e. if , where denotes the jump amplitude of a function *F*=*F*(*x*, *t*) across *Σ* and are assumed to exist, then it follows from system (1.5) that(3.1)where is a constant. Here, a ‘+’ superscript corresponds to , the region into which *Σ* is advancing, while a ‘−’ superscript corresponds to , the region behind *Σ*. What's more, let us assume that in both and that the value of at time *t*=0 is known. Since is non-zero, *Σ* is said to be a *shock wave* (Chen 1973).

Next, we record the following expressions relating the jumps in *ρ* and/or its first derivatives to those in *q* and/or its first derivatives:(3.2)(3.3)(3.4)where is either a constant or (at most) a function of *t* only. Here, the jump relation given in equation (3.2) is a standard result (Bland 1988) while that given in equations (3.3) and (3.4) was obtained by taking the jumps of system (1.5), which is permissible since these equations hold on both sides of *Σ*. Also, we should mention that the term , which appeared on the r.h.s. of equation (3.3) after jumps were taken, was simplified using the jump product rule(3.5)

The final relation needed, known as Hadamard's lemma, is of fundamental importance in singular surface theory. For the purposes of the present study, Hadamard's lemma takes the form (Chen 1973; Bland 1988)(3.6)where the (one-dimensional) displacement derivative *δ*/*δt* denotes the time rate of change measured by an observer travelling with Σ.

Omitting the details, it is a relatively straightforward process, using Hadamard's lemma and equations (3.2)–(3.4), to derive the shock amplitude equation(3.7)where(3.8)

### (b) Evolution of shock amplitude

For simplicity of presentation, in the remainder of this section we limit our attention to the special case in which *ρ*^{+} is constant; specifically, we let , where is a constant. Consequently, it can be shown that the exact solution of equation (3.7), which we observe is of the Bernoulli type, is given by(3.9)where the constant *α*=2*μ*/*γ* is known as the *critical amplitude* (Chen 1973) and *S*(0)(≠0) denotes the value of *S* at time *t*=0.

According to equation (3.9), the temporal evolution of *S* can, from a strictly mathematical point of view, occur in any one of seven possible ways.

For (⇒α≠0):

If

*S*(0)=*α*, then*S*=*α*for*t*≥0.If

*S*(0)>0,*α*>0 and*S*(0)≠*α*, then .If

*S*(0)>0>*α*, then*S*∈(0,*S*(0)], for*t*≥0 and*S*→0+0 monotonically as*t*→∞.If

*S*(0)<0,*α*<0 and |*S*(0)|<|*α*|, then*S*∈[−|*S*(0)|, 0), for*t*≥0 and*S*→0−0 monotonically as*t*→∞.If

*S*(0)<0 and*S*(0)<*α*, then .

For (⇒*α*=0):

If

*S*(0)>0, then*S*∈(0,*S*(0)], for*t*≥0 and*S*→0+0 monotonically as*t*→∞.If

*S*(0)<0, then .

Here, the *breakdown* time is given by(3.10)where we note that *α*>0 (respectively, *α*<0) for (respectively, ).

From cases (v) and (vii) it is evident that if the initial jump amplitude is negative and, simultaneously, its value is less than *α*, then the shock magnitude blows-up in *finite* time *t*=*t*_{∞}. In contrast, if the conditions given in cases (i)–(iv) or (vi) are satisfied, then *S* is bounded for all *t*>0. Clearly, the blow-up of *S* described in cases (v) and (vii) is physically unrealistic and suggests that the mathematical model underlying equation (1.7) breaks down under such conditions.

If , meaning *α*=1, then cases (iii)–(vii) are all *precluded* because the requirements *α*>1 *and* , where denotes the initial value of *ρ*^{−}, are violated either collectively or individually. In other words, if , then only cases (i) and (ii) are possible.

### (c) Stability and bifurcation analysis

Let us now recast equation (3.7) as(3.11)

Recalling that *γ*>0 and *α*∈(−1, 1], it is not difficult to see that the shock amplitude equation undergoes a *transcritical bifurcation* (e.g. Hale & Koçak 1991) at *α*=0. What this means is that as the parameter *α* passes through zero, where *α*=0 is the bifurcation value, the two equilibrium solutions first coalesce at the origin, at which point the origin becomes a non-hyperbolic (Hale & Koçak 1991) equilibrium, and then switch their stability.

The bifurcation diagram given in figure 1 clearly illustrates the situation just described. In particular, we see that is stable (respectively, unstable) when *α*>0 (respectively, *α*<0), while the reverse is true for .

## 4. Acceleration waves

### (a) Derivation and analysis of jump amplitude

In the case of an acceleration wave (Chen 1973), we again have a planar wavefront propagating to the right, where the wavefront's location is given by *x*=*σ*(*t*). Now, however, but at least one of their first derivatives, say *ρ*_{t}, suffers a jump across *σ*. Consequently, in the acceleration wave case Hadamard's lemma gives(4.1)while equations (3.3) and (3.4), respectively, reduce to(4.2)

We now derive, using singular surface theory once again, the general expression for as a function of *t*. To this end, we begin by stating the following equalities:(4.3)all of which follow directly from equations (4.1) and (4.2). Next, differentiating the equations in system (1.5) with respect to *t*, taking jumps, and once more employing the jump product (equation (3.5)) results in(4.4)

(4.5)Here, we recall that *ρ*^{+}=*ρ*^{+}(*t*) and we observe that *ρ*^{+}=*ρ*^{−} since in the case of acceleration waves. Finally, using Hadamard's lemma with along with the above jump relations, we find that the acceleration wave amplitude satisfies the *linear* equation(4.6)the solution of which is immediately found to be(4.7)where denotes the value of at time *t*=0.

If we once again take *ρ*^{+} to be a constant, i.e. we once again set (see §3*b*), then equation (4.7) reduces to(4.8)

From this, it is evident that(4.9)while for all *t*≥0, when *α*=0.

While not occurring in the present study, finite-time blow-up is, of course, possible in the case of acceleration waves (e.g. Chen 1973; Fu & Scott 1991; Müller & Ruggeri 1993; Ostoja-Starzewski & Tre¸bicki 1999; Quintanilla & Straughan 2004; Jordan 2005*b*; Morro 2006; Christov *et al*. 2007; Jaisaardsuetrong & Straughan 2007 and references therein).

### (b) Generic model system: formulation and scheme construction

While the analysis presented in §4*a* has allowed us to determine and , this information, unfortunately, tells us nothing quantitative about *ρ* itself in the region *behind σ*. For example, what is the behaviour of *ρ* in the case where exhibits infinite-time blow-up? To answer this and related questions, as well as illustrate the evolution of *σ*, computational methods and tools will have to be employed.

To this end, we begin by considering the following initial-boundary value problem (IBVP) involving Rosen's equation:(4.10)where *H*(.) denotes the Heaviside unit step function, the constant denotes the amplitude of the input ‘pulse’, is both the duration (or width) of the input pulse and the time required for *σ* to complete its initial transit of the interval , and we note that *σ*(0)=0 and *σ*(*t*_{f})= since *x*_{0}=0.

We now introduce the following dimensionless variables:(4.11)and recast IBVP (4.10) in the slightly simpler (i.e. non-dimensional) formwhere plays the role of a Mach number, *γ*^{*}=*γt*_{f}, and the overbars have been omitted but are understood. Here, we should mention that in the context of IBVP (4.12), equation (4.8) becomes(4.13)where the amplitude of the jump in the time derivative of the boundary condition at *x*=0 across *t*=0 is , and we observe that since .

Setting aside any hope of obtaining an analytical solution, we turn to the calculus of finite differences and consider the following simple discretization of equation (4.12*a*):(4.14)where , and . On solving for , the most advanced time-step approximation, we obtain the (explicit) scheme(4.15)where and the truncation error is .

### (c) Numerical simulations

In figures 2–4, the curves shown in boldface were produced from datasets computed by a simple algorithm which implemented the scheme given in equation (4.15) on a desktop PC running Mathematica (v. 5.0). Interpolations between the points were then accomplished using the built-in cubic spline routine that is a part of this software package. The thin solid line segments, which also appear in these figures, have been included to illustrate the behaviour of the acceleration wave amplitudes; they are the tangents to the solution profiles at *x*=*t*, taken in the limit *x*→*t*−0, and were generated from equation (4.13). To simplify the notation/discussion, the peak (i.e. stationary) values of the solution profiles shown will be denoted by , where and .

The sequences shown in figures 2–4 illustrate the evolution of the normalized relative density profiles versus *x* during *σ*'s initial transit of the interval *x*∈(0, 1) and correspond to *α*<0, *α*=0 and *α*>0, respectively. These three figures were generated using *ϵ*=0.25 and *γ*^{*}=5.0. It should be noted, however, that the choice of these values was based solely on the need for clear, informative graphs; i.e. *ϵ*=0.25 and *γ*^{*}=5.0 may not necessarily correspond to a particular RD system.

In figures 2 and 3, which were generated for *α*=−0.50 and *α*=0, respectively, we see that both profiles suffer positive attenuation. This is, of course, exactly as one should expect for these two cases based on the fact that the coefficient of the term in equation (4.12*a*) is strictly negative for . In figure 2, however, we observe that decreases (exponentially) over time. In contrast, figure 3 illustrates the fact that the profile's slope at the wavefront remains constant, specifically, in the case of this particular IBVP, when *α*=0.

From figure 4, which also depicts the profile versus *x*, but for *α*=1.0, we observe that unlike figures 2 and 3, now experiences negative attenuation, i.e. growth. In particular, we observe that by time *t*=0.50, *P*(*t*) already exceeds the input pulse amplitude *ϵ*(=0.25). From the last frame of figure 4, however, it is clear that *P*(*t*) grows to surpass even (1/2)*α*. This might initially come as a surprise since it appears to contradict equation (4.12*a*); i.e. the notion that, since the coefficient of is positive when , and negative when , the upper bound on *should* be (1/2)*α*, for which the coefficient of is zero. Actually, had we chosen a sufficiently larger value of in figure 4 we would have observed .

The behaviour of *P*(*t*) depicted in figures 2–4 resembles, at least qualitatively, that of *S*(*t*), the shock amplitude, in cases (iii), (vi) and (ii), respectively, with *ϵ* in the role of *S*(0). To see if this similarity extends further, we have, in the same manner as those in figures 2–4, plotted in figure 5 the versus *x* profile, but for *ϵ*=*α*=1.0 and without the tangent lines. Now if this correspondence is to hold, then *P*(*t*) in figure 5 should behave like *S*(*t*) in case (i), where *α*=*S*(0) is assumed. Observing that the broken horizontal line corresponds to , it is clear from the last three frames of figure 5 that this is, in fact, what occurs, i.e. *P*(*t*)=1.0. Finally, since we have only considered *ϵ*∈(0, 1] in the present study, it would be of interest to investigate whether the similarity between *P* and *S* extends to those cases corresponding to *ϵ*<0 (i.e. ) and *ϵ*>1 (i.e. ).

Our simple finite difference scheme (equation (4.14)) has not only accurately captured the behaviour of predicted by equation (4.13), but also the sharp corner at *x*=*t*, which corresponds to *σ*, and the strict positivity of on the interval *x*∈(0, *t*). However, although clearly a desirable feature in numerical schemes for RD equations, it should be pointed out this scheme was *not* specifically designed to be positivity preserving (e.g. Mickens & Jordan 2005).

## 5. Summary and observations

By replacing Fick's diffusion law with the GN type-II flux law, and assuming growth governed by the logistic law, a new application of Rosen's equation has been found in the context of RD, i.e. kinematic wave, phenomena. An exact TWS of this PDE was determined and analysed. A singular surface analysis was also carried out and exact expressions for both shock and acceleration wave amplitudes were obtained. In addition, the question of shock stability was addressed and the breakdown time value for the shock amplitude was determined. Finally, a simple, but effective, finite-difference scheme was constructed and employed to numerically illustrate acceleration waves in the context of a well-established IBVP. Based on an analysis of these findings, we report the following.

Contrary to being associated with ‘dissipationless diffusive transport’, it has been shown that if the source term involved is density-/concentration-dependent, then the GNII flux law

*can*give rise to a transport PDE that exhibits positive attenuation (i.e. dissipation).A well-defined TWS with

*f*in the form of a Taylor shock is obtained only when*ρ*_{1}+*ρ*_{2}=*ρ*_{s}*and v*_{0}>*c*_{∞}, where the inequality means that the propagation speed*v*_{0}is*supersonic*.The minimum (positive) value of the shock thickness

*l*occurs when*ρ*_{2}=0 and is equal to*l*_{0}(see equations (2.8) and (2.9)).Shock formation in the TWS case, which corresponds to

*l*→0, can only occur in the limit*v*_{0}→c_{∞}+0.When

*α*>0, case (i) of §3*b*exactly corresponds to the shock limit of the TWS. This follows from the fact that, with*α*>0, the condition stated in case (i) is equivalent to the Taylor shock requirement given in equation (2.3).A non-zero stable equilibrium solution for

*S*is*only*possible under the conditions of case (ii) or under those of case (i) provided*α*>0.If

*α*=0, then there are no non-zero stable equilibrium solutions, i.e., |*S*| either blows-up in finite time or tends to zero as*t*→∞.Case (ii) describes a situation in which

*Σ*is a*compression*shock (Chen 1973), i.e.,*S*>0, for all*t*≥0. This also applies to case (i) but only when*α*>0.The shock amplitude equation admits a transcritical bifurcation about

*α*=0, where is a stable equilibrium solution only for*α*>0 (see figure 1).Since the characteristic speed is a constant, i.e. , finite-time blow-up of and is

*not*possible under Rosen's equation; see §4*a*.The acceleration wave

*σ*is classified as*compressive*, i.e. , when and*expansive*, i.e. , when ; see the definitions given in Chen (1973).The shock/acceleration wave behaviour reported here in the case of Rosen's equation is very similar to that exhibited by the MC-based PDE3(5.1)where the constant

*v*_{m}(>0) carries units of m s^{−1}, which arises in the study of traffic flow under the continuum assumption (Lighthill & Whitham 1955; Jordan 2005*a*).The evolution of the peak value,

*P*(*t*), of the profiles depicted in figures 2–5 is, qualitatively, identical to that of the shock amplitude*S*(*t*) described in cases (iii), (vi), (ii) and (i), respectively, where*ϵ*in IBVP (4.12) plays the role of*S*(0).

## Acknowledgments

The author would like to thank the anonymous referee for his/her extremely helpful comments and suggestions, as well as for bringing a number of useful references to his attention. The author would also like to thank Prof. Brian Straughan for many enlightening discussions. This research was supported by ONR/NRL funding (PE 061153N).

## Footnotes

↵By which we mean equation (1.1) with different source terms and/or a form of

*ρ*-dependent diffusivity. Also, some authors refer to both equation (1.1) and the PDE*ρ*_{t}−*νρ*_{xx}=*γρ*[1−(*ρ*/*ρ*_{s})^{2}] as Fisher–KPP equations (Ebert & van Saarloos 2000).↵When

*τ*_{0}represents a*thermal*relaxation time, the MC law does not hold exactly with*τ*_{0}constant; thermodynamics requires*τ*_{0}to be a function of temperature (e.g. Franchi & Straughan 1994 and references therein). In contrast, RD theory places no such demand on*τ*_{0}.↵The reader should take note of the fact that, although they may appear to be equivalent at first glance, system (1.5) of the present article and system (3.1) of Jordan (2005

*a*) are, in fact, quite different.- Received September 21, 2006.
- Accepted June 26, 2007.

- © 2007 The Royal Society