The evolution of traveling waves in a KPP reaction‐diffusion model with cut‐off reaction rate. I. Permanent form traveling waves

We consider Kolmogorov‐Petrovskii‐Piscounov (KPP) type models in the presence of a discontinuous cut‐off in reaction rate at concentration u=uc . In Part I, we examine permanent form traveling wave solutions (a companion paper, Part II, is devoted to their evolution in the large time limit). For each fixed cut‐off value 0


INTRODUCTION
Traveling waves arise in a wide range of applications in mathematical chemistry and biology (for example, in combustion 1 and in ecology, epidemiology, and genetics 2,3 ). They describe the invasion of chemical or biological reactions and are usually established as a result of the interaction between molecular diffusion, local growth, and saturation. The simplest model that encapsulates this interaction is the Kolmogorov-Petrovskii-Piscounov (KPP) reaction-diffusion equation (also called Fisher-KPP equation 4,5 ). In one spatial dimension, this describes the evolution of the concentration ( , ) as = + ( ), ( , ) ∈ ℝ × ℝ + , with ( , 0) = 0 ( ), where 0 ∶ ℝ → ℝ is piecewise continuous and smooth with limits 0 and 1 as → ∞ and → −∞, respectively. This is typically supplemented with boundary conditions with these limits being uniform for ∈ [0, ] and any > 0. The function ∶ ℝ → ℝ is a normalized KPP-type reaction function which satisfies conditions that ∈ 1 (ℝ) and An illustration of ( ) against is given in Figure 1A. Another popular example of a KPP reaction function is It is well-known 2,5-7 that the initial-boundary value problem (1) for the KPP equation supports a one-parameter family of nonnegative permanent form traveling wave solutions of the form These remain steady in time in a reference frame moving in the positive direction with speed ≥ 0 to be determined. Their existence and uniqueness (up to linear translation in origin of the independent coordinate ) is established for where denotes the minimum speed of propagation. This is achieved by analyzing the following nonlinear boundary value problem, namely, where the dash denotes differentiation with respect to . This is obtained by inserting (4) into Equation (1a) and using (1b) together with the initial conditions. The analysis is based on examining the existence of a unique heteroclinic orbit connecting the stable fixed point ( , ′ ) = (0, 0) to the unstable fixed point ( , ′ ) = (1, 0) in the ( , ′ ) phase plane of the equivalent twodimensional dynamical system obtained from (6). It is also used to establish that ( ) is monotone decreasing in ∈ ℝ. When translational invariance is fixed by requiring that (0) = 1∕2, then explicit expressions for the behavior of the permanent form traveling wave near the two fixed points are given by and for all ≥ = 2, where with ∞ (>0), ∞ , ∞ (>0), and −∞ (>0) being globally determined constants, dependent on the nonlinearity of the boundary value problem (6) (see, for example, Refs. 2,8).
A key result is that the initial condition in 0 ( ) determines the permanent form traveling wave solution that emerges at large times. When 0 ( ) is sufficiently close to a Heaviside function, specifically, 0 ( ) ≤ ( − ) (meaning ( − ) or ( − )) as → ∞, the solution to the KPP initialboundary value problem (1) converges at large times to the permanent form traveling solution with minimum speed = 2 (see, for example, Refs. 5, 6,9,10) at an algebraic rate determined in Refs. 11-13. The mechanism which selects the speed of propagation of the emerging permanent form traveling wave solution (as well as the rate of convergence) is based on the linearization of the KPP equation (1a) at the leading edge of the traveling wave. There, the concentration is small and the dynamics are unstable. As a result, any modification of the dynamics near the leading edge of the traveling wave would invalidate this speed selection mechanism. This is precisely the case for the cut-off KPP model that Brunet and Derrida 14 proposed and considered. Motivated by the discrete nature of chemical and biological phenomena at the microscopic level, they took a reaction function that is effectively deactivated at points where the concentration lies at or below a threshold value ∈ (0, 1). This case corresponds to the cut-off KPP equation given by with ( , 0) = 0 ( ), which is once more supplemented with the boundary conditions uniformly for ∈ [0, ] for all > 0. The main difference is that the reaction function ∶ ℝ → ℝ in the KPP equation (1) is replaced with a cut-off reaction function ∶ ℝ → ℝ given by where ( ) satisfies the KPP conditions (2). An illustration of ( ) against is given in Figure 1B, with + = ( + ) where ( + ) is the short notation for lim → + ( ). We remark that ( ) exhibits similarities with reaction functions arising in models of combustion in which represents an ignition temperature threshold 1,15 . Focussing on the initial conditions we henceforth refer to this initial-boundary value problem as IVP. Brunet and Derrida 14 proposed (8) as a model of front propagation arising in discrete systems of interacting particles. Such systems are for example, lattice models with discrete particles which make diffusive hops to neighboring sites, and which have some birth-death type of reaction 16 . In the continuum limit, obtained by allowing an arbitrarily large number of particles per lattice site, Brunet and Derrida 14 conjectured that discreteness in concentration values can be represented by an effective cut-off where may be viewed as the effective mass of a single particle. The idea is that for < , diffusion dominates over growth. Although the connection between (8) and discrete systems of interacting particles is phenomenological, model (8) remains useful in providing insight into their behavior. Analyzing the specific example (3b), Brunet and Derrida 14 considered the behavior of permanent form traveling wave solutions for small values of . Their main result is a prediction for the propagation speed * ( ) of the unique permanent form traveling wave given by * ( ) ≈ 2 − 2 (ln ) 2 , as → 0 + , which they obtained using a two-region informal point patching procedure (see also Ref. 17 where (9) was compared against numerical simulations of lattice particle models). This significant result demonstrates the strong influence of a cut-off on the value of * ( ) for small values of . The same approximation to * ( ) has also been obtained via an alternative variational approach in Refs. 18,19. Subsequently, a more rigorous approach was employed by Dumortier et al 20 who used geometric desingularization, to prove the existence and uniqueness of a permanent form traveling wave with * ( ) ∼ 2 − 2 (ln ) 2 + , as → 0 + .
All these results have restricted validity to the small limit with specific choices of cut-off KPPtype reaction function (8d), the most common based on ( ) given by (3b) 1 . Expression (10) was found in Ref. 20 to be generic when considering a slightly more general class of cut-off KPP-type reaction functions, namely, identical to (8d) when ∈ ( , ∞) but has ( ) = (1) uniformly for ∈ [0, ] as → 0 + . There are a number of fundamental questions that remain. The first question concerns the existence and uniqueness of a permanent form traveling wave solution for arbitrary threshold values and KPP reaction functions ( ). The second question concerns the propagation speed of such permanent form traveling wave solutions for arbitrary threshold values . The third question is with regard to the shape of the permanent form traveling wave solution. The fourth question concerns a systematic approach that captures the leading as well as higher-order corrections to the asymptotic behavior of the speed and shape of the permanent form traveling wave solution as → 0 + and → 1 − . The second limit may be less relevant for discrete systems of interacting particles. It is however relevant in models of combustion since the ignition temperature that determines the cut-off is not necessarily small 1 . A final question concerns the evolution in time to the permanent form traveling wave solution via the initial boundary value problem IVP. Part I of this series of papers addresses the first four of these questions while part II addresses the fifth and last question. In particular, we study classical solutions ∶ ℝ × ℝ + → ℝ to IVP for the cut-off KPP equation (8). In this paper, we proceed as follows. In Section 2, we reformulate IVP as a moving boundary problem. We then make a simple coordinate transformation to consider an equivalent initial-boundary value problem that we refer to as QIVP. In Section 3, we examine the possibility that QIVP supports permanent form traveling wave solutions ( ) = ( − ) where ∈ 1 (ℝ) ∩ 2 (ℝ⧵{0}) satisfies the nonlinear boundary value problem, We establish the following theorem.
for some global constant −∞ > 0 (which depends upon ), and In Sections 4 and 5, we use matched asymptotic expansions to develop the detailed asymptotic structure to the permanent form traveling wave solutions as → 0 + and as → 1 − , respectively. These are used to obtain higher-order corrections to (10) and (13) in a systematic manner. In the first limit, the analysis is carried out on the direct problem (rather than the phase plane). It highlights that higher-order corrections are controlled by two global constants ∞ and ∞ associated with the leading behaviour of , the permanent form traveling wave solution to the non cut-off problem (1) with minimal speed = (see equation (7a)). These global constants represent the nonlinearity in the problem when is small. The analysis is readily generalized to degenerate and singular KPP conditions, obtained for example when ′ (0) = 0 or ( ) ∼ 1∕2 as → 0 + , respectively. Section 6 presents numerical examples for the specific Fisher cut-off reaction function (3a). The paper concludes with a discussion in Section 7.

FORMULATION OF EVOLUTION PROBLEM QIVP
Due to the discontinuity in ( ) at = , it is convenient to restructure IVP as a moving boundary problem. To this end, we introduce the domains: and the curve that describes the moving boundary between the two domains. The boundary is expressed in terms of ( ) which satisfies ( ( ), ) = , with ≥ in and ≤ in . In this context, a classical solution will have ∶ ℝ × ℝ + → ℝ and ∶ ℝ + → ℝ such that, The moving boundary problem is then formulated as follows: The situation is illustrated in Figure 2. It is now convenient to make the simple coordinate transformation ( , ) → ( , ) with = − ( ). We then introduce the following domains: with ∶ ℝ × ℝ + → ℝ and ∶ ℝ + → ℝ such that The equivalent problem to (16) is then given by uniformly for ∈ [0, ] for all > 0 and where the dot denotes differentiation with respect to time, . This initial-boundary value problem will henceforth be referred to as QIVP. On using the classical maximum principle and comparison theorem (see, for example, Refs. 2 and 24), together with translational invariance in , and the regularity in (18), we can establish the following basic qualitative properties concerning QIVP, namely, In addition, via the partial differential equation (19a) and the regularity conditions (18), we have with the limits in (20c) and (20d) being uniform for ∈ [ 0 , 1 ] (for any 0 < 0 < 1 ). It follows from (20c) and (20d) that while, using (20b), (20d), and the regularity condition (18), we establish that The remainder of this paper and its companion (part II) concentrates on the analysis of QIVP. Specifically, in this paper we consider the existence and uniqueness of permanent form traveling wave solutions to QIVP including their asymptotic behavior in the limits of → 0 + and → 1 − via the method of matched and regular asymptotic expansions.

PERMANENT FORM TRAVELING WAVES IN QIVP
We anticipate that as → ∞, a permanent form traveling wave solution will develop in the solution to QIVP, advancing with a (nonnegative) propagation speed, allowing for the transition between the fully reacted state, = 1 as → −∞, to the unreacted state, = 0 as → ∞. Therefore, in this section we focus attention on the possibility of QIVP supporting permanent form traveling wave solutions (henceforth referred to as PTW solutions). We begin by establishing the existence and uniqueness of a PTW to QIVP for each fixed ∈ (0, 1), denoting the unique propagation speed by = * ( ). We then consider limiting values of * ( ) as → 0 + and → 1 − . The results established in this section provide proof of Theorem 1 as stated in Section 1.

The existence and uniqueness of a PTW solution to QIVP
A PTW solution to QIVP, with constant speed of propagation ≥ 0, is a steady-state solution to where ∈ 1 (ℝ) ∩ 2 (ℝ ⧵ {0}) and ≥ 0 satisfy the nonlinear boundary value problem, where the dash denotes differentiation with respect to . The nonlinear boundary value problem (23) can be thought of as a nonlinear eigenvalue problem with the eigenvalue being the propagation speed ≥ 0.
It is convenient to consider the ordinary differential equation (23) as the following equivalent autonomous first-order two-dimensional dynamical system, with = and = ′ , namely, We will analyze this dynamical system in the ( , ) phase plane for ≥ 0. In particular, it is straightforward to establish that the existence of a solution to (23) is equivalent to the existence of a heteroclinic connection in the ( , ) phase plane, for the dynamical system (24), which connects the equilibrium point (1,0), as → −∞, to the equilibrium point (0,0), as → ∞ (the translational invariance is then fixed by condition (23c) which requires that (0) = ). From (23b), this heteroclinic connection must remain in the ≥ 0 half plane of the ( , ) phase plane, which we denote We henceforth focus on this region of the ( , ) phase plane. However, before we proceed further, it is first worth considering the effect of introducing the cut-off into the reaction function on the dynamical system (24). To that end, we introduce the function ⃗ ∶ ℝ 2 → ℝ 2 where ⃗ ( , ) is given by to represent the vector field generating the dynamical system (24). We observe that, in the ( , ) phase plane, the effect of the discontinuity in ( ) across the line = is simply to refract the phase paths passing through this line. In particular, for each ∈ ℝ, there is exactly one phase path passing through ( , ), which has tangent vectors, ⃗ ( − , ) = ( , − ) and ⃗ ( + , ) = ( , − − + ). Thus, the refraction vector for the phase paths which cross the line = is We observe that the refraction vector (26) is independent of ( , ) ∈ ℝ × ℝ + and depends continuously on ∈ (0, 1). It follows that uniformly in ( , ) ∈ ℝ × ℝ + . After determining the effect of the discontinuity on the phase paths of the dynamical system (24) in + , we next consider the equilibrium points of (24) in + . These are readily found to be at locations We begin by examining the local phase portrait in the neighborhood of the equilibrium point ⃗ 1 . We find that ⃗ 1 is a hyperbolic equilibrium point. Moreover, ⃗ 1 is a saddle point with eigenvalues The associated local one-dimensional unstable and stable manifolds of ⃗ 1 are, respectively, given by We denote the phase path which forms the part of the (one-dimensional) unstable manifold entering + = {( , ) ∶ 0 < < 1, < 0} as  + 1 . Similarly, we denote as  − 1 the phase path which forms part of the (one-dimensional) unstable manifold entering − = {( , ) ∶ > 1, > 0}. The F I G U R E 3 The local phase portrait for the equilibrium points of the dynamical system (24). The thick black arrows denote the direction of the vector field ⃗ ( , ) (see (25)) along the line segments 0 , 1 and on the boundary of − situation is illustrated in Figure 3. We next determine the local phase portrait of the equilibrium points ⃗ for each ∈ [0, ]. For ∈ (0, ) and > 0, each of the equilibrium points ⃗ is nonhyperbolic with a single (one-dimensional) stable manifold in + given by {( , ) ∶ = − ( − ); 0 ≤ ≤ }. Also, the equilibrium point ⃗ 0 is nonhyperbolic with a single (onedimensional) stable manifold in ℝ + which we will denote by Finally, the equilibrium point ⃗ is again nonhyperbolic, and, for 0 ≤ ≤ , has a single (onedimensional) stable manifold in + given by {( , ) ∶ = − ( − ); 0 ≤ ≤ }. In fact, the collection of phase paths of the dynamical system (27) in the region {( , ) ∶ 0 ≤ ≤ , ≤ 0} is given by the family of curves = − , for each ∈ ℝ. This is illustrated in Figure 3. Next, for the line segment {( , ) ∶ = 1, > 0}, we observe the following: Similarly, for the line segment {( , ) ∶ > 1, = 0}, we observe that Together with the local structure at the equilibrium point ⃗ 1 , we conclude from (32) and (33) that the region − is a strictly positively invariant region for the dynamical system (24).We now examine the line segments 0 = {( , ) ∶ = 1, < 0} and 1 = {( , ) ∶ < < 1, = 0}, we observe that In addition, for > 0, we observe that for all ( , ) ∈ + ∇ ⋅ ⃗ ( , ) = − < 0.
Thus, for any > 0, it follows from the Bendixson negative criterion (see, for example, Ref. 25) that (27) has no periodic orbits, homoclinic orbits, or heteroclinic cycles in + . Finally, we observe that F I G U R E 4 The phase path  0 which forms part of the unstable manifold of the equilibrium point ⃗ 1 = (1, 0) of the dynamical system (24) the vector field ⃗ ( , ) rotates continuously clockwise for increasing ≥ 0. At the equilibrium point ⃗ 1 , the unstable manifold  + 1 rotates clockwise for increasing ≥ 0, as does the stable manifold  0 at the equilibrium point ⃗ 0 . As the phase path  − 1 enters − on leaving ⃗ 1 , and we have established that − is a strictly positively invariant region for the dynamical system (24), we conclude that this cannot correspond to a heteroclinic connection between ⃗ 1 and ⃗ 0 . Thus, at any ≥ 0, the existence of a heteroclinic connection in + connecting ⃗ 1 , as → −∞, to ⃗ 0 , as → ∞, is equivalent to the phase path  + 1 , leaving ⃗ 1 , being coincident with the phase path  0 , entering ⃗ 0 . It also follows that, at those ≥ 0 when such a heteroclinic connection exists, then it is unique.
With ∶ ℝ → ℝ being the associated PTW solution, then from (23), It is convenient, in what follows, to make a shift of origin by introducing the coordinatēviā =̄( ) + ,
Since ∞ > 0, then we must have (recallinĝ0 > 0)̂0 = , for some ∈ ℕ. However, condition (63a), with (62), then requires = 1, and sô0 = . (67) Proceeding to (̄( ) 1 2 ), we find that, on using (67), Thus, via (66), (67), and (68) we have, as → 0 + . It remains to apply the first condition in (63b). On using (62) and (69), the first condition in (63b) becomes as → 0 + . A rearrangement of (70) then gives, as → 0 + . It then follows from (69) and (71) that, as → 0 + . Finally, via (48) and (71), we can construct * ( ) as * ( ) = 2 − 2 (ln ) 2 − as → 0 + . For completeness, we give a schematic diagram of the asymptotic structure for (̄; ) in terms of the coordinatēas → 0 + in Figure 5. Returning to (73) we observe that the approximation is decreasing in as → 0 + , and is in full accord with the rigorous results established in Theorem 1. We see immediately that the approximation derived here, agrees in the first two terms with prediction (9) that Brunet and Derrida 14 first obtained and its third term is consistent with the order of the error term in (10) that Dumortier et al 20 derived. However, the method of matched asymptotic expansions has enabled us to obtain the next correction term in (73), and higher-order terms could be obtained by systematically following this approach (of course it may also be possible to obtain the third-and higher-order terms via extending the approach of Ref. 20). In fact, we may continue the expansion in each region to next order, and after matching, we can readily F I G U R E 5 A schematic diagram of the asymptotic structure of (̄; ) as → 0 + obtain that the higher-order correction to (73) is given by as → 0 + . For brevity, we do not provide a derivation to (74). We now consider the asymptotic structure of the PTW solution to QIVP as → 1 − .

NUMERICAL EXAMPLE
We now focus on the particular case of the cut-off Fisher reaction function, namely, (8d) with (3a). We obtain numerical approximations of the speed * ( ) and PTW solutions ∶ ℝ → ℝ for a range of values of cut-off . This is achieved by solving (11) numerically over an interval ∈ [0, ] for ∈ ℝ + using the MATLAB initial value solver ode45 where (0) = . As "initial" condition we employ (30) with 7(c) to approximate the unstable manifold near the unstable fixed point ( , ′ ) = (1, 0), taking (0) = 1 − and ′ (0) = − + ( ) where = 10 −12 and prescribe an absolute and relative tolerance of 10 −13 . The value of in the second initial condition is not known a priori. We therefore build the initial value solver into a shooting type algorithm, for which we guess the value of , integrate (11) to obtain ′ (0), and then compare ′ (0) to the target value − . The value of is then modified using the bisection method and this integration procedure is iterated until the absolute error satisfies | ′ (0) + | < 10 −13 . We start with a value of close to 1 and take = 2 as initial guess to begin the iteration which leads to the solution with speed * ( ). We then iterate over decreasing values of using the previously determined value of * ( ) as an initial guess to find the next solution.
It is also useful to obtain a numerical approximation of the permanent traveling wave solution for the Fisher reaction function (3a) in the absence of a cut-off. This is readily achieved by solving (6) numerically over an interval ∈ [0, ] for ∈ ℝ + once more using the Matlab initial value solver ode45. As "initial condition" we employ (7b) and (7c) to approximate the unstable manifold near the unstable fixed point ( , ′ ) = (1, 0), taking = 2, (0) = 1 − , and ′ (0) = ( √ 2 − 1) , where = 10 −12 and prescribe an absolute and relative tolerance of 10 −13 . We then determine that value of for which is equal to 1∕2 and then perform a coordinate shift to the origin. Figure 6 contrasts the behavior of against . A direct comparison between and is achieved when and are expressed in terms of̄so that (0) = (0) ≈ 0.5. For small values of , (̄) is close to (̄) in agreement with the asymptotic theory of Section 4. As increases, we observe a strong departure of (̄) from (̄) with the slope of (̄) at = 0 becoming increasingly steep until it reaches a maximum at = 0.5. Beyond this value, the slope satisfies ′ (0) = − * ( )∕2 and therefore becomes increasingly gentle with (since * ( ) decreases with ). When approaches 1, (̄) is in full agreement with the asymptotic prediction, derived from (89) (not shown). Figure 7 examines the behavior of the speed * ( ) and compares it to the various asymptotic expansions obtained as → 0 + and → 1 − (as derived from (74) and (90), respectively). The asymptotic expansion (74) obtained for → 0 + relies on the global constants ∞ and ∞ associated with the leading edge behavior of (̄) (see (56)). We determine the values of ∞ and ∞ by performing a least-squares polynomial fit to the computed (̄)̄for̄≳ 10 from where we obtain to a very good approximation the linear polynomial fit with ∞ ≈ 3.5 and ∞ ≈ −11.2.
(91) Figure 7A demonstrates that the two-term asymptotic expansion of * ( ) as → 1 − accurately captures the speed * ( ) for a wide range of values given by 0.4 ≲ < 1 (when = 0.4, = 1 − = 0.6 associated with expansions (79) is no longer small). Figure 7B focusses on the behavior of the speed obtained for smaller values of . It shows that the curve representing the two-term expansion based on retaining two terms in (74) and corresponding to the asymptotics that Brunet and Derrida 14 first obtained crosses the numerically computed curve representing * ( ) at ≈ 10 −4 and has a monotonic approach to this curve for smaller values of . We therefore anticipate that this two-term expansion only becomes genuinely asymptotic for values of ≪ 10 −4 . This implies that any comparison of the three-term and four-term expansions based on retaining three and four terms in (74), respectively, should only be considered for ≪ 10 −4 . With this in mind, the logarithmic corrections included in our three-and four-term expansions are an improvement over the two-term expansion for ≪ 10 −4 and a reasonable approximation to * ( ) with acceptable accuracy, less than 10 percentage error, for ≲ 8 × 10 −3 and for ≲ 4 × 10 −3 , respectively. However, the higher-order logarithmic corrections that they neglect are significant for larger values of . A B F I G U R E 7 A, Propagation speed * ( ) computed numerically as a function of the cut-off value for the particular case of the cut-off Fisher reaction function (8d) with (3a). Comparison against the asymptotic expansions for * ( ) obtained as → 1 − based on retaining one and two terms in (90) and as → 0 + based on retaining two, three, and four terms in (74). The two-term asymptotic expansion of * ( ) as → 0 + corresponds to prediction (9) first obtained in Brunet and Derrida 14 . B, Same as (A) but focusing on smaller values of

CONCLUSIONS
In this paper, we have considered a canonical evolution problem for a reaction-diffusion process when the reaction function is of standard KPP-type, but experiences a cut-off in the reaction rate below the normalized cut-off concentration ∈ (0, 1). We have formulated this evolution problem in terms of the moving boundary initial-boundary value problem QIVP. In Section 2, we have obtained some very general results concerning the solution to QIVP. In particular, these general results indicate that in the large time, as → ∞, the solution to QIVP will involve the propagation of an advancing nonnegative permanent form traveling wave, effecting the transition from the unreacting state = 0 (ahead of the wave front) to the fully reacted state = 1 (at the rear of the wave front). With this in mind, this paper has concentrated on examining the existence of permanent form traveling wave solutions to QIVP with propagation speed ≥ 0, referred to as PTW solutions. In Section 3, we have used a phase plane analysis of the nonlinear boundary value problem (23) to establish that (a) for each ∈ (0, 1), then QIVP has a unique PTW solution, with propagation speed = * ( ) > 0 and (b) * ∶ (0, 1) → ℝ + is continuous and monotone decreasing, with * ( ) → 0 + as → 1 − , and * ( ) → 2 − as → 0 + . It should be noted that 2 is the minimum propagation speed of permanent form traveling wave solutions for the related KPP-type function in the absence of cut-off. In Section 4, we have developed asymptotic methods to determine the asymptotic forms of * ( ) as → 0 + and → 1 − . The first limit was previously considered by Brunet and Derrida 14 and Dumortier et al 20 . The latter employed geometric desingularization to systematically determine the order of the error in Ref. 14. We have here used matched asymptotics expansions on the direct problem (23) to obtain higher-order corrections in a systematic manner. We show that these are controlled by the detailed structure ahead of the wave front solution traveling with speed 2 for the related KPP problem obtained in the absence of a cut-off. The second limit of → 1 − is motivated by applications in combustion 1 . In this limit, the asymptotic behavior is obtained via the use of regular asymptotic expansions in the phase plane.
We anticipate that the approach developed in this paper, for considering PTW solutions to QIVP, will be readily adaptable to corresponding problems, when the cut-off KPP-type reaction considered here is replaced by a broader class of cut-off reaction functions, such as those considered in Refs. 15, 20-23. In comparing the PTW theory for the cut-off KPP-type reaction function studied here, and its associated KPP-type reaction function without cut-off, we make the observation that, in the absence of cut-off, a PTW solution exists for each propagation speed ∈ [2, ∞), while at each fixed cut-off value ∈ (0, 1), a PTW solution exists only at the single propagation speed = * ( ), with 0 < * ( ) < 2; this observation has been made previously in Ref. 20, although restricted to sufficiently small cut-off values . This will have implications for the development of PTW solutions as large-structures in QIVP, with more general classes of initial data. In the companion paper, we consider the evolution problem QIVP in more detail. Specifically we establish that, as → ∞, the solution to QIVP does indeed involve the formation of the PTW solution considered in this paper, and we give the detailed asymptotic structure of the solution to QIVP as → ∞.
Finally, it is interesting to contrast our results with results obtained for a related problem, the stochastic KPP equation = + ( ) + (̂( )) 1∕2̇, ( , ) ∈ ℝ × ℝ + , with ( , 0) = 0 ( ), wherėis a standard space-time white-noise. Similarly to the cut-off KPP equation (8), Equation (92) arises as a continuum approximation to (microscopic) interacting particle systems. In particular, for a Fisher reaction function (3a), there is an exact relationship between this problem and discrete systems of particles which undergo a birth-coagulation type of reaction in addition to diffusion 27,28 . Rigorous results have been derived for this model too 29,30 , establishing that the average speed of the random traveling wave solutions of (92) is, in the smallor, weak noise limit, given by , aŝ→ 0 + .
Thus, takinĝ= , the difference between (93) and the speed of the PTW solution of the cut-off KPP model (8) only arises in the third term of the asymptotic expansion of (̂) and * ( ) as → 0 + , a conjecture that was initially made by Brunet and Derrida 14,31 . The two models behave very differently when̂can no longer be regarded as small, as might be anticipated. In the large-̂or, strong noise limit 28,32 , find that The behavior in this limit should be contrasted against expression (90) obtained for → 1 − . A comparison suggests that̂and may in this case be related according tô∼ 1∕(1 − ) 2 as → 1 − . It would be interesting to extend this comparison to arbitrary .

A C K N O W L E D G M E N T S
A. Tzella thanks C. Doering and J. Vanneste for useful conversations. All authors thank the referees and J. M. Meyer for constructive comments.