Speed of pulled fronts with a cutoff

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(1)PHYSICAL REVIEW E 75, 051106 共2007兲. Speed of pulled fronts with a cutoff R. D. Benguria and M. C. Depassier Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile 共Received 4 February 2007; published 8 May 2007兲 We study the effect of a small cutoff ⑀ on the velocity of a pulled front in one dimension by means of a variational principle. We obtain a lower bound on the speed dependent on the cutoff, for which the two leading order terms correspond to the Brunet-Derrida expression. To do so we cast a known variational principle for the speed of propagation of fronts in different variables which makes it more suitable for applications. DOI: 10.1103/PhysRevE.75.051106. PACS number共s兲: 05.70.Ln, 47.20.Ky, 05.45.⫺a, 02.30.Xx. I. INTRODUCTION. In several problems arising in physics, population dynamics, chemistry, and other fields, it is found that a small perturbation to an unstable state leads to a propagating front joining the unstable to a stable state. The simplest model of such phenomenon is provided by the scalar reaction diffusion equation ut = uxx + f共u兲, where the reaction term f共u兲 is a nonlinear function with at least two fixed points, one stable and one unstable. Without loss of generality we assume that there is an unstable fixed point at u = 0 and a stable fixed point at u = 1. The reaction term f共u兲 obeys additional requirements depending on the phenomenon under study. In the present work we shall be interested in two generic classes. The first class, which we label type A, is that for which f ⬎ 0 in 共0,1兲, the second class, type B, also called the combustion case, is that for which f = 0 in 共0 , a兲, and f ⬎ 0 in 共a , 1兲. It was proven by Aronson and Weinberger 关1兴 that sufficiently localized initial conditions evolve into a monotonic front joining the stable to the unstable state. In case B there is a unique speed for which a monotonic front exists. In case A, the front propagates with the minimal speed for which monotonic fronts exist. This minimal speed satisfies 2冑 f ⬘共0兲艋 c* ⬍ 2. 冑 sup 共f共u兲/u兲, 0艋u艋1. 共1兲. a result also found by Kolmogorov, Petrovsky, and Piskunov 共KPP兲关2兴. For the classical Fisher-Kolmogorov 关2,3兴 equation ut = uxx + u共1 − u兲, the upper and lower bounds coincide and the speed is exactly the so-called linear or KPP value cKPP = 2冑 f ⬘共0兲. Fronts for which this is the minimal speed are called pulled since this is the speed obtained from linear considerations at the leading edge of the front. In all cases the speed can be calculated from the integral variational principle 关4兴 1539-3755/2007/75共5兲/051106共5兲. 2. c = sup 2 g共u兲. 冕冕. 1. f共u兲g共u兲du. 0 1. ,. 共2兲. g 共u兲/h共u兲du 2. 0. where the supremum is taken over all positive monotonic decreasing functions g共u兲 for which the integrals exist and where h共u兲 = −g⬘共u兲. Moreover, the supremum is always attained for reaction terms of type B, and for reaction terms of type A it is attained whenever c ⬎ cKPP. Reaction diffusion equations of type A are often used to model phenomena in population dynamics, with the assumption that the number of particles or individuals is large. It was noticed by Brunet and Derrida 关5兴 that the effect of a finite number of particles can be modeled by reaction terms of type A with a cutoff ⑀ = 1 / N, where N is the average number of particles at the saturation state of the front. The effect of a cutoff on the fronts was studied for the case f共u兲 = u − u3 and it was found that the speed of the front with a cutoff is given approximately by c⬇2−. ␲2 . 共ln ⑀兲2. This result was obtained by a matching approach. Recently this result has been proven rigorously by a geometric method 关6兴. The precise dependence of the speed of the front on the cutoff is not universal. An example of this nonuniversality was shown by introducing a small region of vanishing slope next to the cutoff 关7兴, the speed in this case turns out to be larger than the KPP value. The effect of a cutoff on reaction diffusion equations is of relevance not only as a model of populations with a large but finite number of individuals, it is also relevant to the study of noisy fronts 关8兴 and to some problems of particle physics. See, for example, Ref. 关9兴 and references therein. The purpose of the present work is to show how the speed of a pulled front with a cutoff can be found from the variational principle 共2兲. It is important to notice that the effect of a cutoff on a reaction term of type A is to transform it into a reaction term of type B, reaction terms for which the supremum in Eq. 共2兲 is attained and for which a unique speed exists. To obtain this result we reformulate the variational principle in a new way better suited to treat the fronts with cut-. 051106-1. ©2007 The American Physical Society.

(2) PHYSICAL REVIEW E 75, 051106 共2007兲. R. D. BENGURIA AND M. C. DEPASSIER. off. We apply this new form of the variational principle to the reaction term considered by Brunet and Derrida, but the results are of more general validity. We find a lower bound on the speed which depends on the cutoff, for which the leading order is the Brunet-Derrida expression. That is, we show that. ␲2 + h.o.t. 共ln ⑀兲2. c 艌 c共⑀兲 ⬇ 2 −. The same result can be obtained by using the alternative variational principle for the speed 关10兴.. F共1兲/s0 + u共s兲. c 艌2. g 共u兲/h共u兲du 2. 0. F共1兲 + s0. 0. if. u − u3 if. 0艋u艋⑀. ⑀ ⬍ u ⬍ 1.. 冎. 冕. g2共u兲 du = h共u兲. if. 0艋u艋⑀. u /2 − u /4 − ⑀ /2 + ⑀ /4 if. ⑀⬍u⬍1. 0 2. 冕. s0. 0. 2. 4. 冊. ␲2 + ¯ . 兩ln ⑀兩2. u共s兲 =. 再. if. s. A冑s cos ␾共s兲 if. 0艋s艋⑀. ⑀ ⬍ s ⬍ s0 ,. 冕冉冊 0. 共4兲. where s0 = 1/⑀,. ␻=. ␾* , 兩ln ⑀兩. ␾共s兲 = ␻ ln共s/⑀兲 − ␾* ,. 冉. A = 冑⑀ 1 +. 1 4␻2. 1 ␾* tan ␾* = 兩ln ⑀兩. 2 du ds. 冎. 冊. 共5兲. 1/2. ,. 共6兲. and where ␾* is the solution of. f共q兲dq.. s0. 冎. For the sake of clarity we postpone until Sec. IV the construction of the trial function. Choose the trial function. F„u共s兲… ds, s2. u. 4. 冉. The denominator becomes. 冕. 0. c 2 艌 c 2共 ⑀ 兲 ⬇ 4 1 −. 0. 1. 再. so that F共1兲 = 1 / 4 − ⑀2 / 2 + ⑀4 / 4. We will show that for a certain trial function the variational formula 共3兲 yields. where s0 = 1 / g共u = 1兲 is an arbitrary parameter and F共u兲 =. 再. s = 1/g,. and consider s as the independent variable in Eq. 共2兲. With this change of variables we find f共u兲g共u兲du =. f共u兲 =. F共u兲 =. It was shown in 关4兴 that the trial function ĝ共u兲 for which equality holds diverges at u = 0, so it is convenient to consider trial functions which in addition to the requirements g共u兲 ⬎ 0, g⬘共u兲 ⬍ 0 also satisfy g共0兲 → ⬁. Since g共u兲 is a monotonic decreasing we may perform the change of variables. 1. In this section we consider the speed of a front for a reaction term with a cutoff. Even though we choose a specific reaction term the results obtained are valid for a larger class of reaction terms. We choose the same reaction term studied previously by Brunet and Derrida, namely. .. 0. 冕. 共du/ds兲 ds. where the supremum is taken over positive increasing functions u共s兲 such that u共0兲 = 0, u共s0兲 = 1 and for which all the integrals in Eq. 共3兲 are finite. In Appendix A, we illustrate how to use this variational principle to show that for profiles satisfying the KPP criterion 关i.e., f共u兲艋 f ⬘共0兲u, for all 0 艋 u 艋 1兴, c = 2冑 f ⬘共0兲.. f共u兲g共u兲du. where. 共3兲. , 2. For this reaction term. 1. 0 1. u = u共s兲,. 0. s0. III. THE SPEED OF THE FRONT WITH A CUTOFF. In this section we introduce different variables which render the variational formula 共2兲 for the speed simpler to apply, particularly to the case of fronts with a cutoff. As an application of this simpler form we show in Appendix A how the linear or KPP value cKPP = 2 is obtained for the FisherKolmogorov equation. The variational expression 共2兲 implies that for any admissible trial function g共u兲,. 冕冕. F„u共s兲…/s2ds. 0. II. A SIMPLER FORM FOR THE VARIATIONAL PRINCIPLE. 2. 冕. c2 = sup 2. 冕. s0. 2. ds.. In this new variable the variational principle becomes. 共7兲. Notice that these definitions imply that in the range ⑀ ⬍ s ⬍ s0, −␾* ⬍ ␾共s兲 ⬍ ␾*, and that, for small ⑀, ␾* ⬍ ␲ / 2. Therefore u共s兲 is positive and monotonic increasing. Having chosen a trial function it is straightforward to obtain a lower bound. The numerator in Eq. 共3兲 is given by 051106-2.

(3) PHYSICAL REVIEW E 75, 051106 共2007兲. SPEED OF PULLED FRONTS WITH A CUTOFF. N共⑀兲 = F共1兲/s0 +. 冕. s0. 0. 冉冊冉. = ⑀F共1兲 + ⑀ − −. 1 4. 冕. 1/⑀. ⑀. u d 2u 2 + ␭ 2 = 0, s ds. F„u共s兲…/s2ds 1 ⑀. 冊冉冊冉. subject to u共0兲 = 0. The solution is of the form u = s␣ where ␣ is given by. ⑀2 ⑀4 A2 − + 共2␾* + sin 2␾*兲 2 4 4␻. 1 u4 2 ds艌 ⑀F共1兲 + ⑀ − s ⑀. ⑀2 ⑀4 − 2 4. 冊. ␣=. 2. +. A 共2␾* + sin 2␾*兲 + O关⑀3/2共ln ⑀兲4兴. 4␻. 共8兲. 关See Eq. 共B5兲 in Appendix B for the details on the estimation of the last integral in Eq. 共8兲兴. The denominator is given by D共⑀兲 =. 冕. s0. 共du/ds兲2ds = ⑀ +. 0. + 共1 − 4␻2兲sin 2␾*兴.. 共9兲. We know then that c 2 艌 c 2共 ⑀ 兲 = 2. N共⑀兲 . D共⑀兲. 冉. d 2u 1 = 0, ds2. 冊. ␲2 + h.o.t . 兩ln ⑀兩2. 共11兲. for 0 ⬍ u ⬍ ⑀ .. u1共0兲 = 0,. Since the variational formula 共3兲 is invariant to scaling in s we may choose, without loss of generality, u1 = s. Moreover, since this is valid for 0 ⬍ u ⬍ ⑀, we conclude that in this region 0 ⬍ s ⬍ ⑀. To sum up we have 0 ⬍ s 艋 ⑀.. u1共s兲 = s, if. 共10兲. The bound above is rigorous and it is explicitly dependent on ⑀. Expanding c共⑀兲 for small ⑀ we obtain in leading order the desired result 关see part 共ii兲 of Appendix B, in particular the derivation of Eq. 共B8兲兴, c 2共 ⑀ 兲 = 4 1 −. As shown in Appendix A, the best bound is obtained for ␣ → 1 / 2 hence the Lagrange multiplier in that limit is ␭ = 1 / 4. Next we construct the appropriate trial function for a pulled front with a cutoff. We must solve. 2. A 关共1 + 4␻2兲2␾* 8␻. 1 ± 冑1 − 4␭. 2. For u ⬎ ⑀ 共s ⬎ ⑀兲 but still small enough for the linear regime to be valid, we must solve u2 d 2u 2 2 + ␭ 2 = 0, with s ds. u2共⑀兲 = u1共⑀兲,. u2⬘共⑀兲 = u1⬘共⑀兲. The solution to this equation is straightforward and is given by u2共s兲 = A冑s cos ␾共s兲,. It is not difficult to obtain higher order terms in the expansion of c共⑀兲 but it is of no interest here. where IV. THE TRIAL FUNCTION. ␾共s兲 = ␻ ln. To find the trial function for which the maximum in the variational formula for the speed is attained we should solve the associated Euler-Lagrange equation. This is not possible in general since there are few exactly solvable cases. In the present situation we are interested in the effect of the cutoff on pulled fronts, that is, on fronts whose speed is determined from linearization at the leading edge, therefore we solve the Euler-Lagrange equation in the linear approximation. The Euler-Lagrange equation for the variational principle 共3兲 is f共u兲 d 2u + ␭ 2 = 0, s ds2 where ␭ is a Lagrange multiplier. Even though it is unrelated to the present discussion, it is worth mentioning that this equation can be obtained by performing the change of variable s = exp共−cz兲 in the ordinary differential equation uzz + cuz + f共u兲 = 0, hence we identify the Lagrange multiplier with 1 / c2. First we obtain the adequate trial function for pulled fronts without a cutoff. In the linear approximation the EulerLagrange equation is. s + ␦, ⑀. ␻=. 1 冑4␭ − 1. 2. The constants A and ␦ are found matching the solution to u1 as indicated above. Applying the matching conditions we obtain. 冉. A = 冑⑀ 1 +. 1 4␻2. 冊. 1/2. ,. ␦ = arctan. −1 . 2␻. 共12兲. We must also require that u共s兲 be positive and monotonic increasing. The first condition implies −␲ / 2 ⬍ ␾共s兲 ⬍ ␲ / 2. The second condition, that u共s兲 be monotonic increasing, implies tan ␾共s兲艋. 1 . 2␻. Since ␾共s兲 is an increasing function of s we know that arctan. 1 −1 = ␾共⑀兲艋 ␾共s兲 ⬍ ␾共s0兲艋 arctan . 2␻ 2␻. It is intuitively evident that the best bound will be obtained when the maximum range for ␾ is allowed. We choose then. 051106-3.

(4) PHYSICAL REVIEW E 75, 051106 共2007兲. R. D. BENGURIA AND M. C. DEPASSIER. ␾共s0兲 ⬅ ␾* = arctan. 1 . 2␻. 共13兲. With this choice, ␦ = ␾共⑀兲 = −␾*. The only free parameter left is the arbitrary parameter s0. To fix s0 we observe that as s → s0 the solution must approach u = 1. Since for pulled fronts without a cutoff ␻ = 0, we expect that for small ⑀, ␻ will be small. Then from Eq. 共13兲 it follows that. ␾* =. ␲ − 2␻ + h.o.t. 2. ACKNOWLEDGMENTS. We acknowledge partial support of Fondecyt 共CHILE兲 Projects No. 106-0627 and No. 106-0651, and CONICYT/ PBCT Proyecto Anillo de Investigación en Ciencia y Tecnología ACT30/2006.. hence u共s0兲 = A冑s0cos ␾* ⬇ A冑s0sin共2␻兲 ⬇ 2␻A冑s0 . From Eq. 共12兲 we see that for small ␻, A⬇. 冑⑀ 2␻. detailed analysis of this situation will be reported elsewhere. In the present work we have studied the effect of a cutoff on a pulled front, the effect of a cutoff on pushed fronts and bistable fronts has received less attention. A specific example is studied in 关11兴. General bounds on the speed have been obtained making use of the variational principle 共2兲 and exact solutions have been constructed for piecewise continuous functions 关12兴. These and other related problems will be the subject of future work.. APPENDIX A. .. Therefore for small ␻, u共s0兲 ⬇ 冑⑀s0. Requiring that u共s0兲 → 1 implies then s0 = 1 / ⑀. With this choice for s0 it follows that. Here we show how to recover the value c = 2冑 f ⬘共0兲 from the variational formulation 共3兲 for the speed of propagation of fronts when the profile f共u兲 satisfies the KPP condition, i.e., when f共u兲艋 f ⬘共0兲u, for all 0 艋 u 艋 1. Let us denote. ␾共s0兲 = ␾* = − 2␻ ln ⑀ − ␾* ,. N=. hence. ␻=. ␾* . 兩ln ⑀兩. 冕. F共1兲 + s0. and. s0. 0. 冕冉冊 s0. Replacing this value of ␻ in Eq. 共13兲 we obtain Eq. 共7兲, with which the construction of the trial function is complete. V. CONCLUSION. The purpose of this work was to study the effect of a cutoff on the speed of pulled fronts making use of the variational formulation for the speed. To do so we have rewritten the variational principle in new variables which simplify the problem. An additional advantage of this reformulation is that the Euler-Lagrange equation for the maximizer is seen easily to be the equation of the traveling front. Reaction terms with a cutoff belong to the class of general reaction terms for which a maximizer always exists and for which the speed is unique. If the original front without a cutoff is a pulled front then, with a cutoff, it is possible to solve the Euler-Lagrange equation in the linear approximation and obtain an upper bound on the speed. This value obtained from the linear equation is valid only for sufficiently small cutoffs. The lower bound on the speed is a complicated function of the cutoff, the first two terms in the series expansion of this bound correspond to the approximate formula found by other approximate means. Here we have obtained not only the first two terms in the expansion but a rigorous bound on the speed. It has been shown that small perturbations of the reaction term close to the cutoff have an important effect, and that the Brunet-Derrida term is not universal for all fronts with a cutoff. This can be expected since the Euler-Lagrange equation in the linear regime will be different in each case. A. D=. 1 F„u共s兲… 2 ds, s. du ds. 0. 共A1兲. 2. 共A2兲. ds.. For the KPP case, F共u兲艋 f ⬘共0兲u2 / 2, hence F共1兲艋 f ⬘共0兲 / 2. Then, it follows from Eq. 共A1兲 that N艋M⬅. f ⬘共0兲 + 2s0. 冕. s0. f ⬘共0兲. 0. u2 ds. 2s2. 共A3兲. Integrating the last term by parts and noticing that u共0兲 = 0, u共s0兲 = 1, and that lims→0 u / s = u⬘共0兲 exists, we get. 冕. s0. 0. f ⬘共0兲. u2 ds = 2s2. 冕. M2 艋. 冉冕. s0. 0. 1 f ⬘共0兲uu⬘ ds s. 冊. f ⬘共0兲. 0. =− therefore. s0. f ⬘共0兲 + 2s0. 冉冊. u2 d 1 ds − 2 ds s. 冕. 2. 艋 f ⬘共0兲2. s0. 0. 冕. 1 f ⬘共0兲uu⬘ ds, s. s0. 0. u2 ds s2. 冕. s0. 0. „u⬘共s兲…2ds, 共A4兲. by Schwarz inequality. However, from Eq. 共A3兲 we have f ⬘共0兲兰s00u2 / 共2s2兲ds 艋 M, and inserting this in Eq. 共A4兲 we finally get M 艋 2f ⬘共0兲. 冕. s0. 0. „u⬘共s兲…2ds.. Now, from Eqs. 共A2兲, 共A3兲, and 共A5兲 we have that. 051106-4. 共A5兲.

(5) PHYSICAL REVIEW E 75, 051106 共2007兲. SPEED OF PULLED FRONTS WITH A CUTOFF. 2. N 艋 4f ⬘共0兲, D. 共A6兲. for all possible trial functions u. Therefore taking the supremum of 2N / D over all u, using Eq. 共3兲 we finally get c 艋 2冑 f ⬘共0兲.. 共A7兲. On the other hand, choosing an appropriate maximizing sequence of functions, using the variational principle 共3兲, we may show that c 艌 2冑 f ⬘共0兲 and thus we can conclude that c = 2冑 f ⬘共0兲 in the KPP case. For that purpose, just consider the family of trial functions u␣ = s␣, which are appropriate trial functions as long as ␣ ⬎ 21 . Evaluating the right side of Eq. 共3兲 with u = u␣ and letting ␣ → 1 / 2, we get c2 艌 2冑 f ⬘共0兲, which combined with Eq. 共A7兲 yields the desired result. Just to illustrate this procedure consider the reaction term f共u兲 = u − u3. Effectively, with this trial function Eq. 共3兲 implies c2 艌. 冉. 冊. 2 2␣ − 1 1 2 ␣ − 1 2␣ + − s . 2 4共4␣ − 1兲 0 ␣2 4s20␣. In the limit ␣ → 1 / 2 we obtain c2 艌 4. APPENDIX B. In this appendix we show some details on how to get the lower bound 共11兲. 共i兲 Estimating the integral I ⬅ 兰⑀s0u4 / s2ds for the trial function u given by Eq. 共4兲. In order to estimate this integral we divide it into two parts, as follows: I1 =. 冕. ⑀−1/2. ⑀. u4 ds s2. I2 =. 冕. ⑀−1/2. u4 ds. s2. 冕. ⑀−1/2. ⑀. A4cos4 ␾共s兲ds 艋 A4共⑀−1/2 − ⑀兲.. 共B3兲. As for the second integral we use that u 艋 1 to get I2 艋冑⑀ − ⑀ .. 共B4兲. Adding up this two integrals and using Eq. 共6兲 we see that I can be estimated from above by a term of order ⑀3/2兩ln ⑀兩4, i.e.,. 冕. 1/⑀. ⑀. u4 ds 艋 O共⑀3/2兩ln ⑀兩4兲, s2. 共B5兲. which is small compared with A2 / ␻ ⬇ ⑀ / 共4␻3兲 = O关⑀共ln ⑀兲3兴, since ⑀1/2共ln ⑀兲 → 0 as ⑀ → 0. We have shown that the contribution of the nonlinear term I can be neglected when ⑀ → 0. To do so we split the integral in two parts. If the reaction term corresponded to that of the Fisher-Kolmogorov equation, f共u兲 = u − u2, a different splitting is necessary. One can show that the contribution of the nonlinear term always vanishes compared to the contribution of the linear term. A different case is that when the slope of the reaction term next to the cutoff vanishes 关7兴, but we are not addressing that problem here. 共ii兲 Estimating the leading order of Eq. 共10兲. From the expressions 共6兲 for A and ␻ we see that the leading order in both Eqs. 共8兲 and 共9兲 are the terms proportional to A2 / ␻. Hence the leading order in Eq. 共11兲 is given by J⬅4. 2␾* + sin 2␾* , 2␾* + sin 2␾* + 4␻2共2␾* − sin 2␾*兲. 共B6兲. which we can write as 共B1兲. and ⑀−1. I1 =. 共B2兲. J=4. 1 , 1 + 4 ␻ 2X. 共B7兲. where X = 共2␾* − sin 2␾*兲 / 共2␾* + sin 2␾*兲. Finally, we observe that 0 ⬍ X ⬍ 1, since 0 ⬍ ␾* ⬍ ␲ / 2 共in fact, ␾* ⬇ ␲ / 2兲, and also that 1 / 共1 + a兲 ⬎ 共1 − a兲 if a ⬎ 0, to conclude that. 冉. J 艌 4共1 − 4␻2X兲艌 4共1 − 4␻2兲 ⬇ 4 1 −. To estimate Eq. 共B1兲 we insert Eq. 共4兲 and we get. 关1兴 D. G. Aronson and H. F. Weinberger, Adv. Math. 30, 33 共1978兲. 关2兴 A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, Selected Works of A. N. Kolmogorov, edited by V. M. Tikhomirov 共Kluwer Academic Publishers, Dordrecht, 1991兲. 关3兴 R. A. Fisher, Annals of Eugenics 7, 355 共1937兲. 关4兴 R. D. Benguria and M. C. Depassier, Phys. Rev. Lett. 77, 1171 共1996兲. 关5兴 E. Brunet and B. Derrida, Phys. Rev. E 56, 2597 共1997兲. 关6兴 F. Dumortier, N. Popovic, and T. J. Kaper, Nonlinearity 20, 855 共2007兲. 关7兴 D. Panja and W. van Saarloos, Phys. Rev. E 66, 015206共R兲. 冊. ␲2 . 共B8兲共ln ⑀兲2. 共2002兲. 关8兴 J. G. Conlon and C. R. Doering, J. Stat. Phys. 120, 421 共2005兲. 关9兴 E. Brunet, B. Derrida, A. H. Mueller, and S. Munier, Phys. Rev. E 73, 056126 共2006兲. 关10兴 R. D. Benguria and M. C. Depassier, Commun. Math. Phys. 175, 221 共1996兲. 关11兴 D. A. Kessler, Z. Ner, and L. M. Sander, Phys. Rev. E 58, 107 共1998兲. 关12兴 V. Mendez, D. Campos, and E. P. Zemskov, Phys. Rev. E 72, 056113 共2005兲.. 051106-5.

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