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Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts. (English) Zbl 0984.35030

The authors analyze the convergence of asymptotically uniformly traveling pulled fronts toward their asymptotic speed \(V^*\). It is shown that when such fronts evolve from “sufficiently steep” initial conditions (initial decay faster than \(e^{-\lambda^*x}\) for \(x\to\infty)\), then they have a universal relaxation behaviour as \(t\to\infty\), i.e., the velocity of the pulled front relaxes algebraically like \[ V(t)=V^* -{3\over 2\lambda^*t} +{3\over 2} \sqrt nD\lambda^*/ (D\lambda^{*2}t)^{3/2} +O(1/t^2). \]

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
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[1] C. Mathis, M. Provansal, L. Boyer, J. Phys. Lett. 45 (1984) L483.; C. Mathis, M. Provansal, L. Boyer, J. Phys. Lett. 45 (1984) L483.
[2] Ahlers, G.; Cannell, D. S., Phys. Rev. Lett., 50, 1583 (1983)
[3] Babcock, K. L.; Ahlers, G.; Cannell, D. S., Phys. Rev. Lett., 67, 3388 (1992)
[4] Fineberg, J.; Steinberg, V., Phys. Rev. Lett., 58, 1332 (1987)
[5] Tsameret, A.; Steinberg, V., Phys. Rev. E, 49, 1291 (1994)
[6] R.C. Ball, R.L.H. Essery, J. Phys. Cond. Mat. 2 (1990) 10303.; R.C. Ball, R.L.H. Essery, J. Phys. Cond. Mat. 2 (1990) 10303.
[7] Limat, L.; Jenffer, P.; Dagens, B.; Touron, E.; Fermigier, M.; Wesfreid, J. E., Physica D, 61, 166 (1992)
[8] Langer, J. S.; Müller-Krumbhaar, H., Phys. Rev. A, 27, 499 (1983)
[9] Powers, T. R.; Goldstein, R. E., Phys. Rev. Lett., 78, 2555 (1997)
[10] Deissler, R. J., Phys. Lett. A, 120, 334 (1987)
[11] Deissler, R. J., J. Stat. Phys., 54, 1459 (1989)
[12] Nozaki, K.; Bekki, N., Phys. Rev. Lett., 51, 2171 (1983)
[13] Salje, E. K.H., J. Phys. Cond. Mat., 5, 4775 (1993)
[14] S.J. Di Bartolo, A.T. Dorsey, Phys. Rev. Lett. 77 (1996) 4442, and references therein.; S.J. Di Bartolo, A.T. Dorsey, Phys. Rev. Lett. 77 (1996) 4442, and references therein.
[15] U. Ebert, W. van Saarloos, C. Caroli, Phys. Rev. Lett. 77 (1996) 4178.; U. Ebert, W. van Saarloos, C. Caroli, Phys. Rev. Lett. 77 (1996) 4178.
[16] Niklas, M.; Lücke, M.; Müller-Krumbhaar, H., Phys. Rev. A, 40, 493 (1989)
[17] van Zon, R.; van Beijeren, H.; Dellago, Ch., Phys. Rev. Lett., 80, 2035 (1998)
[18] Uhlig, C.; Eggers, J., Z. Phys. B, 103, 69 (1997)
[19] J.E. Maclennan, N.A. Cark, T. Carlsson, in: Solitons in Liquid Crystals, Springer, Berlin, 1991.; J.E. Maclennan, N.A. Cark, T. Carlsson, in: Solitons in Liquid Crystals, Springer, Berlin, 1991.
[20] van Saarloos, W.; van Hecke, M.; Hołyst, R., Phys. Rev. E, 52, 1773 (1995)
[21] Tu, Y.; Cross, M. C., Phys. Rev. Lett., 69, 2515 (1992)
[22] Krug, J.; Meakin, P., Phys. Rev. A, 43, 900 (1991)
[23] A. Torcini, P. Grassberger, A. Politi, J. Phys. A 28 (1995) 4533.; A. Torcini, P. Grassberger, A. Politi, J. Phys. A 28 (1995) 4533.
[24] van Hecke, M. L.; de Wit, E.; van Saarloos, W., Phys. Rev. Lett., 75, 3830 (1995)
[25] Couairon, A.; Chomaz, J. M., Physica D, 108, 236 (1997)
[26] M. van Hecke, C. Storm, W. van Saarloos, Physica D 134 (1999) 1.; M. van Hecke, C. Storm, W. van Saarloos, Physica D 134 (1999) 1.
[27] Carpentier, D.; Le Doussal, P., Phys. Rev. Lett., 81, 2558 (1998)
[28] Cross, M. C.; Hohenberg, P. C., Rev. Mod. Phys., 65, 851 (1992)
[29] Ya.B. Zeldovich, G.I. Barenblatt, V.B. Librovich, G.M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, 1985.; Ya.B. Zeldovich, G.I. Barenblatt, V.B. Librovich, G.M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, 1985.
[30] P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, Vol. 28, Springer, Berlin, 1979.; P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, Vol. 28, Springer, Berlin, 1979. · Zbl 0403.92004
[31] R.J. Field, M. Burger (Eds.), Oscillations and Traveling Waves in Chemical Systems, Wiley, New York, 1985.; R.J. Field, M. Burger (Eds.), Oscillations and Traveling Waves in Chemical Systems, Wiley, New York, 1985.
[32] N.F. Britton, Reaction-Diffusion Equations and their Applications to Biology, Academic Press, New York, 1986.; N.F. Britton, Reaction-Diffusion Equations and their Applications to Biology, Academic Press, New York, 1986. · Zbl 0602.92001
[33] Horváth, D.; Petrov, V.; Scott, S. K.; Showalter, K., J. Chem. Phys., 98, 6332 (1993)
[34] Malevanets, A.; Careta, A.; Kapral, R., Phys. Rev. E, 52, 4724 (1995)
[35] Horváth, D.; Tóth, A., J. Chem. Phys., 108, 1447 (1998)
[36] J.A. Murray, Mathematical Biology, Springer, Berlin, 1989.; J.A. Murray, Mathematical Biology, Springer, Berlin, 1989. · Zbl 0682.92001
[37] U. Ebert, W. van Saarloos, Phys. Rep. 337 (2000) 139-156.; U. Ebert, W. van Saarloos, Phys. Rep. 337 (2000) 139-156.
[38] P.C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, SIAM, Philadelphia, PA, 1988.; P.C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, SIAM, Philadelphia, PA, 1988. · Zbl 0684.35001
[39] A. Karma, W.-J. Rappel, , Phys. Rev. E 53 (1996) R3017.; A. Karma, W.-J. Rappel, , Phys. Rev. E 53 (1996) R3017.
[40] Mazenko, G. F.; Valls, O. T.; Ruggiero, P., Phys. Rev. B, 40, 384 (1989)
[41] Valls, O. T.; Lust, L. M., Phys. Rev. B, 44, 4326 (1991)
[42] Lemarchand, A.; Lesne, A.; Perera, A.; Moreau, M.; Mareschal, M., Phys. Rev. E, 48, 1568 (1993)
[43] Breuer, H.; Huber, W.; Petruccione, F., Physica D, 73, 259 (1994)
[44] Riordan, J.; Doering, C. R.; ben-Avraham, D., Phys. Rev. Lett., 75, 565 (1995)
[45] Armero, J.; Sancho, J. M.; Casademunt, J.; Lacasta, A. M.; Ramirez-Piscina, L.; Sagues, F., Phys. Rev. Lett., 76, 3045 (1996)
[46] G. Tripathy, W. van Saarloos, Fluctuation and relaxation properties of pulled fronts: a possible scenario for non-KPZ behavior, cond-mat/000513, to appear in Phys. Rev. Lett.; G. Tripathy, W. van Saarloos, Fluctuation and relaxation properties of pulled fronts: a possible scenario for non-KPZ behavior, cond-mat/000513, to appear in Phys. Rev. Lett.
[47] Armero, J.; Casademunt, J.; Ramrez-Piscina, L.; Sancho, J. M., Phys. Rev. E, 58, 5494 (1998)
[48] A. Rocco, U. Ebert, W. van Saarloos, Phys. Rev. E 62 (2000) R13.; A. Rocco, U. Ebert, W. van Saarloos, Phys. Rev. E 62 (2000) R13.
[49] A. Kolmogoroff, I. Petrovsky, N. Piscounoff, in: P. Pelcé (Ed.), Dynamics of Curved Fronts, Academic Press, San Diego, 1988 [translated from: Bulletin de l’Université d’État à Moscou, Ser. Int., Section A, Vol. 1, 1937].; A. Kolmogoroff, I. Petrovsky, N. Piscounoff, in: P. Pelcé (Ed.), Dynamics of Curved Fronts, Academic Press, San Diego, 1988 [translated from: Bulletin de l’Université d’État à Moscou, Ser. Int., Section A, Vol. 1, 1937].
[50] Fisher, R. A., Ann. Eugenics, 7, 355 (1937)
[51] Aronson, D. G.; Weinberger, H. F., Adv. Math., 30, 33 (1978)
[52] P. Collet, J.P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton University Press, Princeton, NJ, 1990.; P. Collet, J.P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton University Press, Princeton, NJ, 1990. · Zbl 0732.35074
[53] McKean, H. P., Commun. Pure Appl. Math., 28, 323 (1975)
[54] R.D. Benguria, M.C. Depassier, Phys. Rev. Lett. 77 (1996) 1171.; R.D. Benguria, M.C. Depassier, Phys. Rev. Lett. 77 (1996) 1171.
[55] T. Ogiwara, H. Matano, Discr. Cont. Dyn. Syst. 5 (1999) 1.; T. Ogiwara, H. Matano, Discr. Cont. Dyn. Syst. 5 (1999) 1.
[56] A.N. Bers, in: M.N. Rosenbluth, R.Z. Sagdeev (Eds.), Handbook of Plasma Physics, North-Holland, Amsterdam, 1983.; A.N. Bers, in: M.N. Rosenbluth, R.Z. Sagdeev (Eds.), Handbook of Plasma Physics, North-Holland, Amsterdam, 1983.
[57] P. Huerre, in: J.E. Wesfreid, H.R. Brand, P. Manneville, G. Albinet, N. Boccara (Eds.), Propagation in Systems Far from Equilibrium, Springer, New York, 1988.; P. Huerre, in: J.E. Wesfreid, H.R. Brand, P. Manneville, G. Albinet, N. Boccara (Eds.), Propagation in Systems Far from Equilibrium, Springer, New York, 1988.
[58] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, Vol. 10, Pergamon, New York, 1981.; L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, Vol. 10, Pergamon, New York, 1981.
[59] Stokes, A. N., Math. Biosci., 31, 307 (1976)
[60] Dee, G.; Langer, J. S., Phys. Rev. Lett., 50, 383 (1983)
[61] Ben-Jacob, E.; Brand, H. R.; Dee, G.; Kramer, L.; Langer, J. S., Physica D, 14, 348 (1985)
[62] Dee, G., J. Stat. Phys., 39, 705 (1985)
[63] van Saarloos, W., Phys. Rev. A, 37, 211 (1988)
[64] Dee, G.; van Saarloos, W., Phys. Rev. Lett., 60, 2641 (1988)
[65] van Saarloos, W., Phys. Rev. A, 39, 6367 (1989)
[66] van Saarloos, W.; Hohenberg, P. C., Physica D, 56, 303 (1992)
[67] Powell, J. A.; Newell, A. C.; Jones, C. K.R. T., Phys. Rev. A, 44, 3636 (1991)
[68] Paquette, G. C.; Chen, L.-Y.; Goldenfeld, N.; Oono, Y., Phys. Rev. Lett., 72, 76 (1994)
[69] Paquette, G. C.; Oono, Y., Phys. Rev. E, 49, 2368 (1994)
[70] Brunet, E.; Derrida, B., Phys. Rev. E, 56, 2597 (1997)
[71] Swift, J.; Hohenberg, P. C., Phys. Rev. A, 15, 319 (1977)
[72] W. Spruijt, Master’s Thesis, Leiden University, Leiden, 1998, unpublished.; W. Spruijt, Master’s Thesis, Leiden University, Leiden, 1998, unpublished.
[73] C. Storm, W. Spruijt, U. Ebert, W. van Saarloos, Phys. Rev. E 61 (2000) R6063.; C. Storm, W. Spruijt, U. Ebert, W. van Saarloos, Phys. Rev. E 61 (2000) R6063.
[74] Bramson, M., Mem. Am. Math. Soc., 44, 285 (1983)
[75] Kessler, D. A.; Ner, Z.; Sander, L. M., Phys. Rev. E, 58, 107 (1998)
[76] R. Durrett, Stoch. Proc. Appl. 9 (1979) 117.; R. Durrett, Stoch. Proc. Appl. 9 (1979) 117.
[77] Th. Gallay, G. Raugel, http://xxx.lanl.gov/patt-sol/9809007 and 9812007.; Th. Gallay, G. Raugel, http://xxx.lanl.gov/patt-sol/9809007 and 9812007.
[78] T.R. Powers, D. Zhang, R.E. Goldstein, H.A. Stone, Phys. Fluids 10 (1998) 1052.; T.R. Powers, D. Zhang, R.E. Goldstein, H.A. Stone, Phys. Fluids 10 (1998) 1052.
[79] J.S. Langer, in: G. Grinstein, G. Mazenko (Eds.), Directions in Condensed Matter Physics, World Scientific, Singapore, 1986.; J.S. Langer, in: G. Grinstein, G. Mazenko (Eds.), Directions in Condensed Matter Physics, World Scientific, Singapore, 1986.
[80] P.W. Bates, P.C. Fife, R.A. Gardner, C.K.R.T. Jones, Physica D 104 (1997) 1.; P.W. Bates, P.C. Fife, R.A. Gardner, C.K.R.T. Jones, Physica D 104 (1997) 1.
[81] J.D. Buckmaster, G.S.S. Lundford, Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982.; J.D. Buckmaster, G.S.S. Lundford, Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982. · Zbl 0557.76001
[82] Ebert, U.; van Saarloos, W., Phys. Rev. Lett., 80, 1650 (1998)
[83] van Saarloos, W., Phys. Rep., 301, 9 (1998)
[84] G.I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996.; G.I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996.
[85] Canosa, J., IBM J. Res. Dev., 17, 307 (1973)
[86] A. Messiah, Quantum Mechanics, North-Holland, Amsterdam, 1974.; A. Messiah, Quantum Mechanics, North-Holland, Amsterdam, 1974.
[87] Larson, D. A., SIAM J. Appl. Math., 34, 93 (1978)
[88] Sattinger, D. H., J. Diff. Eqn., 25, 130 (1977)
[89] Newman, W. I., J. Theor. Biol., 104, 473 (1983)
[90] Kirchgässner, K., J. Diff. Eqn., 96, 256 (1992)
[91] Eckmann, J.-P.; Wayne, C. E., Commun. Math. Phys., 161, 323 (1994)
[92] Kapitula, T., J. Diff. Eqns., 112, 179 (1994)
[93] V.I. Arnold, Ordinary Differential Equations, MIT Press, Cambridge, MA, 1978.; V.I. Arnold, Ordinary Differential Equations, MIT Press, Cambridge, MA, 1978.
[94] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.; J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. · Zbl 0515.34001
[95] J.D. Gunton, M. San Miguel, P.S. Sahni, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 8, Academic Press, New York, 1983.; J.D. Gunton, M. San Miguel, P.S. Sahni, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 8, Academic Press, New York, 1983.
[96] J.S. Langer, in: C. Godreche (Ed.), Solids Far from Equilibrium, Cambridge University Press, Cambridge, 1992.; J.S. Langer, in: C. Godreche (Ed.), Solids Far from Equilibrium, Cambridge University Press, Cambridge, 1992.
[97] Bray, A. J., Adv. Phys., 43, 357 (1994)
[98] M. Abramowitz, I.S. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1972.; M. Abramowitz, I.S. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1972.
[99] W. Spruijt, U. Ebert, W. van Saarloos, unpublished.; W. Spruijt, U. Ebert, W. van Saarloos, unpublished.
[100] C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.; C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. · Zbl 0417.34001
[101] Peletier, L. A.; Troy, W. C., SIAM J. Math. Anal., 28, 1317 (1997)
[102] U. Ebert, W. van Saarloos, L.A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations, Eur. J. Appl. Math., submitted for publication.; U. Ebert, W. van Saarloos, L.A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations, Eur. J. Appl. Math., submitted for publication. · Zbl 1013.35081
[103] Diekmann, O., J. Diff. Eqs., 33, 58 (1979)
[104] S. Fedotov, cond-mat/9807352.; S. Fedotov, cond-mat/9807352.
[105] D. Potter, Computational Physics, Wiley, New York, 1973.; D. Potter, Computational Physics, Wiley, New York, 1973. · Zbl 0314.70008
[106] D. Carpentier, private communication.; D. Carpentier, private communication.
[107] X.Y. Wang, S. Fan, T. Kyu, Phys. Rev. E 56 (1997) R4931.; X.Y. Wang, S. Fan, T. Kyu, Phys. Rev. E 56 (1997) R4931.
[108] Kaliappan, P., Physica D, 11, 368 (1984)
[109] Otwinowski, M.; Paul, R.; Laidlaw, W. G., Phys. Lett. A, 128, 483 (1988)
[110] G.S. Triantafyllou, K. Kupfer, A. Bers, Phys. Rev. Lett. 59 (1987) 1914.; G.S. Triantafyllou, K. Kupfer, A. Bers, Phys. Rev. Lett. 59 (1987) 1914.
[111] R.A.L. Jones, L.J. Norton, E.J. Kramer, F.S. Bates, P. Wiltzius, Phys. Rev. Lett. 66 (1991) 1326.; R.A.L. Jones, L.J. Norton, E.J. Kramer, F.S. Bates, P. Wiltzius, Phys. Rev. Lett. 66 (1991) 1326.
[112] U. Ebert, W. van Saarloos, C. Caroli, Phys. Rev. E 55 (1997) 1530.; U. Ebert, W. van Saarloos, C. Caroli, Phys. Rev. E 55 (1997) 1530.
[113] A. Torcini, S. Lepri, Phys. Rev. E 55 (1997) R3805.; A. Torcini, S. Lepri, Phys. Rev. E 55 (1997) R3805.
[114] J.D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, New York, 1994.; J.D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, New York, 1994. · Zbl 0834.35001
[115] R.D. Benguria, M.C. Depassier, Phys. Rev. E 57 (1998) 6493.; R.D. Benguria, M.C. Depassier, Phys. Rev. E 57 (1998) 6493.
[116] W. van Saarloos, P.C. Hohenberg, Physica D 69 (1993) 209 [Erratum to Physica D 56 (1992) 303]; W. van Saarloos, P.C. Hohenberg, Physica D 69 (1993) 209 [Erratum to Physica D 56 (1992) 303]
[117] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York, 1983.; V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York, 1983. · Zbl 0507.34003
[118] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986.; W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986. · Zbl 0587.65005
[119] D. Carpentier, P. Le Doussal, unpublished.; D. Carpentier, P. Le Doussal, unpublished.
[120] P.E. Cladis, W. van Saarloos, in: Solitons in Liquid Crystals, Springer, Berlin, 1991.; P.E. Cladis, W. van Saarloos, in: Solitons in Liquid Crystals, Springer, Berlin, 1991.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.