Book Review
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MathSciNet review:
1568193
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Book Information:
Author:
Vitaly A. Volpert Aizik I. Volpert, and Vladimir A. Volpert
Title:
Traveling wave solutions of parabolic systems
Additional book information:
Transl. Math. Monographs, vol. 140, Amer. Math. Soc., Providence, RI, 1994, xii + 448 pp., US$142.00. ISBN 0-8218-4609-4.
J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990), 167–212. MR 1068805
H. Berestycki, B. Larrouturou, and P.-L. Lions, Multi-dimensional travelling-wave solutions of a flame propagation model, Arch. Rational Mech. Anal. 111 (1990), no. 1, 33–49. MR 1051478, DOI 10.1007/BF00375699
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
John W. Evans, Nerve axon equations. III. Stability of the nerve impulse, Indiana Univ. Math. J. 22 (1972/73), 577–593. MR 393890, DOI 10.1512/iumj.1972.22.22048
Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/72), 193–226. MR 287106, DOI 10.1512/iumj.1971.21.21017
Paul C. Fife, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc. 84 (1978), no. 5, 693–726. MR 481405, DOI 10.1090/S0002-9904-1978-14502-9
Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), no. 4, 335–361. MR 442480, DOI 10.1007/BF00250432
Paul C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 981594, DOI 10.1137/1.9781611970180
[9] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), 355-569.
R. Gardner and C. K. R. T. Jones, Stability of travelling wave solutions of diffusive predator-prey systems, Trans. Amer. Math. Soc. 327 (1991), no. 2, 465–524. MR 1013331, DOI 10.1090/S0002-9947-1991-1013331-0
S. P. Hastings, On the existence of homoclinic and periodic orbits for the Fitzhugh-Nagumo equations, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 123–134. MR 393759, DOI 10.1093/qmath/27.1.123
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
Christopher K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984), no. 2, 431–469. MR 760971, DOI 10.1090/S0002-9947-1984-0760971-6
C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), no. 1, 64–88. MR 1268351, DOI 10.1006/jdeq.1994.1025
Todd Kapitula, On the stability of travelling waves in weighted $L^\infty$ spaces, J. Differential Equations 112 (1994), no. 1, 179–215. MR 1287557, DOI 10.1006/jdeq.1994.1100
Hiroshi Kokubu, Yasumasa Nishiura, and Hiroe Oka, Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems, J. Differential Equations 86 (1990), no. 2, 260–341. MR 1064014, DOI 10.1016/0022-0396(90)90033-L
[17] A. N. Komolgorov, I. G. Petrovskii, and N. S. Piskunov, Etude de l'equation de la chaleur avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Moscov. Gos. Univ. Mat. Mekh. 1 (1937), 1-25.
Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108. MR 791863, DOI 10.1090/memo/0328
Christopher McCord and Konstantin Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), no. 1, 397–422. MR 1059711, DOI 10.1090/S0002-9947-1992-1059711-X
Konstantin Mischaikow and Vivian Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal. 24 (1993), no. 4, 987–1008. MR 1226860, DOI 10.1137/0524059
Yasumasa Nishiura and Hiroshi Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal. 18 (1987), no. 6, 1726–1770. Translated in J. Soviet Math. 45 (1989), no. 3, 1205–1218. MR 911661, DOI 10.1137/0518124
Robert L. Pego and Michael I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A 340 (1992), no. 1656, 47–94. MR 1177566, DOI 10.1098/rsta.1992.0055
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI 10.1016/0001-8708(76)90098-0
Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779, DOI 10.1007/978-1-4612-0873-0
[25] A. I. Volpert and V. A. Volpert, Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations, Trans. Moscow Math. Soc. 52 (1990), 59-108.
[26] V. A. Volpert and A. I. Volpert, Existence and stability of travelling waves in chemical kinetics, Dynamics of Chemical and Biological Systems, "Nauka", Novosibirsk, 1989, pp. 56-131.
Eiji Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol. 22 (1985), no. 1, 81–104. MR 802737, DOI 10.1007/BF00276548
- [1]
- J. Alexander, R. Gardner, and C.K.R.T. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990), 167-212. MR 1068805 (92d:58028)
- [2]
- H. Berestycki, B. Larrouturou, and P. L. Lions, Multi-dimensional travelling wave solutions of a flame propagation model, Arch. Rational Mech. Anal. 111 (1990), 33-49. MR 1051478 (91h:35148)
- [3]
- C. C. Conley, Isolated invariant sets and the generalized Morse index, CBMS Regional Conf. Ser. in Math. vol. 38, Amer. Math. Soc., Providence, RI, 1978. MR 511133 (80c:58009)
- [4]
- J. Evans, Nerve axon equations iii: Stability of the nerve impulses, Indiana Univ. Math. J. 22 (1972), 577-594. MR 0393890 (52:14697)
- [5]
- N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971), 193-226. MR 0287106 (44:4313)
- [6]
- P. Fife, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc. 84 (1978), 693-726. MR 0481405 (58:1522)
- [7]
- P. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), 335-361. MR 0442480 (56:862)
- [8]
- P. C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 53, SIAM, Philadelphia, PA, 1988. MR 981594 (90c:80012)
- [9]
- R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), 355-569.
- [10]
- R. A. Gardner and C.K.R.T. Jones, Stability of travelling waves of diffusive predator prey systems, Trans. Amer. Math. Soc. 327 (1991), 465-524. MR 1013331 (92a:92004)
- [11]
- S. Hastings, On the existence of homoclinic and periodic orbits for the Fitzhugh-Ngumo equations, Quart. J. Math. Oxford Ser. (2) 27 (1976), 123-134. MR 0393759 (52:14568)
- [12]
- D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer, New York, 1981. MR 610244 (83j:35084)
- [13]
- C.K.R.T. Jones, Stability of the traveling wave solution of the Fitzhugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984), 431-469. MR 760971 (86b:35011)
- [14]
- C.K.R.T. Jones and N. Kopell, Tracking invariant manifolds with differential forms, J. Differential Equations 108 (1994), 64-88. MR 1268351 (95c:34085)
- [15]
- T. Kapitula, On the weighted stability of travelling waves in weighted
spaces, J. Differential Equations 112 (1994), 179-215. MR 1287557 (95h:35107)
- [16]
- H. Kokubu, Y. Nishiura, and H. Oka, Heteroclinic and homoclinic bifurcations in bistable reaction-diffusion systems, J. Differential Equations 86 (1990), 260-341. MR 1064014 (92a:58128)
- [17]
- A. N. Komolgorov, I. G. Petrovskii, and N. S. Piskunov, Etude de l'equation de la chaleur avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Moscov. Gos. Univ. Mat. Mekh. 1 (1937), 1-25.
- [18]
- T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 328 (1985). MR 791863 (87a:35127)
- [19]
- C. McCord and K. Mishaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), 397-422. MR 1059711 (92k:58193)
- [20]
- K. Mishaikow and V. Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal. 24 (1993), 987-1008. MR 1226860 (94m:92013)
- [21]
- Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions of reaction-diffusion equations, SIAM J. Math. Anal. 18 (1987), 1726-1770. MR 911661 (88j:35089)
- [22]
- R. Pego and M. Weinstein, A class of eigenvalue problems, with applications to instability of solitary waves, Philos. Trans. Roy. Soc. London Ser. A 340 (1991), 47-94. MR 1177566 (93g:35115)
- [23]
- D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976), 312-355. MR 0435602 (55:8561)
- [24]
- J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, Berlin and New York, 1982. MR 1301779 (95g:35002)
- [25]
- A. I. Volpert and V. A. Volpert, Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations, Trans. Moscow Math. Soc. 52 (1990), 59-108.
- [26]
- V. A. Volpert and A. I. Volpert, Existence and stability of travelling waves in chemical kinetics, Dynamics of Chemical and Biological Systems, "Nauka", Novosibirsk, 1989, pp. 56-131.
- [27]
- E. Yanagida, Stability of the fast travelling pulse solution of the Fitzhugh-Nagumo equations, J. Math. Biol. 22 (1985), 1158-1173. MR 802737 (87a:92019)
Review Information:
Reviewer:
Robert Gardner
Journal:
Bull. Amer. Math. Soc.
32 (1995), 446-452
DOI:
https://doi.org/10.1090/S0273-0979-1995-00607-5
