Phantom asymptotic solutions

L. A. Kalyakin

Ufa mathematical journal, Tome 6 (2014) no. 2, pp. 44-65

Résumé

We provide some examples of the problems with small parameter which have formal asymptotic solutions associated with no exact solutions.

Détail

Keywords: small parameter, asymptotics.

Bibliographie

[1] Babich V. M., Buldyrev V. S., “Iskusstvo asimptotiki”, Vestnik LGU, 1977, no. 13, 5–12 | MR | Zbl

[2] M. D. Kruskal, H. Segur, “Asymptotics beyond all orders in a model of crystal growth”, Stud. Appl. Math., 85:2 (1991), 129–181 | MR | Zbl

[3] Anosov D. V., “O ponyatii asimptoticheskogo resheniya”, Mezhdunarodnaya konferentsiya “Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki”, posvyaschennaya 150-letiyu so dnya rozhdeniya V. A. Steklova (14 maya 2013 g.) http://www.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=6787

[4] D. V. Anosov, “Some examples of singular perturbations”, J. Dynam. Control Systems, 2:2 (1996), 289–298 | DOI | MR | Zbl

[5] Andrianov I. V., Barantsev R. G., Manevich L. I., Asimptoticheskaya matematika i sinergetika, URSS, M., 2004

[6] Vishik M. I., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12:5 (1957), 3–122 | MR | Zbl

[7] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR

[8] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[9] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vysshaya shkola, M., 1990 | MR | Zbl

[10] Vasileva A. B., Butuzov V. F., Nefedov N. N., “Kontrastnye struktury v singulyarno vozmuschennykh zadachakh”, Fundamentalnaya i prikladnaya matematika, 4:3 (1998), 799–851 | MR | Zbl

[11] Mitropolskii Yu. A., Khoma G. P., Matematicheskoe obosnovanie asimptoticheskikh metodov nelineinoi mekhaniki, Naukova dumka, Kiev, 1983 | MR

[12] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[13] Laricheva V. V., “O rasshirenii intervala primenimosti osredneniya”, Dokl. AN SSSR, 220:4 (1975), 775–778 | Zbl

[14] Azhotkin V. D., Babich V. M., “O primenenii metoda dvukhmasshtabnykh razlozhenii k odnochastotnoi zadache teorii nelineinykh kolebanii”, Prikladnaya matematika i mekhanika, 49:3 (1985), 377–383 | MR | Zbl

[15] Kuznetsov A. N., “Differentsiruemye resheniya vyrozhdayuschikhsya sistem obyknovennykh uravnenii”, Funkts. analiz i ego prilozh., 6:2 (1972), 41–51 | MR | Zbl

[16] Kuznetsov A. N., “O suschestvovanii vkhodyaschikh v osobuyu tochku reshenii avtonomnoi sistemy, obladayuschei formalnym resheniem”, Funkts. analiz i ego prilozh., 23:4 (1989), 63–74 | MR | Zbl

[17] Arnold V. I., “O povedenii adiabaticheskogo invarianta pri medlennom periodicheskom izmenenii funktsii Gamiltona”, Dokl. AN SSSR, 142:4 (1962), 758–761 | MR | Zbl

[18] Nekhoroshev N. N., “Eksponentsialnaya otsenka vremeni ustoichivosti gamiltonovykh sistem, blizkikh k integriruemym”, Uspekhi matem. nauk, 32:6 (1977), 5–66 | MR | Zbl

[19] Neishtadt A. I., “O razdelenii dvizhenii v sistemakh s bystro vraschayuscheisya fazoi”, Prikladnaya matematika i mekhanika, 48:2 (1984), 197–204 | MR

[20] Arnold V. I., Kozlov V. V., Neishtadt A. I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, VINITI, M., 1985

[21] Kozlov V. V., Furta S. D., Asimptotiki reshenii silno nelineinykh sistem differentsialnykhz uravnenii, Izd-vo Moskov. un-ta, M., 1996 | MR | Zbl

[22] Bryuno A. D., Stepennaya geometriya v algebraicheskikh i differentsialnykh uravneniyakh, Nauka, M., 1998 | MR | Zbl

[23] L. A. Kalyakin, “Justification of asymptotic expansion at infinity”, Journal of Nonlinear Mathematical Physics, 15:3 (2008), 210–216 | MR

[24] Ilin A. M., Melentsov M. A., “Asimptotika reshenii sistem differentsialnykh uravnenii s malym parametrom pri bolshikh znacheniyakh vremeni”, Trudy IMM UrO RAN, 11, no. 1, 2005, 97–110 | MR | Zbl

[25] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[26] E. B. Davies, “Pseudospectra of differential operators”, Operator Theory, 43 (2000), 243–262 | MR | Zbl

[27] L. N. Trefethen, “Pseudospectra of linear operators”, SIAM Review, 39 (1997), 383–406 | DOI | MR | Zbl

[28] L. N. Trefethen, “Wave packet pseudomodes of variable coefficient differential operators”, Proc. R. Soc. A, 461 (2005), 3099–3122 | DOI | MR | Zbl

[29] L. N. Trefethen, M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton, N.J., 2005 | MR | Zbl

[30] Galtsev S. V., Shafarevich A. I., “Spektr i psevdospektr nesamosopryazhennogo operatora Shrëdingera s periodicheskimi koeffitsientami”, Matematicheskie zametki, 80:3 (2006), 356–366 | DOI | MR | Zbl

[31] Nils Dencker, “Spectral Instability of Semiclassical Operators”, Preprint ESI 2187, The Erwin Schrödinger International Boltzmanngasse, 9, Institute for Mathematical Physics A-1090 Wien, Austria Vienna, October 20, 2009 | MR

[32] Lidskii V. B., “K teorii vozmuschenii nesamosopryazhennykh operatorov”, Zhurnal vychisl. matem. i matem. fiziki, 6:1 (1966), 52–60 | MR | Zbl

[33] F. Calogero, “Singular Potentials and Regularization”, Phys. Rev. B, 139 (1965), 602–606 | DOI | MR

[34] F. Calogero, “Simple Examples in Singular Perturbation Theory: Eigenvalues That Do Not Tend to Be Unperturbed Values as the Perturbation Is Switched Off”, Lett. Nuovo Cim., 25:17 (1979), 533–538 | DOI

[35] Mikhailov A. V., Shabat A. B., Yamilov R. I., “Simmetriinyi podkhod k klassifikatsii nelineinykh uravnenii. Polnye spiski integriruemykh sistem”, Uspekhi matem. nauk, 42:4 (1987), 3–53 | MR | Zbl

[36] A. V. Mikhailov, A. B. Shabat, V. V. Sokolov, “The symmetry approach to classification of integrable equations”, What is integrability?, Springer, Berlin–Heidelberg, 1991, 115–184 | DOI | MR | Zbl

[37] Kalyakin L. A., “Dlinnovolnovye asimptotiki”, Uspekhi matem. nauk, 63:5 (2008), 3–72 | DOI | MR | Zbl

[38] G. Schneider, “Validity and limitation of the Newell–Whitehead equation”, Math. Nachr., 176 (1995), 249–263 | DOI | MR | Zbl

[39] G. Schneider, “The validity of generalized Ginzburg–Landau equations”, Math. Methods Appl. Sci., 19 (1996), 717–736 | 3.0.CO;2-Z class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[40] R. Bhatt, A. V. Mikhailov, On the inconsistency of the Camassa–Holm model with the shallow water theory, 10 Oct. 2010, arXiv: 1010.1932v1[nlin.SI]

[41] G. I. Burde, “On the asymptotic solutions of the KdV equation with higher-order corrections”, Nonlinearity, 18 (2005), 1443–1461 | DOI | MR | Zbl

[42] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979 | MR

[43] Ivanov V. V., Vasin V. V., Tanana V. P., Teoriya nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978

[44] Lavrentev M. M., Savelev L. Ya., Teoriya operatorov i nekorrektnye zadachi, Izd-vo IM, Novosibirsk, 1999 | MR

[45] Z. Schuss, Theory and applications of stochastic processes, Applied Mathematical Sciences, 170, Springer, 2010 | DOI | MR | Zbl

[46] Venttsel A. D., Freidlin M. I., Fluktuatsii v dinamicheskikh sistemakh pod deistviem malykh sluchainykh vozmuschenii, Nauka, M., 1979 | MR | Zbl

[47] S. Kamin, “On elliptic perturbation of a first order operator with a singular point of attracting type”, Indiana Univ. Math. J., 27:6 (1978), 935–951 | DOI | MR

[48] Z. Schuss, B. J. Matkowsky, “The exit problem: A new approach to diffusion across potential barriers”, SIAM J. Appl. Math., 35:3 (1979), 604–623 | DOI | MR

[49] M. V. Day, “Mathematical Approaches to the Problem of Noise-Induced Exit”, Stochastic Analysis, Control, Optimization and Applications, A Volume in Honor of W. H. Fleming Series: Systems Control, Foundations Applications, eds. McEneaney William M., Yin G. George, Zhang Qing, Birkhäuser Boston Inc., Boston, 1999, 269–287, 637 pp. | DOI | MR | Zbl

[50] B. Perthame, “Perturbed dynamical system with an attracting singularity and weak viscosity limits in Hamilton-Jacobi equations”, Transactions of the AMS, 317:2 (1990), 723–748 | DOI | MR | Zbl

[51] Vasileva A. B., Tupchiev V. A., “O periodicheskikh resheniyakh sistem differentsialnykh uravnenii s malym parametrom pri proizvodnoi, blizkikh k razryvnym”, Dokl. AN SSSR, 178:4 (1968), 767–770 | MR | Zbl

[52] F. W. J. Olver, Introduction to asymptotics and special functions, Academic Press, New York–London, 1974 | MR | Zbl

[53] Fedoryuk M. V., Metod perevala, Nauka, M., 1977 | MR | Zbl

[54] Moiseev N. N., Asimptoticheskie metody v nelineinoi mekhanike, Nauka, M., 1981 | MR | Zbl

[55] Sternin B. Yu., Shatalov V. E., “Metod perevala i resurgentnyi analiz”, Matem. zametki, 61:2 (1997), 278–296 | DOI | MR | Zbl

[56] O. Costin, M. D. Kruskal, Optimal Uniform Estimates and Rigorous Asymptotics Beyond all Orders for a Class of Ordinary Differential Equations, 16 Aug. 2006, arXiv: math/0608412v1[math.CA] | MR

[57] E. B. Davies, M. Plum, “Spectral pollution”, IMA J. Numer. Anal., 24 (2004), 417–438 | DOI | MR | Zbl

[58] Ilin A. M., “Raznostnaya skhema dlya differentsialnogo uravneniya s malym parametrom pri starshei proizvodnoi”, Matem. zametki, 6:2 (1969), 237–248 | MR | Zbl