Geodesic
The method of generalized Cole–Hopf substitutions and new examples of linearizable nonlinear evolution equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 158 (2009) no. 1, pp. 58-71 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We propose a new approach for constructing nonlinear evolution equations in matrix form that are integrable via substitutions similar to the Cole–Hopf substitution linearizing the Burgers equation. We use this new approach to find new integrable nonlinear evolution equations and their hierarchies.
Keywords: exactly integrable nonlinear equation, Burgers-type equation
Mots-clés : Cole–Hopf substitution. 
@article{TMF_2009_158_1_a3,
     author = {V. M. Zhuravlev},
     title = {The~method of generalized {Cole{\textendash}Hopf} substitutions and new examples of linearizable nonlinear evolution equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {58--71},
     year = {2009},
     volume = {158},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_158_1_a3/}
}
TY  - JOUR
AU  - V. M. Zhuravlev
TI  - The method of generalized Cole–Hopf substitutions and new examples of linearizable nonlinear evolution equations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2009
SP  - 58
EP  - 71
VL  - 158
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2009_158_1_a3/
LA  - ru
ID  - TMF_2009_158_1_a3
ER  - 
%0 Journal Article
%A V. M. Zhuravlev
%T The method of generalized Cole–Hopf substitutions and new examples of linearizable nonlinear evolution equations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2009
%P 58-71
%V 158
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2009_158_1_a3/
%G ru
%F TMF_2009_158_1_a3
V. M. Zhuravlev. The method of generalized Cole–Hopf substitutions and new examples of linearizable nonlinear evolution equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 158 (2009) no. 1, pp. 58-71. http://geodesic.mathdoc.fr/item/TMF_2009_158_1_a3/

[1] J. M. Burgers, The Nonlinear Diffusion Equation. Asymptotic Solutions and Statistical Problems, Reidel, Dordrecht, Holland, 1974 | Zbl

[2] Dzh. Uizem, Lineinye i nelineinye volny, Mir, M., 1978 | MR | Zbl

[3] C. S. Gardner, J. M. Green, M. D. Kruskal, R. M. Miura, Phys. Rev. Lett., 19:19 (1967), 1095–1097 | DOI | Zbl

[4] S. I. Svinolupov, TMF, 65:2 (1985), 303–307 | DOI | MR | Zbl

[5] N. Kh. Ibragimov, Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR | MR | Zbl

[6] A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, UMN, 42:4(256) (1987), 3–53 | DOI | MR | Zbl

[7] V. V. Sokolov, UMN, 43:5(263) (1988), 133–163 | DOI | MR | Zbl

[8] S. I. Svinolupov, V. V. Sokolov, UMN, 47:3(285) (1992), 115–146 | DOI | MR | Zbl

[9] S. I. Svinolupov, V. V. Sokolov, TMF, 100:2 (1994), 214–218 | DOI | MR | Zbl

[10] V. E. Adler, A. B. Shabat, R. I. Yamilov, TMF, 125:3 (2000), 355–424 | DOI | MR | Zbl

[11] S. Ya. Startsev, TMF, 116:3 (1999), 336–348 | DOI | MR | Zbl

[12] S. Ya. Startsev, TMF, 127:1 (2001), 63–74 | DOI | MR | Zbl

[13] F. Calogero, “Why are certain nonlinear PDEs both widely applicable and integrable?”, What is Integrability?, Springer Ser. Nonlinear Dynam., ed. V. E. Zhakharov, Springer, Berlin, 1991, 1–62 | DOI | MR | Zbl

[14] A. I. Zenchuk, P. M. Santini, On the remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf–Cole transformation, arXiv: 0801.3945 | MR

[15] P. M. Santini, Inverse Problems, 8:2 (1992), 285–301 | DOI | MR | Zbl

[16] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov: Metod obratnoi zadachi, Nauka, M., 1980 | MR | MR | Zbl

[17] R. Dodd, Dzh. Eilbek, Dzh. Gibbon, Kh. Morrio, Solitony i nelineinye volnovye uravneniya, Mir, M., 1988 | MR | MR | Zbl

[18] V. M. Zhuravlev, ZhETF, 110:6 (1996), 2243–2262 | DOI | MR

[19] V. M. Zhuravlev, Nelineinye volny v mnogokomponentnykh sistemakh s dispersiei i diffuziei, Izd. UlGU, Ulyanovsk, 2002

[20] V. M. Zhuravlev, A. V. Nikitin, Treti Kurdyumovskie chteniya “Sinergetika v estestvennykh naukakh”, Materialy mezhd. mezhdistsipl. nauch. konf. (Tver, 19–22 aprelya 2007), Tverskoi gos. un-t, Tver, 2007, 60–61

[21] V. M. Zhuravlev, A. V. Nikitin, Nelineinyi mir, 5:9 (2007), 603–611

[22] V. M. Zhuravlev, D. A. Zinovev, Pisma v ZhETF, 87:5 (2008), 314–318

[23] B. A. Uryukov, Teplofizika i aeromekhanika, 6:3 (1999), 421–424

[24] E. Skott, Volny v aktivnykh i nelineinykh sredakh v prilozhenii k elektronike, Sovetskoe radio, M., 1977

[25] V. M. Zhuravlev, ZhETF, 129:3 (2006), 587–604 ; Письма в ЖЭТФ, 75:1 (2002), 11–16 | DOI | DOI

[26] L. D. Landau, E. M. Lifshits, Mekhanika sploshnykh sred, Gostekhizdat, M., 1953 | MR | Zbl

[27] O. V. Rudenko, S. I. Soluyan, Teoreticheskie osnovy nelineinoi akustiki, Nauka, M., 1975 | MR | MR | Zbl