01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
18.06.1946
Keywords:
dynamical systems,
Lie–Backlund groups,
integrability of hyperbolic systems,
equation of Liouville type.
Subject:
Complete lists of Klein-Gordon equations and their generalizations were obtained possessing higher symmetries. An effective criterion of Darboux integrability of differential equations was established in terms of the Laplace invariants. The classical problem of the classification of nonlinear hyperbolik equations possessing integrals was solved. A new method of integration of such equations was suggested.
Biography
Graduated from Faculty of Mathematics and Mechanics of Novosibirsk State University (NSU) in 1969 (department of differential equations). Ph.D. thesis was defended in 1975. D.Sci. thesis was defended in 1994. A list of my works contains more than 50 titles.
A. V. Zhiber, V. V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type”, Russian Math. Surveys, 56:1 (2001), 61–101
2.
A. V. Zhiber, A. B. Shabat, “The Klein–Gordon equation with nontrivial group”, Dokl. Akad. Nauk SSSR, 247:5 (1979), 1103–1107
3.
A. V. Zhiber, “Conservation laws for the equation $u_{tt}-u_{xx}+\sin u=0$”, Funct. Anal. Appl., 11:1 (1977), 55–57
4.
A. V. Zhiber, S. Ya. Startsev, “Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems”, Math. Notes, 74:6 (2003), 803–811
5.
A. V. Zhiber, “Quasilinear hyperbolic equations with an infinite-dimensional symmetry algebra”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 33–54
6.
A. V. Zhiber, R. D. Murtazina, “On the characteristic Lie algebras for equations $u_{xy}=f(u,u_x)$”, J. Math. Sci., 151:4 (2008), 3112–3122
7.
A. M. Gurieva, A. V. Zhiber, “Laplace Invariants of Two-Dimensional Open Toda Lattices”, Theoret. and Math. Phys., 138:3 (2004), 338–355
8.
A. V. Zhiber, R. D. Murtazina, I. T. Habibullin, A. B. Shabat, “Characteristic Lie rings and integrable models in mathematical physics”, Ufimsk. Mat. Zh., 4:3 (2012), 17–85
9.
A. V. Zhiber, A. B. Shabat, “Systems of equations $u_x=p(u,\,v)$, $v_y=q(u,\,v)$ that possess symmetries”, Dokl. Akad. Nauk SSSR, 277:1 (1984), 29–33
10.
A. V. Zhiber, N. H. Ibragimov, A. B. Shabat, “Equations of Liouville type”, Dokl. Akad. Nauk SSSR, 249:1 (1979), 26–29
11.
Mariya N. Kuznetsova, Asli Pekcan, Anatoliy V. Zhiber, “The Klein–Gordon Equation and Differential Substitutions of the Form $v=\varphi(u,u_x,u_y)$”, SIGMA, 8 (2012), 90 , 37 pp., arXiv: 1111.7255
A. V. Zhiber, M. N. Kuznetsova, “Integrals and characteristic Lie rings of semi-discrete systems of equations”, Ufa Math. J., 13:2 (2021), 22–32
13.
A. V. Zhiber, V. V. Sokolov, “New example of a nonlinear hyperbolic equation possessing integrals”, Theoret. and Math. Phys., 120:1 (1999), 834–839
14.
A. V. Zhiber, O. S. Kostrigina, “Goursat problem for nonlinear hyperbolic systems with integrals of the first and second order”, Ufimsk. Mat. Zh., 3:3 (2011), 67–79
15.
Anatoly V. Zhiber, Olga S. Kostrigina, “Characteristic algebras of nonlinear hyperbolic systems of equations”, J. Sib. Fed. Univ. Math. Phys., 3:2 (2010), 173–184
16.
A. V. Zhiber, Yu. G. Mihaylova, “Algorithm of building the solution of hyperbolic system of equation”, Ufimsk. Mat. Zh., 1:3 (2009), 28–45
17.
M. Gürses, A. V. Zhiber, I. T. Habibullin, “Characteristic Lie rings of differential equations”, Ufimsk. Mat. Zh., 4:1 (2012), 53–62
18.
A. V. Zhiber, Yu. G. Mikhailova, “On hyperbolic systems of equations with zero generalized Laplace invariants”, Proc. Steklov Inst. Math. (Suppl.), 261, suppl. 1 (2008), S154–S164
19.
A. V. Zhiber, A. M. Yur'eva, “Special class of of Liouville-type hyperbolic equations”, J. Math. Sci. (N. Y.), 236:6 (2019), 594–602
20.
G. T. Bulgakova, T. A. Faizullin, A. V. Zhiber, “Non-equilibrium two phase filtration”, Matem. Mod., 18:10 (2006), 19–38
21.
A. V. Zhiber, “Systems of equations of $n$-waves and nonlinear Schrödinger equations from the group-theoretical point of view”, Theoret. and Math. Phys., 52:3 (1982), 882–888
22.
A. V. Zhiber, A. B. Shabat, Differ. Uravn., 6:1 (1970), 137–146
23.
Zhiber A. V, Yureva A. M, “On a certain class of hyperbolic equations with second-order intagrals”, Journal of Mathematical Sciences, 252:2 (2021), 168-174
A. V. Zhiber, A. M. Yur'eva, “On a Certain Class of Hyperbolic Equations with Second-order Integrals”, J. Math. Sci. (N. Y.), 252:2 (2021), 168–174
26.
A. V. Zhiber, S. N. Kamaeva, “Construction of exact solution to sine-Gordon equation on the base of its characteristic Lie ring”, Ufa Math. Journal, 8:3 (2016), 49–57
27.
Yu. G. Voronova, A. V. Zhiber, “Symmetries and Goursat problem for system of equations $u_{xy}=e^{u+v}u_y$, $v_{xy}=-e^{u+v}v_y$”, Ufa Math. Journal, 5:3 (2013), 20–27
28.
Zhiber A. V., Tsirelman N. M, “Determining temperature fields in a spatially inhomogeneous nonlinear medium”, Journal of Mathematical Sciences, 257:3 (2021), 305-312
29.
R. N. Gaisina, A. V. Zhiber, “O zakonakh sokhraneniya dlya odnogo klassa evolyutsionnykh sistem uravnenii”, Vestnik BashGU, 25:1 (2020), 4–11
30.
A. V. Zhiber, N. M. Tsirelman, “Determining temperature fields in a spatially inhomogeneous nonlinear medium”, Complex Analysis. Mathematical Physics, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 162, VINITI, Moscow, 2019, 34–41
31.
A. V. Zhiber, D. R. Toshmurodova, “Kharakteristicheskie koltsa Li giperbolicheskikh sistem uravnenii, porozhdennykh uravneniem Penleve II”, Sovremennaya matematika i ee prilozheniya, Materialy Mezhdunarodnoi nauchno-prakticheskoi konferentsii (Ufa, 18–20 maya 2017 g.), RITs BashGU, Ufa, 2017, 328–332
32.
A. V. Zhiber, O. S. Kostrigina, “Kharakteristicheskie koltsa Li i simmetrii differentsialnykh uravnenii Penleve I i Penleve III”, Vestnik Bashkirsk. un-ta, 21:3 (2016), 566-574
33.
A. V. Zhiber, D. R. Toshmurodova, “Vysshie simmetrii giperbolicheskoi sistemy uravnenii, porozhdennoi uravneniem Penleve IV”, MATEMATIChESKOE MODELIROVANIE PROTsESSOV I SISTEM: Materialy V Vseros. nauch.-prakt. konf., priurochennoi k 110-letiyu so dnya rozhdeniya akademika A.N. Tikhonova, Ch. II. (17-19 noyabrya 2016g., g. Sterlitamak.), eds. S.A. Mustafina, IITs Sterlitamakskogo filiala BashGU, 2016, 215–219
34.
N. M. Tsirelman, A. V. Zhiber, Analiticheskoe opredelenie temperaturnykh polei v prostranstvenno-neodnorodnoi i nelineinoi srede, Innovatsionnoe mashinostroenie, Moskva, 2016 , 286 pp.
35.
Yu. G. Voronova, A. V. Zhiber, “On a class of hyperbolic equations with third-order integrals”, Ufa Math. J., 15:2 (2023), 20–30
36.
A. B. Zhiber, G. Z. Murtazina, N. A. Sidelnikova, Osnovnye differentsialnye uravneniya matematicheskoi fiziki, RITs BashGU, Ufa, 2020 , 303 pp.
37.
A. V. Zhiber, G. Z. Mukhametova, N. A. Sidelnikova, Kraevye zadachi dlya osnovnykh tipov uravnenii matematicheskoi fiziki, RITs BashGU, Ufa, 2018 , 295 pp., ISBN 978-5-7477-4861-3
38.
A. V. Zhiber, R. D. Murtazina, I. T. Khabibullin, A. B. Shabat, Uravneniya matematicheskoi fiziki. Nelineinye integriruemye uravneniya, Uchebnoe posobie dlya bakalavriata i magistratury, Universitety Rossii, 2-e izd., Izdatelstvo Yurait, Moskva, 2017 , 375 pp.
39.
V. A. Baikov, A. V. Zhiber, Uravneniya matematicheskoi fiziki, uchebnik i praktikum dlya akademicheskogo bakalavriata, 2-e izd., Izdatelstvo Yurait, Moskva, 2017 , 255 pp. https://www.biblio-online.ru/book/E4CC7C7D-F3F0-4CD2-8080-579C7F19DA97
40.
A. V. Abanin, S. N. Askhabov, A. B. Borisov, R. A. Bostanov, A. V. Zhiber, V. E. Zakharov, S. B. Klimentov, Yu. F. Korobeinik, A. G. Kusraev, S. S. Kutateladze, A. V. Mikhailov, A. P. Soldatov, Kh. G. Umarov, S. M. Umarkhadzhiev, “Aleksei Borisovich Shabat (on the occasion of his 80th anniversary)”, Vladikavkaz. Mat. Zh., 19:3 (2017), 83–85
41.
A. V. Abanin, S. N. Askhabov, A. B. Borisov, R. A. Bostanov, A. V. Zhiber, V. E. Zakharov, S. B. Klimentov, Yu. F. Korobeinik, A. G. Kusraev, S. S. Kutateladze, A. V. Mikhailov, A. P. Soldatov, Kh. G. Umarov, S. M. Umarkhadzhiev, “Shabat Aleksei Borisovich (on his 75th birthday)”, Vladikavkaz. Mat. Zh., 14:2 (2012), 71–73
42.
V. E. Adler, P. Winternitz, R. N. Garifullin, A. V. Zhiber, D. Levi, A. V. Mikhailov, I. Kh. Musin, F. W. Nijhoff, V. V. Sokolov, B. I. Suleimanov, E. V. Ferapontov, A. P. Fordy, I. T. Habibullin, I. Yu. Cherdantsev, R. A. Sharipov, R. S. Yulmukhametov, “In memory of Yamilov Ravil Islamovich”, Ufa Math. J., 12:3 (2020), 119–120
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