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.
Yu. G. Voronova, A. V. Zhiber, “On a class of hyperbolic equations with third-order integrals”, Ufa Math. J., 15:2 (2023), 20–30
2021
2.
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
3.
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
Zhiber A. V., Tsirelman N. M, “Determining temperature fields in a spatially inhomogeneous nonlinear medium”, Journal of Mathematical Sciences, 257:3 (2021), 305-312
2020
5.
R. N. Gaisina, A. V. Zhiber, “O zakonakh sokhraneniya dlya odnogo klassa evolyutsionnykh sistem uravnenii”, Vestnik BashGU, 25:1 (2020), 4–11
6.
A. B. Zhiber, G. Z. Murtazina, N. A. Sidelnikova, Osnovnye differentsialnye uravneniya matematicheskoi fiziki, RITs BashGU, Ufa, 2020 , 303 pp.
7.
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
2019
8.
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
9.
A. V. Zhiber, A. M. Yureva, “Special class of Liouville-type hyperbolic equations”, J. Math. Sci., 236:6 (2019), 594–602
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
2018
11.
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
2019
12.
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
2017
13.
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
14.
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.
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
2016
17.
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
18.
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
19.
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
20.
N. M. Tsirelman, A. V. Zhiber, Analiticheskoe opredelenie temperaturnykh polei v prostranstvenno-neodnorodnoi i nelineinoi srede, Innovatsionnoe mashinostroenie, Moskva, 2016 , 286 pp.
2013
21.
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
2012
22.
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, 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
24.
M. Gürses, A. V. Zhiber, I. T. Habibullin, “Characteristic Lie rings of differential equations”, Ufimsk. Mat. Zh., 4:1 (2012), 53–62
25.
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
2011
26.
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
2010
27.
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
2009
28.
A. V. Zhiber, Yu. G. Mihaylova, “Algorithm of building the solution of hyperbolic system of equation”, Ufimsk. Mat. Zh., 1:3 (2009), 28–45
2008
29.
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
30.
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
2006
31.
G. T. Bulgakova, T. A. Faizullin, A. V. Zhiber, “Non-equilibrium two phase filtration”, Matem. Mod., 18:10 (2006), 19–38
2004
32.
A. M. Gurieva, A. V. Zhiber, “Laplace Invariants of Two-Dimensional Open Toda Lattices”, Theoret. and Math. Phys., 138:3 (2004), 338–355
2003
33.
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
2001
34.
A. V. Zhiber, V. V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type”, Russian Math. Surveys, 56:1 (2001), 61–101
1999
35.
A. V. Zhiber, V. V. Sokolov, “New example of a nonlinear hyperbolic equation possessing integrals”, Theoret. and Math. Phys., 120:1 (1999), 834–839
1995
36.
A. V. Zhiber, “Quasilinear hyperbolic equations with an infinite-dimensional symmetry algebra”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 33–54
1984
37.
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
1982
38.
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
1979
39.
A. V. Zhiber, A. B. Shabat, “The Klein–Gordon equation with nontrivial group”, Dokl. Akad. Nauk SSSR, 247:5 (1979), 1103–1107
40.
A. V. Zhiber, N. H. Ibragimov, A. B. Shabat, “Equations of Liouville type”, Dokl. Akad. Nauk SSSR, 249:1 (1979), 26–29
1977
41.
A. V. Zhiber, “Conservation laws for the equation $u_{tt}-u_{xx}+\sin u=0$”, Funct. Anal. Appl., 11:1 (1977), 55–57
1970
42.
A. V. Zhiber, A. B. Shabat, Differ. Uravn., 6:1 (1970), 137–146
2023
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