group analys differential equations; factorizations of linear and nonlinear differential operators; autonomization and exact linearization of nonlinear equations; nonlinear evolutionary equations; dynamical systems; nonstationary problems of celestial mechanics; integration of finite terms; application of computer algebra.
Subject:
1) All mathematical laws of a modification of a mass in classical non-stationary problem of celestial mechanics (problem of Gylden–Meshcherskii) are found.
2) First systematic exposition of a method of factorization differential operators in synthesis with a method of transformation of variables. Thev outcomes are applied to an integration of the linear differential equations with variables coefficients. The solution of classical problems of G.-H. Halphen about equivalence and classification linear equations of order $n>2$ is given.
3) The exact invariant solutions for the equation Kolmogorov–Petrovskii–Piskunov and some other nonlinear heat and diffusion equations. For want of it methods of the group analysis and factorization ODE.
4) The general class of nonlinear nonautonomous ODE of order $n$ is constructed, supposing a factorization through nonlinear differential operators of the first order. This class simultaneously supposes a exact linearization to the autonomous equations. The constructed class of the equations is applied to some integrable dynamical systems: Liouvillian's systems, and some systems of a hydrodynamical type, and systems of a right body.
5) The generalization of inhering S. Lie of a nonlinear principle of superposition for solutions of nonlinear ODE is considered. The new nonlinear superposition principle is applied for effective integration of nonlinear evolutionary equations of mathematical physics.
Biography
Graduated from Faculty of Mathematics and Mechanics of V. I. Ulyanov-Lenin Kazan State University in 1958 (department of algebra). Ph.D. thesis was defended in 1967. D.Sci. thesis was presented in 1998. A list of my works contains more than 100 titles. I have led the research seminar at SSU on group analysis of differential equations.
Academician of Academy of Nonlinear Science — 2000.
Main publications:
On the method of exact linearization of autonomous ordinary differential equations. Max-Planck-Institut fuer Mathematik, Bonn, Preprint Series 2001 (73), 24 pp.
L. M. Berkovich, “Factorization as a method for finding exact invariant solutions of the Kolmogorov–Petrovskii–Piskunov equation and related equations of Semenov and Zel'dovich”, Dokl. Akad. Nauk, 322:5 (1992), 823–827; Dokl. Math., 45:1 (1992), 162–167
L. M. Berkovich, “Transformation of variables as a method for finding exact invariant solutions of the Kolmogorov–Petrovskii–Piskunov equation and related nonlinear heat equations”, Dokl. Akad. Nauk, 322:4 (1992), 635–640; Dokl. Math., 45:1 (1992), 146–151
1989
3.
L. M. Berkovich, “Second-order kindred linear differential equations”, Differ. Uravn., 25:2 (1989), 192–201; Differ. Equ., 25:2 (1989), 137–144
1987
4.
L. M. Berkovich, “Third-order reducible ordinary linear differential equations and related nonlinear equations”, Differ. Uravn., 23:5 (1987), 887–890
1986
5.
L. M. Berkovich, V. G. Tsirulik, “The differential resultant and some of its applications”, Differ. Uravn., 22:5 (1986), 750–757
6.
L. M. Berkovich, “The Halphen problem on equivalence of ordinary linear differential equations”, Uspekhi Mat. Nauk, 41:1(247) (1986), 183–184; Russian Math. Surveys, 41:1 (1986), 221–222
B. Rashayski, L. M. Berkovich, “Application of tangent transformations to first-order ordinary differential equations”, Differ. Uravn., 21:12 (1985), 2177–2181
1982
8.
L. M. Berkovich, “Transformation of Sturm–Liouville differential equations”, Funktsional. Anal. i Prilozhen., 16:3 (1982), 42–44; Funct. Anal. Appl., 16:3 (1982), 190–192
L. M. Berkovich, “The Gylden–Meshcherskii problem and the laws of variation of mass”, Dokl. Akad. Nauk SSSR, 250:5 (1980), 1088–1091
1972
10.
L. M. Berkovich, N. Kh. Rozov, “Some remarks on differential equations of the form $y^{''}+a_0(x)y=\varphi(x)y^\alpha$”, Differ. Uravn., 8:11 (1972), 2076–2079
L. M. Berkovich, “Transformations of ordinary nonlinear differential equations”, Differ. Uravn., 7:2 (1971), 353–356
1970
12.
L. M. Berkovich, “Certain classes of difference and differential equations with variable coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 1970, no. 7, 13–25
1969
13.
L. M. Berkovich, “Selfadjoint differential equations and a certain class of Euler equations”, Izv. Vyssh. Uchebn. Zaved. Mat., 1969, no. 8, 3–9
1968
14.
L. M. Berkovich, V. I. Kval'vasser, “Operator identities and certain ordinary linear differential equations of higher orders which are integrable in closed form”, Izv. Vyssh. Uchebn. Zaved. Mat., 1968, no. 5, 3–16
1967
15.
L. M. Berkovich, “Factorization of ordinary linear differential operators that can be transformed into operators with constant coefficients. II”, Izv. Vyssh. Uchebn. Zaved. Mat., 1967, no. 12, 3–14
L. M. Berkovich, “Factorization of ordinary differential operators that can be transformed into operators with constant coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 1965, no. 4, 8–16
L. M. Berkovich, N. A. Izobov, V. A. Il'in, I. T. Kiguradze, V. A. Pliss, N. Kh. Rozov, “Otakar Boruvka [1899–1995]”, Differ. Uravn., 31:10 (1995), 1770–1771; Differ. Equ., 31:10 (1995), 1744–1745
1993
18.
L. M. Berkovich, “Correction to: “Factorization as a method for finding exact invariant solutions of the Kolmogorov–Petrovskii–Piskunov equation and related equations of Semenov and Zel'dovich” [Dokl. Akad. Nauk 322 (1992), no. 5, 823–827]”, Dokl. Akad. Nauk, 329:5 (1993), 688
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