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Shabat, Alexey Borisovich
(1937–2020)
Professor
Doctor of physico-mathematical sciences (1974)
Speciality: 01.01.00 (Mathematics)
Birth date: 8.08.1937
Keywords: integrable systems, solitons.
UDC: 517.9, 517.958
   
Main publications:
  1. V.E. Zakharov, A.B. Shabat,, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Zh. Eksp. Teor. Fiz., 61:1 (1971), 12
  2. A.B. Shabat, “Inverse scattering problem for a system of differential equations”, Funct. Anal. Appl. (English translation), 9:3 (1975), 5
  3. V.E. Adler, A.B. Shabat, R.I. Yamilov, “Symmetry approach to the integrability problem”, Theor. Math. Phys., 125:3 (2000), 9

https://www.mathnet.ru/eng/person18749
List of publications on Google Scholar
https://mathscinet.ams.org/mathscinet/MRAuthorID/212223
https://elibrary.ru/author_items.asp?authorid=103607

Publications in Math-Net.Ru Citations
2020
1. R. Ch. Kulaev, A. B. Shabat, “Darboux system and separation of variables in the Goursat problem for a third order equation in $\mathbb {R}^3$”, Izv. Vyssh. Uchebn. Zaved. Mat., 2020, no. 4,  43–53  mathnet; Russian Math. (Iz. VUZ), 64:4 (2020), 35–43  isi  scopus 2
2. A. E. Artisevich, B. S. Bychkov, A. B. Shabat, “Chebyshev polynomials, Catalan numbers, and tridiagonal matrices”, TMF, 204:1 (2020),  3–9  mathnet  mathscinet  elib; Theoret. and Math. Phys., 204:1 (2020), 837–842  isi  scopus 2
3. A. A. Allahverdyan, A. B. Shabat, “Products of eigenfunctions and Wronskians”, Ufimsk. Mat. Zh., 12:2 (2020),  3–9  mathnet; Ufa Math. J., 12:2 (2020), 3–9  isi
4. A. E. Artisevich, A. B. Shabat, “Three theorems on Vandermond matrices”, Vladikavkaz. Mat. Zh., 22:1 (2020),  5–12  mathnet 1
2019
5. V. E. Adler, A. B. Shabat, “Some exact solutions of the Volterra lattice”, TMF, 201:1 (2019),  37–53  mathnet  mathscinet  elib; Theoret. and Math. Phys., 201:1 (2019), 1442–1456  isi  scopus 7
6. R. Ch. Kulaev, A. B. Shabat, “Conservation laws for Volterra chain with initial step-like condition”, Ufimsk. Mat. Zh., 11:1 (2019),  61–67  mathnet; Ufa Math. J., 11:1 (2019), 63–69  isi  scopus 1
2018
7. R. Ch. Kulaev, A. K. Pogrebkov, A. B. Shabat, “Darboux system as three-dimensional analog of Liouville equation”, Izv. Vyssh. Uchebn. Zaved. Mat., 2018, no. 12,  60–69  mathnet; Russian Mathematics, 62:12 (2018), 50–58  isi  scopus 1
8. V. E. Adler, A. B. Shabat, “Volterra chain and Catalan numbers”, Pis'ma v Zh. Èksper. Teoret. Fiz., 108:12 (2018),  834–837  mathnet  elib; JETP Letters, 108:12 (2018), 825–828  isi  scopus 4
9. R. Ch. Kulaev, A. K. Pogrebkov, A. B. Shabat, “Darboux system: Liouville reduction and an explicit solution”, Trudy Mat. Inst. Steklova, 302 (2018),  268–286  mathnet  mathscinet  elib; Proc. Steklov Inst. Math., 302 (2018), 250–269  isi  scopus 3
10. A. B. Shabat, V. E. Adler, “Cartan matrices in the Toda–Darboux chain theory”, TMF, 196:1 (2018),  22–29  mathnet  mathscinet  elib; Theoret. and Math. Phys., 196:1 (2018), 957–964  isi  scopus
2017
11. A. B. Shabat, “Constructive scattering theory”, TMF, 193:1 (2017),  15–24  mathnet  mathscinet  elib; Theoret. and Math. Phys., 193:1 (2017), 1420–1428  isi  scopus
12. R. Ch. Kulaev, A. B. Shabat, “Some properties of Jost functions for Schrödinger equation with distribution potential”, Ufimsk. Mat. Zh., 9:4 (2017),  60–73  mathnet  elib; Ufa Math. J., 9:4 (2017), 59–71  isi  scopus 1
13. A. B. Shabat, M. Kh. Efendiev, “On applications of Faà-di-Bruno formula”, Ufimsk. Mat. Zh., 9:3 (2017),  132–137  mathnet  elib; Ufa Math. J., 9:3 (2017), 131–136  isi  scopus 3
2016
14. V. V. Sokolov, A. B. Shabat, “Rational solutions of a Riccati equation”, Uspekhi Mat. Nauk, 71:4(430) (2016),  189–190  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 71:4 (2016), 787–789  isi  scopus
15. A. B. Shabat, “Functional Cantor equation”, TMF, 189:3 (2016),  355–361  mathnet  mathscinet  elib; Theoret. and Math. Phys., 189:3 (2016), 1712–1717  isi  scopus 1
16. M. Sh. Badakhov, A. B. Shabat, “Darboux transformations in the inverse scattering problem”, Ufimsk. Mat. Zh., 8:4 (2016),  43–52  mathnet  elib; Ufa Math. J., 8:4 (2016), 42–51  isi  scopus 2
2015
17. A. B. Shabat, “Inverse spectral problem for delta potentials”, Pis'ma v Zh. Èksper. Teoret. Fiz., 102:9 (2015),  705–708  mathnet  elib; JETP Letters, 102:9 (2015), 620–623  isi  scopus 3
18. A. B. Shabat, “Difference Schrödinger equation and quasisymmetric polynomials”, TMF, 184:2 (2015),  200–211  mathnet  mathscinet  elib; Theoret. and Math. Phys., 184:2 (2015), 1067–1077  isi  scopus 3
19. A. B. Shabat, “Scattering theory for delta-type potentials”, TMF, 183:1 (2015),  105–119  mathnet  mathscinet  elib; Theoret. and Math. Phys., 183:1 (2015), 540–552  isi  scopus 4
2014
20. A. B. Shabat, “Rational interpolation and solitons”, TMF, 179:3 (2014),  303–316  mathnet  mathscinet  elib; Theoret. and Math. Phys., 179:3 (2014), 637–648  isi  scopus 5
21. M. Sh. Badakhov, O. Y. Veremeenko, A. B. Shabat, “Asymptotic behavior of generalized eigenvalues of the Schrödinger operator”, Vladikavkaz. Mat. Zh., 16:4 (2014),  9–15  mathnet 2
2013
22. A. B. Shabat, “Periodic solutions of the Hopf equation”, TMF, 177:2 (2013),  222–230  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 177:2 (2013), 1471–1478  isi  elib  scopus 1
2012
23. A. B. Shabat, Z. S. El'kanova, A. B. Urusova, “Two-sided Darboux transformations”, TMF, 173:2 (2012),  207–218  mathnet  mathscinet  elib; Theoret. and Math. Phys., 173:2 (2012), 1507–1517  isi  elib  scopus 4
24. V. E. Adler, V. G. Marikhin, A. B. Shabat, “Quantum tops as examples of commuting differential operators”, TMF, 172:3 (2012),  355–374  mathnet  mathscinet  elib; Theoret. and Math. Phys., 172:3 (2012), 1187–1205  isi  elib  scopus 1
25. R. A. Gabiev, A. B. Shabat, “Differential operators commuting in the principal part”, TMF, 171:1 (2012),  18–25  mathnet  mathscinet  elib; Theoret. and Math. Phys., 171:1 (2012), 435–441  isi  elib  scopus 1
26. 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  mathnet  mathscinet 18
27. A. B. Shabat, “Symmetrical polynomials and conservation laws”, Vladikavkaz. Mat. Zh., 14:4 (2012),  83–94  mathnet 5
2011
28. M. S. Akbasheva, A. B. Shabat, “Theorem on commutation in the principal part”, Ufimsk. Mat. Zh., 3:4 (2011),  3–7  mathnet  zmath
29. A. B. Shabat, Z. S. Elkanova, “Commuting differential operators in two-dimension”, Ufimsk. Mat. Zh., 3:2 (2011),  91–98  mathnet  zmath; Ufa Math. J., 3:2 (2011), 89–95 2
2010
30. A. B. Shabat, Z. S. El'kanova, “Commuting differential operators”, TMF, 162:3 (2010),  334–344  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 162:3 (2010), 276–285  isi  scopus 7
2009
31. R. N. Garifullin, A. B. Shabat, “The structure of polynomial conservation laws”, TMF, 161:3 (2009),  318–326  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 161:3 (2009), 1590–1598  isi  scopus 1
2008
32. M. Kh. Chankaev, A. B. Shabat, “A nonlinear eigenvalue problem”, TMF, 157:2 (2008),  175–187  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 157:2 (2008), 1514–1524  isi  scopus 20
2007
33. V. E. Adler, A. B. Shabat, “Model equation of the theory of solitons”, TMF, 153:1 (2007),  29–45  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 153:1 (2007), 1373–1387  isi  scopus 20
2006
34. Vsevolod E. Adler, Alexey B. Shabat, “On the One Class of Hyperbolic Systems”, SIGMA, 2 (2006), 093, 17 pp.  mathnet  mathscinet  zmath  isi  scopus 3
35. V. E. Adler, A. B. Shabat, “Dressing chain for the acoustic spectral problem”, TMF, 149:1 (2006),  32–46  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 149:1 (2006), 1324–1337  isi  scopus 8
2005
36. F. Musso, A. B. Shabat, “Elementary Darboux Transformations and Factorization”, TMF, 144:1 (2005),  143–152  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 144:1 (2005), 1004–1013  isi 1
2004
37. L. Martínez Alonso, A. B. Shabat, “Hydrodynamic Reductions and Solutions of a Universal Hierarchy”, TMF, 140:2 (2004),  216–229  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 140:2 (2004), 1073–1085  isi 85
2003
38. A. B. Shabat, “Universal Models of Soliton Hierarchies”, TMF, 136:2 (2003),  197–208  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 136:2 (2003), 1066–1076  isi 5
2001
39. V. E. Adler, V. G. Marikhin, A. B. Shabat, “Lagrangian Chains and Canonical Bäcklund Transformations”, TMF, 129:2 (2001),  163–183  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 129:2 (2001), 1448–1465  isi 23
2000
40. V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, TMF, 125:3 (2000),  355–424  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 125:3 (2000), 1603–1661  isi 128
1999
41. A. B. Shabat, “Third version of the dressing method”, TMF, 121:1 (1999),  165–176  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 121:1 (1999), 1397–1408  isi 20
42. V. G. Marikhin, A. B. Shabat, “Integrable lattices”, TMF, 118:2 (1999),  217–228  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 118:2 (1999), 173–182  isi 15
1998
43. V. E. Adler, A. B. Shabat, “First integrals of generalized Toda chains”, TMF, 115:3 (1998),  349–357  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 115:3 (1998), 639–646  isi 5
1997
44. V. E. Adler, A. B. Shabat, “Generalized Legendre transformations”, TMF, 112:2 (1997),  179–194  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 112:2 (1997), 935–948  isi 28
45. V. E. Adler, A. B. Shabat, “On the one class of the Toda chains”, TMF, 111:3 (1997),  323–334  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 111:3 (1997), 647–657  isi 27
46. V. E. Adler, I. T. Habibullin, A. B. Shabat, “Boundary value problem for the KdV equation on a half-line”, TMF, 110:1 (1997),  98–113  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 110:1 (1997), 78–90  isi 23
1995
47. A. P. Fordy, A. B. Shabat, A. P. Veselov, “Factorization and Poisson correspondences”, TMF, 105:2 (1995),  225–245  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 105:2 (1995), 1369–1386  isi 12
48. A. B. Shabat, “On the Laplace–Darboux theory of transformations”, TMF, 103:1 (1995),  170–175  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 103:1 (1995), 482–485  isi 14
1994
49. A. Degasperis, A. B. Shabat, “Construction of reflectionless potentials with infinite discrete spectrum”, TMF, 100:2 (1994),  230–247  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 100:2 (1994), 970–984  isi 18
50. A. B. Shabat, “Discrete symmetries and solitons”, TMF, 99:3 (1994),  537–544  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 99:3 (1994), 783–789  isi
1993
51. A. P. Veselov, A. B. Shabat, “Dressing Chains and Spectral Theory of the Schrödinger Operator”, Funktsional. Anal. i Prilozhen., 27:2 (1993),  1–21  mathnet  mathscinet  zmath; Funct. Anal. Appl., 27:2 (1993), 81–96  isi 223
1990
52. A. B. Shabat, R. I. Yamilov, “Symmetries of nonlinear lattices”, Algebra i Analiz, 2:2 (1990),  183–208  mathnet  mathscinet  zmath; Leningrad Math. J., 2:2 (1991), 377–400 38
1988
53. V. P. Gerdt, A. Yu. Zharkov, S. I. Svinolupov, A. B. Shabat, “The use of computer algebra to investigate the integrability of nonlinear evolution systems”, Zh. Vychisl. Mat. Mat. Fiz., 28:11 (1988),  1674–1684  mathnet  mathscinet  zmath; U.S.S.R. Comput. Math. Math. Phys., 28:6 (1988), 50–57
1987
54. A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “On an extension of the module of invertible transformations”, Dokl. Akad. Nauk SSSR, 295:2 (1987),  288–291  mathnet  mathscinet  zmath; Dokl. Math., 36:1 (1988), 60–63 1
55. A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems”, Uspekhi Mat. Nauk, 42:4(256) (1987),  3–53  mathnet  mathscinet  zmath; Russian Math. Surveys, 42:4 (1987), 1–63  isi 190
1986
56. A. V. Mikhailov, A. B. Shabat, “Integrability conditions for systems of two equations of the form $u_t=A(u)u_{xx}+F(u,u_x)$. II”, TMF, 66:1 (1986),  47–65  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 66:1 (1986), 31–44  isi 15
1985
57. A. V. Mikhailov, A. B. Shabat, “Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I”, TMF, 62:2 (1985),  163–185  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 62:2 (1985), 107–122  isi 22
1984
58. 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  mathnet  mathscinet  zmath 18
1982
59. A. N. Leznov, V. G. Smirnov, A. B. Shabat, “The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems”, TMF, 51:1 (1982),  10–21  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 51:1 (1982), 322–330  isi 72
1980
60. N. Kh. Ibragimov, A. B. Shabat, “Infinite Lie–Beklund algebras”, Funktsional. Anal. i Prilozhen., 14:4 (1980),  79–80  mathnet  mathscinet  zmath; Funct. Anal. Appl., 14:4 (1980), 313–315  isi 49
61. V. V. Sokolov, A. B. Shabat, “$(L,A)$-Pairs and a Ricatti type substitution”, Funktsional. Anal. i Prilozhen., 14:2 (1980),  79–80  mathnet  mathscinet  zmath; Funct. Anal. Appl., 14:2 (1980), 148–150 21
62. N. Kh. Ibragimov, A. B. Shabat, “Evolutionary equations with nontrivial Lie–Bäcklund group”, Funktsional. Anal. i Prilozhen., 14:1 (1980),  25–36  mathnet  mathscinet  zmath; Funct. Anal. Appl., 14:1 (1980), 19–28 67
1979
63. A. V. Zhiber, N. H. Ibragimov, A. B. Shabat, “Equations of Liouville type”, Dokl. Akad. Nauk SSSR, 249:1 (1979),  26–29  mathnet  mathscinet  zmath 19
64. A. V. Zhiber, A. B. Shabat, “The Klein–Gordon equation with nontrivial group”, Dokl. Akad. Nauk SSSR, 247:5 (1979),  1103–1107  mathnet  mathscinet 50
65. N. H. Ibragimov, A. B. Shabat, “Korteweg–de Vries equation from the group standpoint”, Dokl. Akad. Nauk SSSR, 244:1 (1979),  57–61  mathnet  mathscinet 16
66. A. B. Shabat, “An inverse scattering problem”, Differ. Uravn., 15:10 (1979),  1824–1834  mathnet  mathscinet  zmath 31
67. V. E. Zakharov, A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II”, Funktsional. Anal. i Prilozhen., 13:3 (1979),  13–22  mathnet  mathscinet  zmath; Funct. Anal. Appl., 13:3 (1979), 166–174 543
1975
68. A. B. Shabat, “Inverse-scattering problem for a system of differential equations”, Funktsional. Anal. i Prilozhen., 9:3 (1975),  75–78  mathnet  mathscinet  zmath; Funct. Anal. Appl., 9:3 (1975), 244–247 40
1974
69. V. E. Zakharov, A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I”, Funktsional. Anal. i Prilozhen., 8:3 (1974),  43–53  mathnet  mathscinet  zmath; Funct. Anal. Appl., 8:3 (1974), 226–235 622
1973
70. A. B. Shabat, “On the Korteweg–de Vries equation”, Dokl. Akad. Nauk SSSR, 211:6 (1973),  1310–1313  mathnet  mathscinet  zmath 13
1972
71. A. B. Shabat, “On a class of equations of Wiener-Hopf type”, Dokl. Akad. Nauk SSSR, 205:3 (1972),  546–549  mathnet  mathscinet  zmath
72. A. B. Shabat, “A one-dimensional scattering theory. I”, Differ. Uravn., 8:1 (1972),  164–178  mathnet  mathscinet  zmath
1971
73. A. B. Shabat, “A generalized Fourier transform and asymptotic behavior of the solutions of the telegrapher's equation”, Differ. Uravn., 7:1 (1971),  157–165  mathnet  mathscinet  zmath
1970
74. A. V. Zhiber, A. B. Shabat, “The Cauchy problem for the nonlinear Schrödinger equation”, Differ. Uravn., 6:1 (1970),  137–146  mathnet  mathscinet  zmath 2
1966
75. A. B. Shabat, “An analog of the Rado–Cartan theorem for solutions of elliptic equations”, Dokl. Akad. Nauk SSSR, 166:6 (1966),  1304–1307  mathnet  mathscinet  zmath
1965
76. A. B. Shabat, “On a property of solutions of elliptic equations of second order”, Dokl. Akad. Nauk SSSR, 163:2 (1965),  303–305  mathnet  mathscinet  zmath
77. A. B. Shabat, “The removability of level sets for solutions to elliptic equations”, Dokl. Akad. Nauk SSSR, 160:5 (1965),  1032–1035  mathnet  mathscinet  zmath
1963
78. A. B. Shabat, “On two pasting problems”, Dokl. Akad. Nauk SSSR, 150:6 (1963),  1242–1245  mathnet  mathscinet  zmath 2
79. Yu. P. Krasovskii, M. A. Lavrent'ev, N. N. Moiseev, A. M. Ter-Krikorov, A. B. Shabat, “Математические вопросы гидродинамики жидкости со свободными границами”, Prikl. Mekh. Tekh. Fiz., 4:4 (1963),  3–16  mathnet
1962
80. A. B. Shabat, “Об одной схеме плоского движения жидкости при наличии на дне траншеи”, Prikl. Mekh. Tekh. Fiz., 3:4 (1962),  68–80  mathnet
81. A. B. Shabat, “Boundary-value problems with small parameter for ordinary linear differential equations”, Uspekhi Mat. Nauk, 17:1(103) (1962),  235–241  mathnet  mathscinet  zmath 2
1999
82. A. A. Galeev, E. A. Kuznetsov, A. V. Mikhailov, Yu. E. Nesterikhin, S. P. Novikov, R. Z. Sagdeev, Ya. G. Sinai, G. I. Smirnov, V. E. Fortov, I. M. Khalatnikov, B. V. Chirikov, A. B. Shabat, “Vladimir Evgen'evich Zakharov (on his sixtieth birthday)”, UFN, 169:10 (1999),  1161–1162  mathnet; Phys. Usp., 42:10 (1999), 1063–1064  isi
1996
83. A. I. Bobenko, A. M. Il'in, S. Yu. Dobrokhotov, A. R. Its, L. A. Kalyakin, V. B. Matveev, V. Yu. Novokshenov, A. B. Shabat, “Ramil' Faritovich Bikbaev (obituary)”, Uspekhi Mat. Nauk, 51:1(307) (1996),  133–136  mathnet  mathscinet; Russian Math. Surveys, 51:1 (1996), 129–133  isi

Presentations in Math-Net.Ru
1. Cartan matrices in the theory of Toda-Darboux chains
A. B. Shabat
Seminar of the Department of Theoretical Physics, Steklov Mathematical Institute of RAS
February 21, 2018 14:00
2. Система Дарбу и ее обобщения
A. B. Shabat
Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS (Novikov Seminar)
October 4, 2017 18:30
3. О конструктивной теории рассеяния
A. B. Shabat
Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS (Novikov Seminar)
January 18, 2017 14:00
4. О собственных функциях операторов Эйлера
A. B. Shabat
Seminar of the Department of Theoretical Physics, Steklov Mathematical Institute of RAS
January 22, 2014 14:00
5. Об одной нелинейной задаче на собственные значения
A. B. Shabat
Seminar of the Department of Theoretical Physics, Steklov Mathematical Institute of RAS
February 20, 2008 14:00

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