Geodesic
Self-dual Yang–Mills fields in $d=4$ and integrable systems in $1\leq d\leq 3$
Teoretičeskaâ i matematičeskaâ fizika, Tome 102 (1995) no. 3, pp. 384-419
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Ward correspondence between self-dual Yang–Mills fields and holomorphic vector bundles is used to develop a method for reducing the Lax pair for the self-duality equations of the Yang–Mills model in $d=4$ with respect to the action of continuous symmetry groups. It is well known that reductions of the self-duality equations lead to systems of nonlinear differential equations in dimension $1\leq d\leq 3$. For the integration of the reduced equations, it is necessary to find a Lax pair whose compatibility conditions is these equations. The method makes it possible to obtain systematically a Lax representation for the reduced self-duality equations. This is illustrated by a large number of examples. 
@article{TMF_1995_102_3_a7,
     author = {T. A. Ivanova and A. D. Popov},
     title = {Self-dual {Yang{\textendash}Mills} fields in $d=4$ and integrable systems in~$1\leq d\leq 3$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {384--419},
     year = {1995},
     volume = {102},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_102_3_a7/}
}
TY  - JOUR
AU  - T. A. Ivanova
AU  - A. D. Popov
TI  - Self-dual Yang–Mills fields in $d=4$ and integrable systems in $1\leq d\leq 3$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1995
SP  - 384
EP  - 419
VL  - 102
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_1995_102_3_a7/
LA  - ru
ID  - TMF_1995_102_3_a7
ER  - 
%0 Journal Article
%A T. A. Ivanova
%A A. D. Popov
%T Self-dual Yang–Mills fields in $d=4$ and integrable systems in $1\leq d\leq 3$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1995
%P 384-419
%V 102
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_1995_102_3_a7/
%G ru
%F TMF_1995_102_3_a7
T. A. Ivanova; A. D. Popov. Self-dual Yang–Mills fields in $d=4$ and integrable systems in $1\leq d\leq 3$. Teoretičeskaâ i matematičeskaâ fizika, Tome 102 (1995) no. 3, pp. 384-419. http://geodesic.mathdoc.fr/item/TMF_1995_102_3_a7/

[1] Atiyah M. F., Classical Geometry of Yang-Mills Fields, Scuola Normale Superiore, Pisa, 1979 ; Ward R. S., Wells R. O. Jr, Twistor Geometry and Field Theory, Cambridge University Press, Cambridge, 1990 | MR | Zbl | MR

[2] Ablowitz M. J., Clarkson P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambrige University Press, Cambridge, 1991 | MR | Zbl

[3] Ward R. S., Phil. Trans. R. Soc. Lond. A, 315 (1985), 451–457 ; Lect. Notes in Phys, 280, 1987, 106–116 ; Ward R. S., Twistors in Mathematics and Physics, London Mathematical Society Lecture Note Series, 156, eds. T. N. Bailey and R. J. Baston, Cambridge University Press, Cambridge, 1990, 246–259 | DOI | MR | Zbl | DOI | Zbl | MR

[4] Mason L. J., Sparling G. A. J., Phys. Lett. A, 137:1–2 (1989), 29–33 ; J. Geom. and Phys., 8 (1992), 243–271 | DOI | MR | DOI | MR | Zbl

[5] Bakas I., Depireux D. A., Mod. Phys. Lett. A, 6:5 (1991), 399–409 ; 17, 1561–1573 | DOI | MR | MR | Zbl

[6] Ablowitz M. J., Chakravarty S., Clarkson P. A., Phys. Rev. Lett., 65:9 (1990), 1085–1087 ; Chakravarty S., Ablowitz M. J., Painlevé Transcendents, their Asymptotics and Physical Applications, NATO ASI Series B, 278, eds. D. Levi and P. Winternitz, Plenum Press, New York, 1992, 331–343 | DOI | MR | Zbl | DOI | MR | Zbl

[7] Chakravarty S., Kent S., Newman E. T., J. Math. Phys., 31:9 (1990), 2253–2257 ; 33:1 (1992), 382–387 ; Strachan I. A. B., Phys. Lett. A, 154:3–4 (1991), 123–126 ; Ivanova T. A., Popov A. D., Lett. Math. Phys., 23 (1991), 29–34 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl

[8] Ivanova T. A., Popov A. D., Phys. Lett. A, 170:4 (1992), 293–299 | DOI | MR

[9] Woodhouse N. M. J., Class. Quant. Grav., 4 (1987), 799–814 ; Woodhouse N. M. J., Mason L. J., Nonlinearity, 1 (1988), 73–114 ; Fletcher J., Woodhouse N. M. J., Twistors in Mathematics and Physics, London Mathematical Society Lecture Note Series, 156, eds. T. N. Bailey, R. J. Baston, Cambridge University Press, Cambridge, 1990, 260–282 | DOI | MR | Zbl | DOI | MR | Zbl | MR

[10] Mason L. J., Woodhouse N. M. J., Nonlinearity, 6 (1993), 569–581 | DOI | MR | Zbl

[11] Tafel J., J. Math. Phys., 34:5 (1993), 1892–1907 | DOI | MR | Zbl

[12] Kovalyov M., Legaré M., Gagnon L., J. Math. Phys., 34:7 (1993), 3245–3268 | DOI | MR | Zbl

[13] Ward R. S., Nucl. Phys. B, 236:2 (1984), 381–396 ; Ablowitz M. J., Costa D. J., Tenenblat K., SIAM, 77:1 (1987), 37–46 ; Popov A. D., JINR Rapid Commun., 6 (1992), 57–62; Strachan I. A. B., J. Math. Phys., 34:1 (1993), 243–259 | DOI | MR | MR | Zbl | DOI | MR | Zbl

[14] Nakamura Y., J. Math. Phys., 29:1 (1988), 244–245 ; 32:2 (1991), 382–385 ; Takasaki K., Commun. Math. Phys., 127:2 (1990), 225–238 ; Schiff J., Preprint IASSNS–HEP–92/34, 1992; Strachan I. A. B., J. Math. Phys., 33:7 (1992), 2477–2482 ; Ablowitz M. J., Chakravarty S., Takhtajan L. A., Commun. Math. Phys., 158:2 (1993), 289–320 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR

[15] Belavin A. A., Zakharov V. E., Phys. Lett. B, 73:1 (1978), 53–57 ; Ward R. S., Phys. Lett. A, 61:2 (1977), 81–82 | DOI | MR | DOI | MR | Zbl

[16] Legare M., Popov A. D., Pisma v ZhETF, 59:12 (1994), 845–850 | MR

[17] Ivanova T. A., Popov A. D., Pisma v ZhETF, 61:2 (1995), 142–148 | MR

[18] Manakov S. V., Zakharov V. E., Lett. Math. Phys., 5:3 (1981), 247–253 | DOI | MR

[19] Ward R. S., J. Math. Phys., 29:2 (1988), 386–389 ; Nonlinearity, 1 (1988), 671–679 ; Commun. Math. Phys., 128:2 (1990), 319–332 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[20] Ooguri H., Vafa C., Nucl. Phys. B, 367:1 (1991), 83–104 ; Nishino H., Gates S. J. Jr., Mod. Phys. Lett. A, 7:27 (1992), 2543–2554 ; Ketov S. V., Nishino H., Gates S. J. Jr., Phys. Lett. B, 307:3–4 (1993), 331–338 ; Nucl. Phys. B, 393:1–2 (1993), 149–210 | DOI | MR | DOI | MR | MR | DOI | MR | Zbl

[21] Kobayashi S., Nomizu K., Foundations of Differential Geometry, Vol. I, Interscience Publ., 1963 ; Vol. II, John Wiley Sons, New York, 1969 | MR | Zbl | Zbl

[22] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya: Metody i prilozheniya, Nauka, M., 1986 | MR

[23] Lichnerowicz A., Geometrie des groupes de transformations, Dunod, Paris, 1958 ; Kobayashi S., Transformation groups in differential geometry, Springer-Verlag, Berlin, 1972 | MR | Zbl | MR | Zbl

[24] Griffiths P., Harris J., Principles of algebraic geometry, John Wiley Sons, New York, 1978 | MR | Zbl

[25] Atiyah M. F., Hitchin N. J., Singer I. M., Proc. R. Soc. Lond. A, 362 (1978), 425–461 ; Woodhouse N. M. J., Class. Quantum Grav., 2:3 (1985), 257–291 | DOI | MR | Zbl | DOI | MR | Zbl

[26] Sergeev A. G., “Teoriya tvistorov i klassicheskie kalibrovochnye polya: Obzor”, Monopoli: topologicheskie i variatsionnye metody, Mir, M., 1989, 492–555

[27] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 ; Olver P. J., Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986 | MR | MR

[28] Forgács P., Manton N. S., Commun. Math. Phys., 72:1 (1980), 15–35 | DOI | MR

[29] WinternitzP., Partially Integrable Evolution Equations in Physics, NATO ASI series C, 310, eds. R. Conte and N. Boccara, Kluwer Academic Publ., 1990, 515–567 | MR

[30] Nyuell A., Solitony v matematike i fizike, Mir, M., 1989 | MR

[31] Belinskii V. A., Zakharov V. E., ZhETF, 75:12 (1978), 1953–1971 | MR

[32] Burtsev S. P., Zakharov V. E., Mikhailov A. V., TMF, 70:3 (1987), 323–341 | MR | Zbl

[33] Calogero F., Degasperis A., Commun. Math. Phys., 63:2 (1978), 155–176 | DOI | MR | Zbl

[34] Maison D., Phys. Rev. Lett., 41 (1978), 521–522 ; J. Math. Phys., 20:5 (1979), 871–877 ; Preprint MPI–PAE/ PTh–3/82, 1982 | DOI | MR | DOI | MR

[35] Mikhailov A. V., Yaremchuk A. I., Nucl. Phys. B, 202:3 (1982), 508–522 | DOI | MR

[36] Leese R., J. Math. Phys., 30:9 (1989), 2072–2077 ; Попов А. Д., ЯФ, 55:12 (1992), 3371–3374 | DOI | MR | Zbl | MR

[37] Semenov-Tyan-Shanskii M. A., Faddeev L. D., Vestn. Leningr. un-ta, 1977, no. 13, 81–88 | MR

[38] Takhtadzhyan L. A., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[39] Fordy A., Kulish P. P., Commun. Math. Phys., 89:3 (1983), 427–443 ; Fordy A., Wojciechowski S., Marshall I., Phys. Lett. A, 113:8 (1986), 395–400 | DOI | MR | Zbl | DOI | MR

[40] Wadati M., Kamijo T., Progr. Theor. Phys., 52:2 (1974), 397–414 | DOI | MR | Zbl

[41] Calogero F., Degasperis A., Nuovo Cimento B, 39:1 (1977), 1–54 | DOI | MR

[42] Athorne C., Fordy A. P., J. Phys. A, 20:6 (1987), 1377–1386 | DOI | MR | Zbl

[43] Winternitz P., Painlevé Transcendents, their Asymptotics and Physical Applications, NATO ASI Series B, 278, eds. D. Levi and P. Winternitz, Plenum Press, New York, 1992, 425–431 | DOI | MR | Zbl

[44] Burbaki N., Gruppy i algebry Li, Mir, M., 1972 | MR | Zbl

[45] Leznov A. N., Savelev M. V., Gruppovye metody integrirovaniya nelineinykh dinamicheskikh sistem, Nauka, M., 1985 | MR | Zbl

[46] Bogoyavlenskii O. I., Izv. AN SSSR. Ser. matem., 52:2 (1988), 243–266 ; УМН, 46:3 (1991), 3–48 | MR | MR | Zbl

[47] Bobenko A. I., Reyman A. G., Semenov-Tian-Shansky M. A., Commun. Math. Phys., 122:2 (1989), 321–354 | DOI | MR

[48] Trofimov V. V., Fomenko A. T., UMN, 39:2(236) (1984), 3–56 ; “Геометрия скобок Пуассона и методы интегрирования по Лиувиллю систем на симметрических пространствах”, Совр. пробл. матем. Новейшие достижения, 29, ВИНИТИ, M., 1986, 3–108 | MR | Zbl | MR | Zbl