Geodesic
Matrix extension of multidimensional dispersionless integrable hierarchies
Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 1, pp. 3-15 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consistently develop a recently proposed scheme of matrix extensions of dispersionless integrable systems in the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax pairs, Lax–Sato equations, matrix equations on the background of vector fields, and the dressing scheme. Reductions, the construction of solutions, and connections to geometry are discussed. We separately consider the case of an Abelian extension, for which the Riemann–Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of the Penrose formula in curved space.
Keywords: dispersionless integrable system, gauge field, self-dual Yang–Mills equations. 
@article{TMF_2021_209_1_a0,
     author = {L. V. Bogdanov},
     title = {Matrix extension of multidimensional dispersionless integrable hierarchies},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--15},
     year = {2021},
     volume = {209},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2021_209_1_a0/}
}
TY  - JOUR
AU  - L. V. Bogdanov
TI  - Matrix extension of multidimensional dispersionless integrable hierarchies
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2021
SP  - 3
EP  - 15
VL  - 209
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2021_209_1_a0/
LA  - ru
ID  - TMF_2021_209_1_a0
ER  - 
%0 Journal Article
%A L. V. Bogdanov
%T Matrix extension of multidimensional dispersionless integrable hierarchies
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2021
%P 3-15
%V 209
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2021_209_1_a0/
%G ru
%F TMF_2021_209_1_a0
L. V. Bogdanov. Matrix extension of multidimensional dispersionless integrable hierarchies. Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 1, pp. 3-15. http://geodesic.mathdoc.fr/item/TMF_2021_209_1_a0/

[1] L. V. Bogdanov, “SDYM equations on the self-dual background”, J. Phys. A, 50:19 (2017), 19LT02, 9 pp. | DOI | MR

[2] L. V. Bogdanov, “Matrichnoe rasshirenie sistemy Manakova–Santini i integriruemaya kiralnaya model na fone geometrii Einshteina–Veilya”, TMF, 201:3 (2019), 337–346 | DOI | DOI | MR

[3] L. V. Bogdanov, “Bezdispersionnye integriruemye sistemy i uravneniya Bogomolnogo na fone geometrii Einshteina–Veilya”, TMF, 205:1 (2020), 41–54 | DOI | DOI | MR

[4] V. E. Zakharov, A. B. Shabat, “Integrirovanie nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya. II”, Funkts. analiz i ego pril., 13:3 (1979), 13–22 | DOI | MR | Zbl

[5] L. V. Bogdanov, V. S. Dryuma, S. V. Manakov, “Dunajski generalization of the second heavenly equation: dressing method and the hierarchy”, J. Phys. A: Math. Theor., 40:48 (2007), 14383–14393 | DOI | MR

[6] L. V. Bogdanov, “O klasse mnogomernykh integriruemykh ierarkhii i ikh reduktsiyakh”, TMF, 160:1 (2009), 15–22, arXiv: 0810.2397 | DOI | DOI | MR | Zbl

[7] M. Dunajski, E. V. Ferapontov, B. Kruglikov, “On the Einstein–Weyl and conformal self-duality equations”, J. Math. Phys., 56:8 (2015), 083501, 10 pp., arXiv: 1406.0018 | DOI | MR

[8] M. Dunajski, “Anti-self-dual four-manifolds with a parallel real spinor”, Proc. Roy. Soc. London Ser. A, 458:2021 (2002), 1205–1222, arXiv: math/0102225 | DOI | MR

[9] J. F. Plebañski, “Some solutions of complex Einstein equations”, J. Math. Phys., 16:12 (1975), 2395–2402 | DOI | MR

[10] L. V. Bogdanov, B. G. Konopelchenko, “On the $\bar{\partial}$-dressing method applicable to heavenly equation”, Phys. Lett. A, 345:1–3 (2005), 137–143, arXiv: nlin/0504062 | DOI | MR

[11] K. Takasaki, “An infinite number of hidden variables in hyper-Kähler metrics”, J. Math. Phys., 30:7 (1989), 1515–1521 | DOI | MR

[12] L. V. Bogdanov, “Interpoliruyuschie differentsialnye reduktsii mnogomernykh integriruemykh ierarkhii”, TMF, 167:3 (2011), 354–363 | DOI | DOI | MR

[13] R. Penrose, “Solutions of the zero-rest-mass equations”, J. Math. Phys., 10:1 (1969), 38–39 | DOI

[14] M. Dunajski, Solitons, Instantons, and Twistors, Oxford Graduate Texts in Mathematics, 19, Oxford University Press, Oxford, 2010 | MR