Geodesic
Tensor fields associated with integrable systems of chiral type
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 405-412 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie algebra.
Keywords: chiral-type systems, integrable systems, Lax representation, covariantly constant field. 
@article{SVMO_2019_21_4_a0,
     author = {A. V. Balandin},
     title = {Tensor fields associated with integrable systems of chiral type},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {405--412},
     year = {2019},
     volume = {21},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a0/}
}
TY  - JOUR
AU  - A. V. Balandin
TI  - Tensor fields associated with integrable systems of chiral type
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2019
SP  - 405
EP  - 412
VL  - 21
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a0/
LA  - ru
ID  - SVMO_2019_21_4_a0
ER  - 
%0 Journal Article
%A A. V. Balandin
%T Tensor fields associated with integrable systems of chiral type
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2019
%P 405-412
%V 21
%N 4
%U http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a0/
%G ru
%F SVMO_2019_21_4_a0
A. V. Balandin. Tensor fields associated with integrable systems of chiral type. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 405-412. http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a0/

[1] M. Marvan, “On zero-curvature representations of partial differential equations”, 5th International Conference on Differential Geometry and Its Applications, Opava, 1993, 103–122 | MR | Zbl

[2] A. V. Balandin, “Characteristics of conservation laws of chiral-type systems”, Letters in Mathematical Physics, 105:1 (2015), 27–43 | DOI | MR | Zbl

[3] A. V. Balandin, “Tensor fields defined by Lax representations”, Journal of Nonlinear Mathematical Physics, 23:3 (2016), 323–334 | DOI | MR | Zbl

[4] E. V. Ferapontov, W. K. Schief, “Surfaces of Demoulin: differential geometry, Backlund transformation and integrability”, Journal of Geometry and Physics, 30 (1999), 343–363 | DOI | MR | Zbl

[5] D. K. Demskoi, V. G. Marikhin, A. G. Meshkov, “Lax representations for triplets of two-dimensional scalar fields of the chiral type”, Theoretical and Mathematical Physics, 148:2 (2006), 1034–1048 (In Russ.) | DOI | DOI | MR | Zbl