Geodesic
Explicit solutions of $O(3)$ and $O(2,1)$ chiral models and the associated equations of the two-dimensional Toda chain and the Ernst equation when the solutions are parametrized by arbitrary functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 57 (1983) no. 2, pp. 238-248 
M. G. Tseitlin. Explicit solutions of $O(3)$ and $O(2,1)$ chiral models and the associated equations of the two-dimensional Toda chain and the Ernst equation when the solutions are parametrized by arbitrary functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 57 (1983) no. 2, pp. 238-248. http://geodesic.mathdoc.fr/item/TMF_1983_57_2_a6/
@article{TMF_1983_57_2_a6,
     author = {M. G. Tseitlin},
     title = {Explicit solutions of~$O(3)$ and~$O(2,1)$ chiral models and the associated equations of the two-dimensional {Toda} chain and the {Ernst} equation when the solutions are parametrized by arbitrary functions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {238--248},
     year = {1983},
     volume = {57},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1983_57_2_a6/}
}
TY  - JOUR
AU  - M. G. Tseitlin
TI  - Explicit solutions of $O(3)$ and $O(2,1)$ chiral models and the associated equations of the two-dimensional Toda chain and the Ernst equation when the solutions are parametrized by arbitrary functions
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1983
SP  - 238
EP  - 248
VL  - 57
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1983_57_2_a6/
LA  - ru
ID  - TMF_1983_57_2_a6
ER  - 
%0 Journal Article
%A M. G. Tseitlin
%T Explicit solutions of $O(3)$ and $O(2,1)$ chiral models and the associated equations of the two-dimensional Toda chain and the Ernst equation when the solutions are parametrized by arbitrary functions
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1983
%P 238-248
%V 57
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1983_57_2_a6/
%G ru
%F TMF_1983_57_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

By means of elliptic solutions of the $O(3)$ and $O(2,1)$ $\sigma$ models parametrized by arbitrary holomorphie functions (generalization of a singular harmonic mapping) and the previously considered [1] correspondence between chiral models and systems with exponential interaction, elliptic solutions are obtained for one of the two-dimensional Toda chains corresponding to the Kac–Moody algebra parametrized by a holomorphie or an antiholomorphic function. Solutions of the sinh-Gordon equation are given. For the Ernst equation, a solution is generated by the meron sector of the $O(2,1)$ $\sigma$ model which is parametrized by two real functions (cylindrical waves) or a holomorphic function (stationary axisymmetric solutions). A solution of Liouville's equation on a torus is given.

[1] Tseitlin M. G., Zap. nauchn. semin. LOMI, 101 (1981), 194–201 ; Bytsenko A. A., Zeitlin M. G., Phys. Lett., 88A:6 (1982), 275–278 ; Быценко А. А., Цейтлин М. Г., ТМФ, 52:1 (1982), 63–72 | MR | DOI | MR | MR

[2] Leznov A. N., Saveliev M. V., Commun. Math. Phys., 74:2 (1980), 111–118 | DOI | MR | Zbl

[3] Eells J., Lemaire L., Bull. London Math. Soc., 10:1 (1978), 1–58 | DOI | MR

[4] Gross D., Nucl. Phys., B132:3 (1978), 439–456 | DOI | MR

[5] Ghika G., Visinescu M., Z. Phys., C11:4 (1982), 353–357 | MR

[6] Abbott R. B., Z. Phys., C15:1 (1982), 51–59 | MR

[7] Belavin A. A., Polyakov A. M., Pisma v ZhETF, 22:10 (1975), 503–506

[8] Maison D., Phys. Rev. Lett., 41:8 (1978), 521–522 | DOI | MR

[9] Papanicolaou N., J. Math. Phys., 20:10 (1979), 2069–2077 | DOI | MR

[10] Chinea F. J., Phys. Rev., D24:4 (1981), 1053–1055 ; Chinea F. J., Guil F., J. Phys., 15A:8 (1982), 2349–2354 | MR | MR | Zbl

[11] Richard J. L., Rouet A., Nucl. Phys., B211:3 (1983), 447–464 | DOI | MR