Geodesic
On integrability of Rikkati-type systems of hyperbolic equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 113 (1997) no. 2, pp. 261-275 
A. A. Bormisov; E. S. Gudkova; F. Kh. Mukminov. On integrability of Rikkati-type systems of hyperbolic equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 113 (1997) no. 2, pp. 261-275. http://geodesic.mathdoc.fr/item/TMF_1997_113_2_a2/
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     title = {On integrability of {Rikkati-type} systems of hyperbolic equations},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Equations of the type $U_{xy}=U*U_x$ are considered. Here $U(x,y)$ is a $T$-mapped function and $T$ is an algebra over the field $\mathbb C$. It is shown that there are two characteristic Lie algebras $L_x$ and $L_y$ connected with each such equation. A definition of the $\mathbb Z$-graded Lie algebra $\mathfrak G$ corresponding to the equation is given. It is proved that for each of the equations the corresponding algebra $\mathfrak G$ can be taken as a sum of vector spaces $L_x$ and $L_y$ with a commutator between elements of $L_x$ and $L_y$ given by zero-curvature relations. Under assumption that the algebra $T$ has no left ideals, the classifications of the equations with finite dimensional characteristic algebras $L_x$ and $L_y$ is given. All of the equations are Darboux-integrable.

[1] H. Kakuhata, K. Konno, A Generalization of Coupled Integrable, Dispersionless System, Preprint NUP–A–95–12, 1995, P. 1–7 | MR

[2] K. Konno, H. Kakuhata, J. Phys. Soc. Japan, 64 (1995), 2707–2709 | DOI

[3] K. Konno, H. Kakuhata, Novel Solitonic Evolutions in a Coupled Integrable, Dispersionless System, Preprint NUP–A–95–5, 1995, P. 1–23 | MR

[4] K. Konno, H. Oono, J. Phys. Soc. Japan, 63 (1994), 377–378 | DOI

[5] E. S. Gudkova, F. Kh. Mukminov, V. V. Sokolov, “Zadacha Gursa dlya integriruemoi sistemy nelineinykh uravnenii giperbolicheskogo tipa”, Differentsialnye uravneniya. Integralnye uravneniya. Spetsialnye funktsii, Mezhdunarodnaya nauchnaya konferentsiya, posvyaschennaya 90-letiyu so dnya rozhdeniya professora S. P. Pulkina. Tezisy dokladov (Samara), 1997, 23–24

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

[7] V. G. Kats, Izv. AN SSSR, ser. matem., 32 (1968), 1323–1367 | MR | Zbl

[8] A. N. Leznov, V. G. Smirnov, A. B. Shabat, TMF, 51 (1982), 10–21 | MR | Zbl

[9] A. B. Shabat, R. I. Yamilov, Eksponentsialnye sistemy tipa 1 i matritsy Kartana, Preprint BNTs UO AN SSSR, Institut matematiki, Ufa, 1981, S. 1–22

[10] A. V. Zhiber, F. Kh. Mukminov, “Kvadratichnye sistemy, simmetriya, kharakteristicheskie i polnye algebry”, Sb. rabot Inst. mat. BNTs UO AN SSSR, Institut matematiki, Ufa, 1991

[11] V. G. Kats, Beskonechnomernye algebry Li, Mir, M., 1993 | MR | Zbl

[12] K. McCrimmon, Bull. of the Amer. Matem. Soc., 84:4 (1978), 612–627 | DOI | MR | Zbl