Geodesic
Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I
Teoretičeskaâ i matematičeskaâ fizika, Tome 62 (1985) no. 2, pp. 163-185 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Necessary conditions for the existence of nontrivial conservation laws for systems of nonlinear equations of the form $u_t=a(u,v)u_{xx}+b(u,v)v_{xx}+f(u,v,u_x,v_x),$ $-v_t=c(u,v)u_{xx}+d(u,v)v_{xx}+g(u,v,u_x,v_x)$. are found. They take the form of densities of local conservation laws constructed in a definite manner from the coefficients of the system. The conditions can be readily verified in each specific case. A module of simple invertible substitutions that makes it possible to reduce the system to a canonical form when some integrability conditions are satisfied is discussed. 
@article{TMF_1985_62_2_a0,
     author = {A. V. Mikhailov and A. B. Shabat},
     title = {Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)${.~I}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {163--185},
     year = {1985},
     volume = {62},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1985_62_2_a0/}
}
TY  - JOUR
AU  - A. V. Mikhailov
AU  - A. B. Shabat
TI  - Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1985
SP  - 163
EP  - 185
VL  - 62
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1985_62_2_a0/
LA  - ru
ID  - TMF_1985_62_2_a0
ER  - 
%0 Journal Article
%A A. V. Mikhailov
%A A. B. Shabat
%T Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1985
%P 163-185
%V 62
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1985_62_2_a0/
%G ru
%F TMF_1985_62_2_a0
A. V. Mikhailov; A. B. Shabat. Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I. Teoretičeskaâ i matematičeskaâ fizika, Tome 62 (1985) no. 2, pp. 163-185. http://geodesic.mathdoc.fr/item/TMF_1985_62_2_a0/

[1] Ablowitz M., Segur H., Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981 | MR | Zbl

[2] Sokolov V. V., Shabat A. B., Soviet Sci. Rev. C, 4 (1984), 221–280 | MR | Zbl

[3] Estabrook F. B., Wahlquist H. D., Z. Math. Phys., 16 (1975), 1–7 | DOI | MR | Zbl

[4] Estabrook F. B., Wahlquist H. D., J. Math. Phys., 17 (1976), 1293–1297 | DOI | MR | Zbl

[5] Ibragimov N. Kh., Shabat A. B., Funkts. analiz i ego prilozh., 14:4 (1980), 79–80 | MR | Zbl

[6] Golubev V. V., Lektsii po analiticheskoi teorii differentsialnykh uravnenii, Gostekhizdat, M.–L., 1950 | MR

[7] Zhiber A. V., Shabat A. B., DAN SSSR, 247:5 (1979), 1103–1105 | MR

[8] Svinolupov S. I., Sokolov V. V., Funkts. analiz i ego prilozh., 16:4 (1982), 86–87 | MR | Zbl

[9] Yamilov R. I., UMN, 38:6 (1983), 155–156

[10] Lax P. D., Commun. Pure and Appl. Math., 21 (1968), 467–490 | DOI | MR | Zbl

[11] Magri F., J. Math. Phys., 19:5 (1978), 1156–1162 | DOI | MR | Zbl

[12] Zakharov V. E., Schulman E. P., Physica, 4D (1982), 270–274 | Zbl

[13] Fordi A. P., Kulish P. P., Commun. Math. Phys., 89 (1983), 427–443 | DOI | MR

[14] Sklyanin E. K., On complete integrability of the Landau-Lifshitz equation, Preprint LOMI E-3-1979, LOMI, L., 1979 | MR

[15] Wadati M., Konno K., Ichikawa Y. H., J. Phys. Soc. Japan, 47 (1979), 1968–1700 | DOI | MR

[16] Gelfand I. M., Dikii L. A., Funkts. analiz i ego prilozh., 10:4 (1976), 13–29 | MR | Zbl

[17] Manin Yu. I., Sovremennye problemy matematiki, 11, VINITI, M., 1978, 5–152 | MR