Geodesic
Nonlocal symmetries in two-field divergent evolutionary systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 3, pp. 351-363 Cet article a éte moissonné depuis la source Math-Net.Ru

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We evaluate nonlocal symmetries for third-order exactly integrable two-field divergent evolutionary equations. These symmetries, regarded as evolutionary equations, commute with higher analogues of the underlying original equations and seem to be exactly integrable. By differentiating nonlocal systems and changing the variables, we obtain local hyperbolic systems and third-order nonevolutionary systems. We find a zero-curvature representation for some of the new systems.
Keywords: conserved density, weakly nonlocal variable, nonlocal symmetry, negative flow, exact integrability. 
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A. G. Meshkov. Nonlocal symmetries in two-field divergent evolutionary systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 3, pp. 351-363. http://geodesic.mathdoc.fr/item/TMF_2008_156_3_a2/

[1] A. G. Meshkov, Fundament. i prikl. matem., 12:7 (2006), 141–161 | MR

[2] S. Kumei, J. Math. Phys., 16:12 (1975), 2461–2468 | DOI | MR

[3] Zhijun Qiao, Negative order MKdV hierarchy and a new integrable Neumann-like system, arXiv: nlin/0201065

[4] I. M. Gelfand, L. A. Dikii, UMN, 30:5 (1975), 67–100 ; Функц. анализ и его прил., 11:2 (1977), 11–27 | DOI | MR | Zbl | DOI | MR | Zbl

[5] V. E. Zakharov, A. B. Shabat, ZhETF, 61:1 (1971), 118–134 | MR

[6] I. Sh. Akhatov, R. K. Gazizov, N. Kh. Ibragimov, “Nelokalnye simmetrii. Evristicheskii podkhod”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 34, ed. R. V. Gamkrelidze, VINITI, M., 1989, 3–83 ; В. В. Соколов, С. И. Свинолупов, Матем. заметки, 48:6 (1990), 91–97 | MR | Zbl | DOI | MR | Zbl

[7] V. G. Drinfeld, V. V. Sokolov, Dokl. AN SSSR, 284:1 (1985), 29–33 | MR | Zbl

[8] A. B. Shabat, R. I. Yamilov, Phys. Lett. A, 130:4–5 (1988), 271–275 ; А. Б. Шабат, Р. И. Ямилов, Алгебра и анализ, 2:2 (1990), 183–208 ; А. П. Веселов, А. Б. Шабат, Функц. анализ и его прил., 27:2 (1993), 1–21 ; A. B. Shabat, R. I. Yamilov, Phys. Lett. A, 227:1–2 (1997), 15–23 | DOI | MR | MR | DOI | MR | Zbl | DOI | MR | Zbl

[9] A. B. Borisov, S. A. Zykov, TMF, 115:2 (1998), 199–215 | DOI | MR | Zbl

[10] P. Olver, Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989 ; Н. Х. Ибрагимов, Применение групп преобразований в математической физике, Наука, М., 1983 ; А. В. Михайлов, А. Б. Шабат, В. В. Соколов, “Симметрийный подход к классификации интегрируемых уравнений”, Интегрируемость и кинетические уравнения для солитонов, ред. В. Г. Бахтарьяр, В. Е. Захаров, В. М. Черноусенко, Наукова думка, Киев, 1990, 213–279 | MR | MR | Zbl | MR | MR | Zbl | MR | MR | Zbl

[11] A. Sergyeyev, “On recursion operators and nonlocal symmetries of evolution equations”, Proc. Sem. Diff. Geom., Math. Publ., 2, ed. D. Krupka, Silesian Univ. Opava, Opava, 2000, 159–173 | MR | Zbl

[12] V. G. Drinfeld, V. V. Sokolov, “Algebry Li i uravneniya tipa Kortevega–de Friza”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 24, ed. R. V. Gamkrelidze, VINITI, M., 1984, 81–180 | MR | Zbl

[13] A. G. Meshkov, Nonlocal symmetries of integrable two-field divergent evolutionary systems, arXiv: 0803.1425 | MR