Meshkov, A. G. Vector hyperbolic equations with higher symmetries. (English. Russian original) Zbl 1184.35016 Theor. Math. Phys. 161, No. 2, 1471-1484 (2009); translation from Teor. Mat. Fiz. 161, No. 2, 176-190 (2009). Summary: We list eleven vector hyperbolic equations that have third-order symmetries with respect to both characteristics. This list exhausts the equations with at least one symmetry of a divergence form. We integrate four equations in the list explicitly, bring one to a linear form, and bring four more to nonlinear ordinary nonautonomous systems. We find the Bäcklund transformations for six equations. Cited in 1 Document MSC: 35B06 Symmetries, invariants, etc. in context of PDEs 35L76 Higher-order semilinear hyperbolic equations 58J70 Invariance and symmetry properties for PDEs on manifolds Keywords:Bäcklund transformation; exact integrability PDF BibTeX XML Cite \textit{A. G. Meshkov}, Theor. Math. Phys. 161, No. 2, 1471--1484 (2009; Zbl 1184.35016); translation from Teor. Mat. Fiz. 161, No. 2, 176--190 (2009) Full Text: DOI References: [1] N. Kh. Ibragimov, Groups of Transformations in Mathematical Physics [in Russian], Nauka, Moscow (1983); English transl.: N. H. Ibragimov Transformation Groups Applied in Mathematical Physics, Reidel, Dordrecht (1985). [2] P. J. Olver, Applications of Lie Groups to Differential Equations (Grad. Texts in Math., Vol. 107), Springer, New York (1986). · Zbl 0588.22001 [3] A. S. Fokas, Stud. Appl. Math., 77, 253–299 (1987). [4] A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Russ. Math. Surveys, 42, No. 4, 1–63 (1987). · Zbl 0646.35010 · doi:10.1070/RM1987v042n04ABEH001441 [5] A. V. Mikhailov, A. B. Shabat, and V. V. Sokolov, ”Symmetry approach to classification of integrable equations,” in: Integrability and Kinetic Equations for Solitons [in Russian] (V. G. Bakhtar’yar, V. E. Zakharov, and V. M. Chernousenko, eds.), Naukova Dumka, Kiev (1990), pp. 213–279; A. V. Mikhajlov, A. B. Shabat, and V. V. Sokolov, ”The symmetry approach to the classification of integrable equations,” in: What is Integrability? (V. E. Zakharov, ed.), Springer, Berlin (1991), pp. 115–184. [6] S. C. Anco and T. Wolf, J. Nonlinear Math. Phys., 12Suppl. 1, 13–31 (2005). · Zbl 1362.35180 · doi:10.2991/jnmp.2005.12.s1.2 [7] A. V. Zhiber and V. V. Sokolov, Russ. Math. Surveys, 56, 61–101 (2001). · Zbl 1003.35093 · doi:10.1070/RM2001v056n01ABEH000357 [8] A. G. Meshkov and V. V. Sokolov, Theor. Math. Phys., 139, 609–622 (2004). · Zbl 1178.37083 · doi:10.1023/B:TAMP.0000026179.16504.6c [9] A. G. Meshkov and V. V. Sokolov, Comm. Math. Phys., 232, 1–18 (2002). · Zbl 1012.37049 · doi:10.1007/s00220-002-0737-9 [10] R. M. Miura, ed., Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications (Lect. Notes Math., Vol. 515), Springer, Berlin (1976). · Zbl 0317.00006 [11] A. G. Meshkov, J. Math. Sci. (New York), 151, 3167–3181 (2008). · Zbl 1149.37319 · doi:10.1007/s10958-008-9022-6 [12] M. Yu. Balakhnev, Theor. Math. Phys., 154, 220–226 (2008). · Zbl 1158.37309 · doi:10.1007/s11232-008-0021-z [13] G. L. Lamb Jr., Elements of Soliton Theory, Wiley, New York (1980). · Zbl 0445.35001 [14] M. Yu. Balakhnev, Math. Notes, 82, 448–450 (2007). · Zbl 1156.35456 · doi:10.1134/S0001434607090180 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.