zbMATH — the first resource for mathematics

On the characteristic Lie algebras for equations \(u_{xy} = f(u, u_x)\). (English. Russian original) Zbl 1158.35312
J. Math. Sci., New York 151, No. 4, 3112-3122 (2008); translation from Fundam. Prikl. Mat. 12, No. 7, 65-78 (2006).
Summary: A new approach to classification of integrable nonlinear equations is proposed. The method is based on description of the structure of the characteristic algebra. A basis of the characteristic algebra is constructed for the sinh-Gordon equation.

35A30 Geometric theory, characteristics, transformations in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
58J70 Invariance and symmetry properties for PDEs on manifolds
35Q58 Other completely integrable PDE (MSC2000)
Full Text: DOI
[1] A. A. Bormisov, E. C. Gudkova, and F. H. Mukminov, ”On the intergrability of hyperbolic systems of the Riccati equation type,” Theor. Math. Phys., 113, No. 2, 1418–1430 (1998). · doi:10.1007/BF02634167
[2] A. A. Bormisov and F. H. Mukminov, ”Symmetries of hyperbolic systems Riccati equation type,” Theor. Math. Phys., 127, No. 1, 446–459 (2001). · Zbl 0990.35011 · doi:10.1023/A:1010307823974
[3] I. T. Habibullin, ”Characteristic algebras of fully discrete hyperbolic type equations,” SIGMA Symmetry Integrability Geom. Meth. Appl., 1 (2005). · Zbl 1099.37058
[4] A. N. Leznov, V. G. Smirnov, and A. B. Shabat, ”Internal symmetry group and integrability conditions for two-dimensional dynamical systems,” Teor. Mat. Fiz., 51, No. 1, 10–21 (1982). · Zbl 0504.35023
[5] A. B. Shabat and R. I. Yamilov, Exponential systems of type I and the Cartan matrices, Ufa (1981).
[6] A. V. Zhiber and F. H. Mukminov, ”Quadratic systems, symmetries, characteristic and full algebras,” in: Problems in Mathematical Physics and Asymptotics of Their Solutions [in Russian], Ufa (1991), pp. 14–33.
[7] A. V. Zhiber and A. B. Shabat, ”The Klein-Gordons equations with nontrivial groups,” Dokl. Akad. Nauk SSSR, 247, No. 5, 1103–1107 (1979).
[8] A. V. Zhiber and A. B. Shabat, ”Systems of equations u x = p(u, v), v y = q(u, v) that possess symmetries,” Dokl. Akad. Nauk SSSR, 277, No. 1, 29–33 (1984). · Zbl 0599.35023
[9] A. V. Zhiber and V. V. Sokolov, ”Exactly integrable hyperbolic equations of Liouville type,” Russ. Math. Surv., 56, No. 1, 61–101 (2001). · Zbl 1003.35093 · doi:10.1070/RM2001v056n01ABEH000357
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.