Approximate symmetry analysis of nonlinear Rayleigh-wave equation. (English) Zbl 1390.74103


74J15 Surface waves in solid mechanics
86A15 Seismology (including tsunami modeling), earthquakes
58J70 Invariance and symmetry properties for PDEs on manifolds
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
70S10 Symmetries and conservation laws in mechanics of particles and systems
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[1] Atanackovic, T. M.; Konjik, S.; Pilipovic, S.; Simic, S., Variational problems with fractional derivatives: invariance conditions and noether’s theorem, Nonlinear Anal., 71, 1504-1517, (2009) · Zbl 1163.49022
[2] Avdonina, E. D.; Ibragimov, N. H.; Khamitova, R., Exact solutions of gas dynamic equations obtained by the method of conservation laws, Commun. Nonlinear Sci. Numer. Simul., 18, 9, 2359-2366, (2013) · Zbl 1304.35528
[3] V. A. Baikov, R. K. Gazizov and N. H. Ibragimov, Approximate symmetries of equations with a small parameter, Mat. Sb.136 (1988) 435-450 (in Russian) Math USSR Sb.64 (1989) 427-441.
[4] Baikov, V. A.; Gazizov, R. K.; Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, 3, Approximate transformation groups and symmetry Lie algebras, 31-67, (1996), CRC Press, Boca Raton, FL
[5] Bluman, G. W.; Cheviakov, A. F.; Anco, C., Construction of conservation laws: how the direct method generalizes noether’s theorem, Proc. 4th Workshop “Group Analysis of Differential Equations and Integrability”, 1-23, (2009), National Academy of the Science, Ukraine · Zbl 1253.35007
[6] Bluman, G. W.; Cheviakov, A. F.; Anco, C., Application of Symmetry Methods to Partial Differential Equations, (2000), Springer, New York
[7] Fushchich, W. L.; Shtelen, W. M., On approximate symmetry and approximate solutions on the nonlinear wave equation with a small parameter, J. Phys. A: Math. Gen., 22, 18, L887, (1989) · Zbl 0711.35086
[8] Frederico, G. S. F.; Odzijewicz, T.; Torres, D. F. M., Noethers theorem for non-smooth extremals of variational problems with time delay, Appl. Anal., 96, 153-170, (2014) · Zbl 1287.49018
[9] Galaktionov, V.; Svirshchevskii, S., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, (2007), CRC Press · Zbl 1153.35001
[10] Hydon, P. E., Symmetry Method for Differential Equations, (2000), Cambridge University Press, Cambridge, UK · Zbl 1035.35005
[11] Ibragimov, N. H.; Aksenov, A. V.; Baikov, V. A.; Chugunov, V. A.; Gazizov, R. K.; Meshkov, A. G., CRC Handbook of Lie Group Analysis of Differential Equations, 2, (1995), CRC Press, Boca Raton
[12] Ibragimov, N. H.; Anderson, R. L.; Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations: Symmetries, Exact Solutions and Conservation Laws, 1, Lie theory of differential equations, 7-14, (1994), CRC Press, Boca Raton · Zbl 0864.35001
[13] Ibragimov, N. H.; Kovalev, V. F., Approximate and Renormgroup Symmetries Nonlinear Physical Science, (2009), Higher Education Press, Beijing, China · Zbl 1170.22001
[14] Ibragimov, N. H.; Avdonina, E. D., Nonlinear self-adjointness, conservation laws, and the construction of solutions of partial differential equations using conservation laws, Russian. Math. Surveys, 68, 5, 889-921, (2013) · Zbl 1286.35013
[15] Kara, A. H.; Mahomed, F. M.; Unal, G., Approximate symmetries and conservation laws with applications, Int. J. Theor. Phys., 38, 9, 2389-2399, (1999) · Zbl 0989.37076
[16] Krasilshchik, I. S.; Vinogradov, A. M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, (1999), American Mathematical Society, Providence, RI
[17] Liu, H.; Li, J., Lie symmetry analysis and exact solutions for the short pulse equation, Nonlinear Anal., 71, 2126-2133, (2009) · Zbl 1244.35003
[18] Olver, P. J., Applications of Lie Groups to Differential Equations, 107, (1993), Springer-Verlag, New York · Zbl 0785.58003
[19] Olver, P. J., Equivalence, Invariant and Symmetry, (1995), Cambridge University Press, Cambridge
[20] Hejazi, S. Reza, Lie group analysis, Hamiltonian equations and conservation laws of Born-Infeld equation, Asian-European J. Math., 7, 3, 1450040, (2014) · Zbl 1307.37030
[21] Wang, G. W.; Liu, X. Q.; Zhang, Y. Y., Symmetry reduction, exact solutions and conservation laws of a new fifth-order non-linear integrable equation, Commun. Nonlinear Sci. Numer. Simul., 18, 2313-2320, (2013) · Zbl 1304.35623
[22] Zhdanov, R. Z., Separation of variables in the nonlinear wave equation, J. Phys. A: Math. Gen., 27, 9, L291, (1994) · Zbl 0843.35003
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