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Group classifications, symmetry reductions and exact solutions to the nonlinear elastic rod equations. (English) Zbl 1241.35198
Summary: A Lie symmetry analysis is performed on the three nonlinear elastic rod (NER) equations. The complete group classification of the generalized nonlinear elastic rod equations is obtained. The symmetry reductions and exact solutions to the equations are presented. Furthermore, by means of dynamical system and power series methods, the exact explicit solutions to the equations are investigated. It is shown that the combination of Lie symmetry analysis and dynamical system methods is a feasible approach to deal with symmetry reductions and exact solutions to nonlinear PDEs.

35Q74PDEs in connection with mechanics of deformable solids
35B06Symmetries, invariants, etc. (PDE)
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
35C07Traveling wave solutions of PDE
22E70Applications of Lie groups to physics; explicit representations
Full Text: DOI
[1] Ablowitz M.J., Segur H.: Solition and the inverse scattering transform. SIAM, Philadelphia (1981) · Zbl 0472.35002
[2] Gardner C. et al.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095--1097 (1967) · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[3] Matveev V.B., Salle M.A.: Darboux transformations and solitions. Springer, Berlin (1991) · Zbl 0744.35045
[4] Y. S. Li, Soliton and integrable systems. In: Advanced series in nonlinear science, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1999 (in Chinese).
[5] Hirota R., Satsuma J.: A variety of nonlinear network equations generated from the Bäcklund transformation for the Tota lattice. Suppl. Prog. Theor. Phys. 59, 64--100 (1976) · Zbl 1079.35536 · doi:10.1143/PTPS.59.64
[6] Liu H., Li J., Chen F.: Exact periodic wave solutions for the hKdV equation. Nonlinear Anal. 70, 2376--2381 (2009) · Zbl 1162.35312 · doi:10.1016/j.na.2008.03.019
[7] Olver P.J. (1993) Applications of Lie groups to differential equations. In: Graduate texts in Mathematics, vol.107, Springer, New York, 1993. · Zbl 0785.58003
[8] G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for Differential Equations. In: Applied Mathematical Sciences, vol. 154, Springer-Verlag, New York, 2002. · Zbl 1013.34004
[9] B. J. Cantwell, Introduction to Symmetry Analysis. Cambridge University Press, 2002. · Zbl 1082.34001
[10] Qu C., Huang Q.: Symmetry reductions and exact solutions of the affine heat equation. J. Math. Anal. Appl. 346, 521--530 (2008) · Zbl 1149.35306 · doi:10.1016/j.jmaa.2008.05.082
[11] Liu H., Li J.: Lie symmetry analysis and exact solutions for the short pulse equation. Nonlinear Anal. TMA 71, 2126--2133 (2009) · Zbl 1244.35003 · doi:10.1016/j.na.2009.01.075
[12] Liu H., Li J., Zhang Q.: Lie symmetry analysis and exact explicit solutions for general Burgers’ equation. J. Comput. Appl. Math. 228, 1--9 (2009) · Zbl 1166.35033 · doi:10.1016/j.cam.2008.06.009
[13] Liu H., Li J., Liu L.: Lie group classifications and exact solutions for two variable-coefficient equations. Appl. Math. Comput. 215, 2927--2935 (2009) · Zbl 1232.35173 · doi:10.1016/j.amc.2009.09.039
[14] Liu H., Li J., Liu L.: Lie symmetries, optimal systems and exact solutions to the fifth-order KdV type equations. J. Math. Anal. Appl. 368, 551--558 (2010) · Zbl 1192.35011 · doi:10.1016/j.jmaa.2010.03.026
[15] Liu H., Li J., Liu L.: Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations. Nonlinear Dyn., 59, 497--502 (2010) · Zbl 1183.35236 · doi:10.1007/s11071-009-9556-2
[16] Liu H., Li J.: Lie symmetry analysis and exact solutions for the extended mKdV equation. Acta Appl. Math. 109, 1107--1119 (2010) · Zbl 1223.37079 · doi:10.1007/s10440-008-9362-8
[17] Liu H., Li J.: Lie Symmetries, Conservation Laws and Exact Solutions for Two Rod Equations. Acta Appl. Math. 110, 573--587 (2010) · Zbl 1276.35013 · doi:10.1007/s10440-009-9462-0
[18] Clarkson P., Kruskal M.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201--2213 (1989) · Zbl 0698.35137 · doi:10.1063/1.528613
[19] Clarkson P.: Painlevé analysis and the complete integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation. IMA J. Appl. Math. 44, 27--53 (1990) · Zbl 0719.35083 · doi:10.1093/imamat/44.1.27
[20] Kuang Y., He X., Chen C., Li G.: Anaysis of nonlinear vibrations of doublewalled carbon nanotubes conveying fluid. Comput. Mater. Sci. 45, 875--880 (2009) · doi:10.1016/j.commatsci.2008.12.007
[21] Zhuang W., Zhang S.: The strain solitary waves in a nonlinear elastic rod. Acta Mechanica Sinica 20, 58--67 (1988) (in Chinese)
[22] Zhuang W., Zhang G.: The propagation of solitary waves in a nonlinear elastic rod. Appl. Math. Mech. 7, 615--626 (1986) · Zbl 0602.73025 · doi:10.1007/BF01895973
[23] Duan W., Zhao J.: Solitary waves in a quadratic nonlinear elastic rod. Chaos, Soliton and Fractals 11, 1265--1267 (2000) · Zbl 0961.74036 · doi:10.1016/S0960-0779(99)00014-4
[24] Li J., Zhang Y.: Exact traveling wave solutions in a nonlinear elastic rod equation. Appl. Math. Comput. 202, 504--510 (2008) · Zbl 1143.74033 · doi:10.1016/j.amc.2008.02.027
[25] Lv K. et al.: Perturbation analysis for wave equation of nonlinear elastic rod. Appl. Math. Mech. 27, 1233--1238 (2006) · Zbl 1258.74018 · doi:10.1007/s10483-006-0910-z
[26] Byrd P.F., Fridman M.D.: Handbook of elliptic integrals for engineers and sciensists. Springer, Berlin (1971)
[27] Z.X. Wang, D. R. Guo, Introduction to special functions. In: The series of advanced physics of Peking University, Peking University Press, Beijing, 2000 (in Chinese).
[28] Guckenheimer J., Holmes P.J.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag, Berlin (1983) · Zbl 0515.34001