Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections
Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations (ODEs) are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times. Along with solutions with time-dependent singularities, we also obtain solutions which do not exhibit time-dependent singularities.
Key wordsgroup invariant solutions Lie symmetries nonlinear elasticity equations partial differential equations
Chinese Library ClassificationO151.23
2000 Mathematics Subject Classification76M60 73C50 74J30 74H05
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- Symmetries, exact solutions and conservation laws (ed. Ibragimov, N. H.). CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, CRC Press, Boca Raton (1994)Google Scholar
- Applications in engineering and physical sciences (ed. Ibragimov, N. H.). CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 2, CRC Press, Boca Raton (1995)Google Scholar
- New trends in theoretical developments and computational methods (ed. Ibragimov, N. H.). CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3, CRC Press, Boca Raton (1996)Google Scholar
- Fermi, E., Pasta, J., and Ulam, S. Studies of nonlinear problems. I. Los Alamos Report LA-1940 (1955); Collected Papers by Entico Fermi (ed. Segré, E.), Unveristy of Chicago Press, Chicago (1965).Google Scholar