Abstract
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.
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References
Polyanin, A. D. and Zaitsev, V. F. Handbook of Nonlinear Partial Differential Equations, CRC Press, Boca Raton (2004)
Ovsiannikov, L. V. Group Analysis of Differential Equations, Academic Press, New York (1982)
Ames, W. F. Nonlinear Partial Differential Equations in Engineering, Vols. I & II, Academic Press, New York (1965 & 1972)
Baumann, G. Symmetry Analysis of Differential Equations with Mathematica, Springer-Verlag, New York (2000)
Bluman, G. W. and Cole, J. D. Similarity Methods for Differential Equations, Springer-Verlag, New York (1974)
Bluman, G. W. and Kumei, S. Symmetries and Differential Equations, Springer-Verlag, New York (1989)
Euler, N. and Steeb, W. H. Continuous Symmetries, Lie Algebras and Differential Equations, Bibliographisches Institut, Mannheim (1992)
Hansen, A. G. Similarity Analyses of Boundary Value Problems in Engineering, Prentice Hall, Englewood Cliffs (1964)
Hydon, P. E. Symmetry Methods for Differential Equations, Cambridge University Press, Cambridge (2000)
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)
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)
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)
Ibragimov, N. H. Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, Chichester (1999)
Miller, W. Symmetry and Separation of Variables, Addison Wesley, Reading, Massachusetts (1977)
Olver, P. J. Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986)
Stephani, H. Differential Equations: Their Solution Using Symmetries, Cambridge University Press, Cambridge (1989)
Kosevich, Y. A. Nonlinear sinusoidal waves and their superposition in anharmonic lattices. Phys. Rev. Lett. 71, 2058–2061 (1993)
Kosevich, Y. A. Nonlinear sinusoidal waves and their superposition in anharmonic lattices-reply. Phys. Rev. Lett. 79, 4716–4716 (1997)
Rodriguez-Achach, M. and Perez, G. Nonlinear sinusoidal waves and their superposition in anharmonic lattices-comment. Phys. Rev. Lett. 79, 4715–4715 (1997)
Pouget, J. Lattice-dynamics and stability of modulated-strain structures for elastic phasetransitions in alloys. Phys. Rev. B 48(2), 864–875 (1993)
Alfinito, E., Causo, M. S., Profilo, G., and Soliani, G. A class of nonlinear wave equations containing the continuous Toda case. J. Phys. A 31, 2173–2189 (1998)
Apostol, B. F. On a non-linear wave equation in elasticity. Phys. Lett. A 318(6), 545–552 (2003)
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).
Bokhari, A. H., Kara, A. H., and Zaman, F. D. Exact solutions of some general nonlinear wave equations in elasticity. Nonlinear Dynamics 48(1–2), 49–54 (2007)
Gradshteyn, I. S. and Ryzhik, I. M. Table of Integrals, Series and Products, Academic Press, New York (1980)
Slater, L. J. Generalized Hypergeometric Functions, Cambridge University Press, Cambridge (1966)
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Mustafa, M.T., Masood, K. Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections. Appl. Math. Mech.-Engl. Ed. 30, 1017–1026 (2009). https://doi.org/10.1007/s10483-009-0808-z
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DOI: https://doi.org/10.1007/s10483-009-0808-z