Abstract
Nonclassical symmetries of the fourth-order nonlinear partial differential equation with dispersion and dissipation are obtained and are used as a basis for deriving new exact solutions that are invariant with respect to these symmetries. The equation describes the propagation of nonlinear long-wavelength longitudinal deformations in an elastic rod placed in an external dissipative medium, the waves at the surface of a viscous liquid, etc. The solutions describing running waves are investigated based on the classical symmetries of a reduced version of the basic equation. It is shown that such solutions can be constructed within the class of elliptic functions.
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Translated from Zhurnal Tekhnichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Fiziki, Vol. 73, No. 11, 2003, pp. 1–5.
Original Russian Text Copyright © 2003 by Gurski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \), Samsonov, Schwarz.
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Gurskii, V.V., Samsonov, A.M. & Schwarz, F. Classical and nonclassical symmetries of the nonlinear equation with dispersion and dissipation. Tech. Phys. 48, 1359–1363 (2003). https://doi.org/10.1134/1.1626765
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DOI: https://doi.org/10.1134/1.1626765