Classical and nonclassical symmetries of the nonlinear equation with dispersion and dissipation

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. (C) 2003 MAIK "Nauka/Interperiodica".