Classical and Nonclassical Symmetries of a Nonlinear Differential Equation for Describing Waves in a Liquid with Gas Bubbles

In this paper, we consider a nonlinear differential equation for describing nonlinear waves in a liquid with gas bubbles if the liquid viscosity and the interphase heat exchange are accounted for. Classical and nonclassical symmetries of this partial differential equation are investigated. We show that it is invariant under shift transformations in space and time. At an additional restriction on the parameters, this equation is also invariant under the Galilean transformation. Nonclassical symmetries of the equation in question are found by the Bluman-Cole method. Both regular and singular cases of nonclassical symmetries are considered. Five families of nonclassical symmetries admitted by this equation are specified. Invariant reductions corresponding to these families are obtained. With their use, families of exact solutions of the considered equation are found. These solutions are expressed in terms of rational, trigonometric, and special functions.