Group classification of one-dimensional equations of fluids with internal inertia

One-dimensional flows of fluids with internal inertia are studied in the manuscript. The given equations include such models as the non -linear one-velocity model of a bubbly fluid (with incompressible liquid phase) at small volume concentration of gas bubbles (Zh. Prikl. Tekh. Fiz. 1960; N3:102-111, Dokl. AN USSR 1961; 137:1331-1333, J. Fluid Mech. 1968; 33:465-474), and the dispersive shallow water model (J. Fluid Mech. 1976; 78:237-246, Lectures on Geophysical Fluid Dynamics. Oxford University Press: New York, 1998). These models are obtained for special types of the function W(p,p). The group classification separates these models in 10 different classes. Optimal systems of subalgebras are constructed for all models. The knowledge of optimal systems of admitted subalgebras allows constructing essentially different invariant solutions.