Traveling-wave solutions to thin-film equations

Thin films can be effectively described by the lubrication approximation, in which the equation of motion is ${\mathit{h}}_{\mathit{t}}$+(${\mathit{h}}^{\mathit{n}}$ ${\mathit{h}}_{\mathit{x}\mathit{x}\mathit{x}}$ ${)}_{\mathit{x}}$=0. Here h is a necessarily positive quantity which represents the height or thickness of the film. Different values of n, especially 1, 2, and 3 correspond to different physical situations. This equation permits solutions in the form of traveling disturbances with a fixed form. If u is the propagation velocity, the resulting equation for the disturbance is ${\mathit{uh}}_{\mathit{x}}$=(${\mathit{h}}^{\mathit{n}}$ ${\mathit{h}}_{\mathit{x}\mathit{x}\mathit{x}}$ ${)}_{\mathit{x}}$. Here, quantitative and qualitative solutions to the equation are presented. The study has been limited to the intervals in x where the solutions are positive. It is found that transitions between different qualitative behaviors occur at n=3, 2, 3/2, and 1/2. For example, if u is not zero, solitonlike solutions defined on a finite interval are only possible for n<3. More specific results can be obtained. In the case in which the velocity is zero, solitons occur for n<2. For n=1, the region 3/2<n is characterized by the presence of advancing-front solutions, with support on (-∞,t). For n>1/2, single-minimum solutions diverging at ±∞ are possible. The generic solution, present for all positive values of n, is a receding front, which diverges at finite x for n<0.