TRAVELING-WAVE SOLUTIONS TO THIN-FILM EQUATIONS

Thin films can be effectively described by the lubrication approximati on, in which the equation of motion is h(1)+(h(n)h(xxx))(x) =0. Here h is a necessarily positive quantity which represents the height or thi ckness of the film. Different values of n, especially 1, 2, and 3 corr espond to different physical situations. This equation permits solutio ns in the form of traveling disturbances with a fixed form. If u is th e propagation velocity, the resulting equation for the disturbance uh( x)=(hnh(xxx))(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 betw een different qualitative behaviors occur at n=3,2,3/2, and 1/2. For e xample, if u is not zero, solitonlike solutions defined on a finite in terval 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 advanc ing-front solutions, with support on (-infinity,t). For n > 1/2, singl e-minimum solutions diverging at +/- infinity are possible. The generi c solution, present for all positive values of n, is a receding front, which diverges at finite x for n <O.