Heteroclinic orbits, mobility parameters and stability for thin film type equations

TL;DR

This paper numerically investigates the stability and heteroclinic orbits of solutions to a class of thin film equations, analyzing how mobility exponents influence the existence and nature of steady states and singularities.

Contribution

It provides new insights into the nonlinear stability, heteroclinic connections, and the effects of mobility exponents on singularity formation in thin film equations.

Findings

Heteroclinic orbits are perturbed but not broken when mobility exponents change.

The exponent n influences the occurrence and number of touch-down singularities.

Stability of steady states depends on the parameters and exponents.

Abstract

We study numerically the phase space of the evolution equation h_t = -(h^n h_{xxx})_x - B (h^m h_x)_x . Here h(x,t) is nonnegative, n>0 and m is real, and the Bond number B is positive. We pursue three goals: to investigate the nonlinear stability of the positive periodic and constant steady states; to locate heteroclinic connecting orbits between these steady states and the compactly supported 'droplet' steady states; and to determine how these orbits change when the 'mobility' exponents n and m are changed. For example, we change the mobility coefficients in such a way that the steady states are unchanged and find evidence that heteroclinic orbits between steady states are perturbed but not broken. We also find that when there appear to be touch-down singularities, the exponent n affects whether they occur in finite or infinite time. It also can affect whether there is one touch-down or two touch-downs per period.