Papers Published
Abstract:
Recent studies of liquid films driven by competing forces
due to surface tension gradients and gravity reveal
that undercompressive traveling waves play an important role
in the
dynamics when the competing forces are comparable.
In this paper we provide a theoretical framework for
assessing the spectral
stability of compressive and undercompressive traveling
waves
in thin film models.
Associated with the linear stability problem is an Evans
function
which vanishes precisely at eigenvalues of the linearized
operator.
The structure of an index related to the Evans function
explains computational
results for stability of compressive waves.
A new formula for the index in the undercompressive case
yields results consistent with stability.
In considering stability of undercompressive waves to transverse perturbations, there is an apparent inconsistency between long-wave asymptotics of the largest eigenvalue and its actual behavior. We show that this paradox is due to the unusual structure of the eigenfunctions and we construct a revised long-wave asymptotics. We conclude with numerical computations of the largest eigenvalue, comparisons with the asymptotic results, and several open problems associated with our findings.