Stability of compressive and undercompressive thin film travelling waves

Recent studies of liquid films driven by competing forces due to surface tension gradients and gravity reveal that undercompressive travelling 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 travelling 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 behaviour. 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.