Professor Davis works in the area of interfacial dynamics and stability. The interfaces can be in small-scale hydrodynamics in which two immiscible fluids are separated by an interface having surface tension, e.g. thin films, spreading of liquid on solid, and thermocapillary effects. The interfaces can be in systems with phase transformations in which, say, a liquid and its frozen counterpart are separated by a front having surface energy.

The main questions to be answered involve the nonlinear dynamic states of the system, their stability, nonlinear evolution, and pattern selection. Interfacial waves in fluids propagate, steepen, and evolve into three-dimensional complex wave systems, perhaps chaotic. Interfaces in solidification become cellular or dendritic and can lead to oscillatory states. Thin, continuous solid films of semi-conductors formed by vapor deposition can break up into islands as a result of crystal-mismatch stresses.

The means of analyzing such systems involves modelling, asymptotic and numerical methods. The answers have both practical and intrinsic interest.