The interaction of two lump solitons described by the Kadomtsev-Petviashvili I (KPI) equation is analysed using both exact and numerical methods. The numerical method is based on a third order Runge-Kutta method, and a Crank-Nicholson scheme. The main characteristic of a direct interaction when the two lumps are initially aligned along the x-axis is that they may separate in the y-direction, but then come back to the x-axis after collision; the dependence of the maximum separation in the y-direction on the relative velocity difference is described. Two lumps may also experience an abrupt phase change in the case of an oblique interaction.

The development of spontaneous stationary equilibrium patterns in thin liquid films is investigated by means of a hexagonal-planform weakly nonlinear stability analysis applied to the appropriate governing evolution equation for this phenomenon. In the long-wavelength limit the mathematical system modeling the liquid film can be reduced to such a single nonlinear partial differential timeevolution equation describing the layer thickness on an unbounded two-dimensional spatial domain and including the effects of gravity, intermolecular forces, and temperature-dependent surface tension. The main result of this analysis is that supercritical equilibrium patterns can occur for an interval of mean layer thickness with subcritical rupture occurring outside that interval. These patterns consist of surface ridges and hexagonal network-like cells or close-packed configurations of nanodroplets separated by relatively flat ultra thin films. In particular those morphological phase separation patterns are generated by the coupling between the long-range attractive and short-range repulsive intermolecular forces with cells being stable for the thicker layers; nanodroplets, for the thinner ones; and ridges, for layers of intermediate thickness. These theoretical predictions are in accord with both relevant experimental evidence involving thin liquid polymer, crystal, and metal films coating a solid substrate and numerical simulations of similar model equations as well as being consistent with dewetting-type rupture occurring for such situations by hole formation in relatively thick layers but by drop formation in thinner ones.

The development of stationary patterns on a thin polymer surface subject to an electric field is studied by means of the hexagonal-planform weakly nonlinear stability analysis and numerical simulations. The time evolution of the interface between the air and the polymer film on the unbounded spatial domain is described by a thin film equation, incorporating the electric driving force and the surface diffusion. The nonlinear interfacial growth includes the amplitude equations and superposition of one-dimensional structures at regular orientations. The pattern selection is driven by the subcritical instability mechanism in which the relative thickness of the polymer film plays a critical role.

A major upgrade to the SLC final focus was installed in 1994 to eliminate the dominant third-order aberration of the system, and thereby to reduce the vertical beam size at the IP by a factor of two. At low current, the optimal beam size of about 400 nm is now routinely established, and its sensitivity to orbit variations, to changes of emittance and energy spread, and to other beam parameters has been studied. For intensities above 3 × 10 10 particles per bunch, tuning is more difficult due to increased fluctuations of energy, orbit, and emittances. Nonetheless, the expected beam size of about 600 nm has been observed. New procedures and diagnostics allow easier tuning and optimization of the final focus, and also a first measurement of the emittance increase in the arcs.