  The presence of slight azimuthal asymmetry in the initial shape of an
underwater bubble entirely alters the final breakup dynamics. Here I examine
the influence of initial asymmetry on the final breakup by simulating the
bubble surface evolution as a Hamiltonian evolution corresponding to an
inviscid, two-dimensional, planar implosion. I find two types of breakups: a
previously reported coalescence mode in which distant regions along the
air-water surface curve inwards and eventually collide with finite speed, and a
hitherto unknown cusp-like mode in which the surface develops sharp tips whose
radii of curvature are much smaller than the average neck radius. I present
three sets of results that characterize the nature of this cusp mode. First, I
show that the cusp mode corresponds to a saddle-node. In other words, an
evolution towards a cross-section shape with sharp tips invariably later
evolves away from it. In phase space, this saddle-node separates coalescence
modes whose coalescence planes lie along different spatial orientations.
Second, I show that the formation of the sharp tips can be interpreted as a
weakly first-order transition which becomes second-order, corresponding to the
formation of a finite-time curvature singularity, in the limit that the initial
perturbation amplitude approaches zero. Third I show that, as the curvature
singularity is approached, the maximum surface curvature diverges approximately
as $(t_c - t)^{-0.8}$, where $t_c$ is the onset time of the singularity and the
maximum velocity diverges approximately as $(t_c-t)^{-0.4}$. In practice, these
divergences imply that viscous drag and compressibility of the gas flow, two
effects not included in my analysis, become significant as the interface
evolves towards the curvature singularity.
