  Recent progress in the theory and computation for the Novikov-Veselov (NV)
equation is reviewed with initial potentials decaying at infinity, focusing
mainly on the zero-energy case. The inverse scattering method for the
zero-energy NV equation is presented in the context of Manakov triples,
treating initial data of conductivity type rigorously. Special closed-form
solutions are presented, including multisolitons, ring solitons, and breathers.
The computational inverse scattering method is used to study zero-energy
exceptional points and the relationship between supercritical, critical, and
subcritical potentials.
