  Using the inverse scattering method, we construct global solutions to the
Novikov-Veselov equation for real-valued decaying initial data q with the
property that the associated Schrodinger operator with potential q is
nonnegative. Such initial data are either critical (an arbitrarily small
perturbation of the potential makes the operator nonpositive) or subcritical
(sufficiently small perturbations of the potential preserve non-negativity of
the operator). Previously, Lassas, Mueller, Siltanen and Stahel proved global
existence for critical potentials, also called potentials of "conductivity
type." We extend their results to include the much larger class of subcritical
potentials. We show that the subcritical potentials form an open set and that
the critical potentials form the nowhere dense boundary of this open set. Our
analysis draws on previous work of the first author and on ideas of P. G.
Grinevich and S. V. Manakov.
