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

Global solutions for the zero-energy Novikov-Veselov equation by inverse scattering