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

TL;DR

This paper extends the inverse scattering method to construct global solutions for the Novikov-Veselov equation with a broader class of initial potentials, including subcritical potentials, and analyzes their stability and properties.

Contribution

It generalizes previous results by including subcritical potentials, showing they form an open set, and critical potentials form its boundary, thus broadening the class of initial data for global solutions.

Findings

Subcritical potentials form an open set.

Critical potentials form the boundary of this set.

Global solutions exist for a larger class of initial data.

Abstract

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.