Using the inverse scattering method, we construct global solutions to the Novikov-Veselov equation for real-valued decaying initial data q0 with the property that the associated Schrödinger operator 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 Grinevich and Manakov.
|State||Published - Jun 6 2018|
Bibliographical noteFunding Information:
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1208778.
We are grateful to the referee for a careful reading of this paper, for helpful suggestions, and for pointing out an error in a previous version of the manuscript. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1208778.
© 2018 IOP Publishing Ltd & London Mathematical Society.
- Novikov-Veselov equation
- inverse scattering
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy (all)
- Applied Mathematics