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

Abstract

Using the inverse scattering method, we construct global solutions to the Novikov-Veselov equation for real-valued decaying initial data q 0 with the property that the associated Schrödinger operator −∂ x ∂ x + q 0 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.

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

“…In particular, the solutions Q n,0 do not satisfy the conditions imposed in Music and Perry's [33] work.…”

“…Those spectral properties are satisfied, in particular, under some small-norm assumptions on initial data. Very recently, Music and Perry [33] showed that in the case of zero energy, if the initial datum v 0 has enough regularity in weighted Sobolev spaces and is such that the associated Schrödinger operator −∆ x,y +v 0 is critical or subcritical (i.e., nonnegative), then N V 0 (E = 0) has a global solution. Schottdorf [39] showed in his Ph.D. thesis that the modified NV equation has small global solutions in L 2 (R 2 ), by making use of suitable Koch-Tataru spaces (see also a previous work of Perry [36]).…”

Low regularity local well-posedness for the zero energy Novikov-Veselov equation

“…The precise meaning of this statement is the subject of a lively discussion, see e.g. [11,12,27,28,30,32]. As for (KdV), smooth and -in case of R 2 being their domain -rapidly decreasing solutions of (NV) satisfy a whole sequence of conservation laws: Integration of the equation over R 2 or T 2 gives that u(x, y, t)dxdy = const.…”

“…The first is the inverse scattering method, which has the great advantage of leading to some global existence theorems and to a solution formula. To the best of our knowledge, the most advanced results in this direction are those of Perry [32,Theorem 1.6] and of Music and Perry [30,Theorem 1.2], who built on earlier works [15,29] of Music and of Grinevich and Manakov. The data are assumed to belong to some weighted Sobolev space of fairly high regularity and to lie in the image of the Miura map, or to satisfy a certain (sub-)criticality condition, see Definition 1.1 in [30].…”

“…To the best of our knowledge, the most advanced results in this direction are those of Perry [32,Theorem 1.6] and of Music and Perry [30,Theorem 1.2], who built on earlier works [15,29] of Music and of Grinevich and Manakov. The data are assumed to belong to some weighted Sobolev space of fairly high regularity and to lie in the image of the Miura map, or to satisfy a certain (sub-)criticality condition, see Definition 1.1 in [30]. Unfortunately, uniqueness and hence continuous dependence remains open in this approach.…”