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

TL;DR

This paper establishes local well-posedness for the two-dimensional Novikov-Veselov equation at positive energy using dispersive estimates, Fourier analysis, and $X^{s,b}$ spaces, revealing stability across energy signs.

Contribution

It introduces new dispersive estimates for rational symbols and proves local well-posedness of NV at positive energy, extending understanding of its behavior across energy regimes.

Findings

NV is locally well-posed for s > 1/2 in H^s

Dispersive estimates are derived for positive energy case

Explicit solutions at zero energy show interesting large-time behavior

Abstract

In this paper we continue our study on the Cauchy problem for the two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schr\"odinger operator at a fixed energy parameter. This work is concerned with the case of positive energy. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining two different frequency regimes. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. We combine the linear estimates with the Fourier decomposition method and $ X^{s,b} $ spaces to obtain local well-posedness of NV at positive energy in $H^s$, $s>\frac12$. Our result implies, in particular, that {\it at least} for $s>\frac12$, NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev-Petviashvili equations. As a supplement, we provide some new explicit solutions of NV at zero energy which exhibit an interesting behavior at large times.