Large data global well-posedness for the modified Novikov-Veselov system

TL;DR

This paper proves the global well-posedness and scattering of solutions for the large data modified Novikov-Veselov system using inverse scattering, extending previous small data results and introducing new inequalities.

Contribution

It establishes the first large data global well-posedness and scattering results for the mNV system using inverse scattering methods.

Findings

Proves global well-posedness for large $L^2$ data.

Shows solutions scatter as time approaches infinity.

Introduces a new nonlinear Gagliardo-Nirenberg inequality.

Abstract

The modified Novikov-Veselov system (mNV) is a cubic third order dispersive evolution in two space dimensions. It is also completely integrable, belonging to the same hierarchy as the defocusing Davey-Stewartson II (DS II) system. The mNV system is $L^2$ critical. Some time ago, Schottdorf proved that for small $L^2$ initial data, the mNV equation is globally well-posed. In this article, we consider instead the large data problem, using inverse scattering methods. Our main result asserts that the mNV system is globally well-posed for large $L^2$ data, with the solutions scattering as time goes to $\pm \infty$. One key ingredient in the proof, which is of independent interest, is a new nonlinear Gagliardo-Nirenberg inequality for the associated scattering transform. As a byproduct of our main result, we are also able to prove a global well-posedness result for the closely related Novikov-Veselov problem at the critical $\dot H^{-1} + L^1$ level, for a range of data which can heuristically be described as soliton-free. Here we use the associated Miura map to connect the mNV and the NV flows. In order to characterize the range of the Miura map, we prove another result of independent interest, namely a sharp, scale invariant form of the Agmon-Allegretto-Piepenbrink principle in the critical case of two space dimensions.