The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis

TL;DR

This paper proves the absence of exceptional points at zero energy for the Novikov-Veselov equation with certain initial data, establishing the well-definedness and properties of the inverse scattering method without smallness assumptions.

Contribution

It demonstrates the absence of exceptional points at zero energy for a class of initial data, ensuring the inverse scattering method is well-defined for the NV equation without smallness constraints.

Findings

Absence of exceptional points at zero energy for specific initial data.

Inverse scattering evolution is well-defined and preserves conductivity type.

No smallness assumption needed on initial data.

Abstract

The Novikov-Veselov (NV) equation is a (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. Solution of the NV equation using the inverse scattering method has been discussed in the literature, but only formally (or with smallness assumptions in case of nonzero energy) because of the possibility of exceptional points, or singularities in the scattering data. In this work, absence of exceptional points is proved at zero energy for evolutions with compactly supported, smooth and rotationally symmetric initial data of the conductivity type: $q_0=\gamma^{-1/2}\Delta\gamma^{1/2}$ with a strictly positive function $\gamma$. The inverse scattering evolution is shown to be well-defined, real-valued, and preserving conductivity-type. There is no smallness assumption on the initial data.