The modified Novikov-Veslov Equation and the Inverse Scattering Transform

TL;DR

This paper corrects previous errors in the inverse scattering solutions of the modified Novikov-Veselov equation, ensuring the mathematical validity of the solution method for this integrable PDE.

Contribution

It provides corrected formulas and proofs for solving the modified Novikov-Veselov equation via inverse scattering, clarifying its relation to the Davey-Stewartson II equation.

Findings

Corrected the nonlinearity formula for the mNV equation

Validated inverse scattering as a solution method for mNV

Ensured the solution yields a classical solution

Abstract

This paper corrects several errors in the author's previous papers (Journal of Spectral Theory 2016, Analysis and PDE 2014) on the Davey-Stewartson II (DS II) and modified Novikov-Veselov (mNV) equations. In each of these papers a proof was given that the solution by inverse scattering yields a classical solution to the PDE. The mNV equation lies in the integrable hierarchy of the DS II equation, so the same scattering transform may be used in both cases. In the 2014 paper, an incorrect formula is given for the nonlinearity in the mNV equation. Here we correct errors in the proof and obtain a correct statement of the mNV equation as solved by inverse scattering.