Miura Maps and Inverse Scattering for the Novikov-Veselov Equation

TL;DR

This paper applies inverse scattering techniques and Miura maps to construct solutions for the Novikov-Veselov equation, linking it to the modified NV equation and the defocusing Davey-Stewartson II equation, addressing a problem posed by prior researchers.

Contribution

It introduces a method to solve the NV equation via inverse scattering and Miura maps, connecting it to the mNV and Davey-Stewartson II equations for the first time.

Findings

Constructed classical solutions for the NV equation using inverse scattering.

Identified the range of the Miura map as the class of initial data considered.

Linked the solution of the mNV equation to the defocusing Davey-Stewartson II equation.

Abstract

We use the inverse scattering method to construct classical solutions for the Novikov-Veselov (NV) equation, solving a problem posed by Lassas, Mueller, Siltanen, and Stahel. We exploit Bogadanov's Miura-type map which transforms solutions of the modified Novikov-Veselov (mNV) equation into solutions of the NV equation. We show that the Cauchy data of conductivity type considered by Lassas, Mueller, Siltanen, and Stahel correspond precisely to the range of the Miura map, so that it suffices to study the mNV equation. We solve the mNV equation using the scattering transform associated to the defocussing Davey-Stewartson II equation.