The Novikov-Veselov Equation: Theory and Computation

TL;DR

This paper reviews recent advances in the theory and computational methods for the Novikov-Veselov equation at zero energy, including inverse scattering, special solutions, and potential classifications.

Contribution

It introduces a rigorous inverse scattering approach for zero-energy NV equation with conductivity-type data and presents new explicit solutions and computational insights.

Findings

Explicit multisoliton, ring soliton, and breather solutions

Analysis of zero-energy exceptional points

Relationship between potential types and spectral properties

Abstract

Recent progress in the theory and computation for the Novikov-Veselov (NV) equation is reviewed with initial potentials decaying at infinity, focusing mainly on the zero-energy case. The inverse scattering method for the zero-energy NV equation is presented in the context of Manakov triples, treating initial data of conductivity type rigorously. Special closed-form solutions are presented, including multisolitons, ring solitons, and breathers. The computational inverse scattering method is used to study zero-energy exceptional points and the relationship between supercritical, critical, and subcritical potentials.