A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data

TL;DR

This paper investigates the long-time behavior of solutions to the Novikov-Veselov equation at negative energy, showing they decay as t^{-3/4} and providing evidence for the optimality of this decay rate.

Contribution

It establishes the large time asymptotics for solutions of the Novikov-Veselov equation at negative energy with non-singular scattering data, extending understanding of its long-term behavior.

Findings

Solution decays as t^{-3/4} at large times

Asymptotics are optimal based on presented arguments

Results apply to non-singular scattering data

Abstract

In the present paper we are concerned with the Novikov--Veselov equation at negative energy, i.e. with the $ (2 + 1) $--dimensional analog of the KdV equation integrable by the method of inverse scattering for the two--dimensional Schr\"odinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non--singular scattering data behaves asymptotically as $ \frac{\const}{t^{3/4}} $ in the uniform norm at large times $ t $. We also present some arguments which indicate that this asymptotics is optimal.