A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy

TL;DR

This paper investigates the large time behavior of solutions to the Novikov-Veselov equation at positive energy, focusing on reflectionless potentials, and finds no soliton-like waves emerge asymptotically, contrasting with KdV behavior.

Contribution

It provides the first analysis of large time asymptotics for the Novikov-Veselov equation with reflectionless potentials at positive energy.

Findings

No isolated soliton waves in large time asymptotics

Reflectionless potentials exhibit different asymptotic behavior than KdV

Contrast with classical KdV soliton results

Abstract

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov--Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data.