Absence of solitons with sufficient algebraic localization for the Novikov-Veselov equation at nonzero energy

TL;DR

This paper proves that the Novikov-Veselov equation, a 2+1 dimensional analog of KdV, does not admit solitons with strong algebraic localization at nonzero energies, highlighting limitations in soliton solutions for this equation.

Contribution

It establishes the nonexistence of highly localized solitons for the Novikov-Veselov equation at nonzero energies, extending understanding of its solution structure.

Findings

No solitons with localization stronger than O(|x|^{-3}) exist at nonzero energies.

Results apply to both positive and negative energy cases.

Provides insights into the limitations of soliton solutions in higher-dimensional integrable systems.

Abstract

We show that the Novikov--Veselov equation (an analog of KdV in dimension 2 + 1) at positive and negative energies does not have solitons with the space localization stronger than O(|x|^{-3}) as |x| \to \infty.