Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation

TL;DR

This paper numerically investigates the evolution, stability, and blow-up phenomena of line solitons in the Novikov-Veselov equation, revealing behaviors that interpolate between the KP equation limit and intermediate regimes.

Contribution

It provides the first detailed numerical analysis of blow-up mechanisms and stability of solitons in the Novikov-Veselov equation across different energy regimes.

Findings

NV behaves like KP equation as |E|→∞

Blow-ups occur at intermediate energy levels

Mechanism of blow-up is analyzed in detail

Abstract

We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the "energy" parameter $ E $. We show that as $ |E| \to \infty $, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when $ | E | $ is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.