Two-dimensional rational solitons and their blow-up via the Moutard transformation

TL;DR

This paper develops a method using the Moutard transformation to explicitly construct two-dimensional Schrödinger operators with rational, rapidly decaying potentials and degenerate kernels, demonstrating finite-time blow-up solutions for the Novikov-Veselov equation.

Contribution

It introduces a novel procedure for constructing explicit rational potentials with degenerate kernels and analyzes their blow-up behavior in the context of the Novikov-Veselov equation.

Findings

Constructed explicit examples of rational decaying potentials

Showed finite-time blow-up for solutions with these potentials

Extended previous sketches of the construction method

Abstract

By using the Moutard transformation of two-dimensional Schroedinger operators we derive a procedure for constructing explicit examples of such operators with rational fast decaying potentials and degenerate $L_2$-kernels (this construction was sketched in arXiv:0706.3595) and show that if we take some of these potentials as the Cauchy data for the Novikov-Veselov equation (a two-dimensional version of the Korteweg-de Vries equation), then the corresponding solutions blow up in a finite time