On a formation of singularities of solutions to soliton equations represented by L,A,B-triples

TL;DR

This paper investigates how singularities form in solutions to certain 2+1-dimensional soliton equations, linking blow-up phenomena to spectral properties and presenting exact solutions with indeterminacies.

Contribution

It introduces a framework using L,A,B-triples for analyzing singularity formation in soliton equations and characterizes the nature of these singularities as indeterminacies.

Findings

Singularities correspond to non-conservation of the zero spectral level.

Exact solutions depend on two functional parameters.

All singularities are indeterminacies with directional limits.

Abstract

We discuss the mechanism of formation of singularities of solutions to the Novikov-Veselov, modified Novikov-Veselov, and Davey-Stewartson II (DSII) equations obtained by the Moutard type transformations. These equations admit the L,A,B-triple presentation, the generalization of the L,A-pairs for 2+1-soliton equations. We relate the blow-up of solutions to the non-conservation of the zero level of discrete spectrum of the L-operator. We also present a class of exact solutions, of the DSII system, which depend on two functional parameters, and show that all possible singularities of solutions to DSII equation obtained by the Moutard transformation are indeterminancies, i.e., points when approaching which in different spatial directions the solution has different limits.