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

I. A. Taimanov

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.

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

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.
Paper Structure (6 sections, 2 theorems, 74 equations)

This paper contains 6 sections, 2 theorems, 74 equations.

Table of Contents

  1. $L,A,B$-triples
  2. $L,A$-pairs
  3. Two-dimensional soliton equations represented by $L,A,B$-triples
  4. Formation of singularities of solutions and the discrete spectrum of $L$-operators
  5. The Moutard transformation for the DSII equa-tion
  6. Remarks

Key Result

Theorem 1

1) Given for the functions satisfy the Novikov--Veselov equation and 2) Given for the function satisfies the modified Novikov--Veselov equation and where and the polynomial $|a|^2+|b|^2$ vanishes if and only if $C=t$. Moreover 3) For the function satisfies the Davey--Stewartson II equation and where Moreover

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2