Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 2 (2014), 311-343.

Miura maps and inverse scattering for the Novikov–Veselov equation

Peter Perry

Full-text: Open access

Abstract

We use the inverse scattering method to solve the zero-energy Novikov–Veselov (NV) equation for initial data of conductivity type, solving a problem posed by Lassas, Mueller, Siltanen, and Stahel. We exploit Bogdanov’s Miura-type map which transforms solutions of the modified Novikov–Veselov (mNV) equation into solutions of the NV equation. We show that the Cauchy data of conductivity type considered by Lassas, Mueller, Siltanen, and Stahel lie in the range of Bogdanov’s Miura-type map, so that it suffices to study the mNV equation. We solve the mNV equation using the scattering transform associated to the defocussing Davey–Stewartson II equation.

Article information

Source
Anal. PDE, Volume 7, Number 2 (2014), 311-343.

Dates
Received: 9 November 2012
Revised: 18 December 2013
Accepted: 10 February 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731491

Digital Object Identifier
doi:10.2140/apde.2014.7.311

Mathematical Reviews number (MathSciNet)
MR3218811

Zentralblatt MATH identifier
06322734

Subjects
Primary: 37K15: Integration of completely integrable systems by inverse spectral and scattering methods
Secondary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx] 78A46: Inverse scattering problems

Keywords
Novikov–Veselov equation Miura map Davey–Stewartson equation

Citation

Perry, Peter. Miura maps and inverse scattering for the Novikov–Veselov equation. Anal. PDE 7 (2014), no. 2, 311--343. doi:10.2140/apde.2014.7.311. https://projecteuclid.org/euclid.apde/1513731491


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