Analysis & PDE
- Anal. PDE
- Volume 7, Number 2 (2014), 311-343.
Miura maps and inverse scattering for the Novikov–Veselov equation
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.
Anal. PDE, Volume 7, Number 2 (2014), 311-343.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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