Miura maps and inverse scattering for the Novikov–Veselov equation

Perry, Peter

doi:10.2140/apde.2014.7.311

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Miura maps and inverse scattering for the Novikov–Veselov equation

Peter A. Perry

Vol. 7 (2014), No. 2, 311–343

DOI: 10.2140/apde.2014.7.311

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.

Keywords

Novikov–Veselov equation, Miura map, Davey–Stewartson equation

Mathematical Subject Classification 2010

Primary: 37K15

Secondary: 35Q53, 47A40, 78A46

Milestones

Received: 9 November 2012

Revised: 18 December 2013

Accepted: 10 February 2014

Published: 30 May 2014

Authors

Peter A. Perry
Mathematics Department University of Kentucky Lexington, KY 40506-0027 United States

Vol. 7, No. 2, 2014