Miura Maps and Inverse Scattering for the Novikov-Veselov Equation

We use the inverse scattering method to construct classical solutions for the Novikov-Veselov (NV) equation, solving a problem posed by Lassas, Mueller, Siltanen, and Stahel. We exploit Bogadanov'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 correspond precisely to the range of the Miura 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.