Miura Maps and Inverse Scattering for the Novikov-Veselov Equation

Abstract

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.

30 pages

References in corpus (6)

• Exceptional circles of radial potentials
• Estimates for a family of multi-linear forms
• Absence of exponentially localized solitons for the Novikov--Veselov equation at negative energy
• Gauge-invariant description of some (2+1)-dimensional integrable nonlinear evolution equations
• Large time asymptotics for the Grinevich-Zakharov potentials
• Third order semilinear dispersive equations related to deep water waves

Cited by in corpus (9)

• Exceptional circles of radial potentials
• Nonlinear Fourier analysis for discontinuous conductivities: computational results
• Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
• Global Well-Posedness and Long-time Asymptotics for the Defocussing Davey-Stewartson II Equation in $H^{1,1}(R^2)$
• Faddeev eigenfunctions for two-dimensional Schrodinger operators via the Moutard transformation
• The Cauchy problem for the generalized hyperbolic Novikov-Veselov equation via the Moutard symmetries
• Low regularity local well-posedness for the zero energy Novikov-Veselov equation
• The Cauchy problem for the generalized hyperbolic Novikov-Veselov equation
• Absence of sufficiently localized traveling wave solutions for the Novikov-Veselov equation at zero energy