The modified Novikov-Veslov Equation and the Inverse Scattering Transform
Peter A. Perry
TL;DR
This work corrects prior errors in the inverse-scattering treatment of the modified Novikov-Veselov (mNV) and DS II equations, establishing the correct nonlinearity for the mNV equation solved by inverse scattering. Building on Beals-Coifman–style direct/inverse scattering maps $\mathcal{R}$ and $\mathcal{I}$ and their associated $\,\overline{\partial}$-problems, the authors derive a real-phase evolution framework and compute the mNV nonlinear terms via large-$k$ asymptotics of the scattering data. The main result is the corrected nonlocal cubic nonlinearity $N_{mNV}(u)$ and the evolution equation $u_t + (\partial^3 + \overline{\partial}^3)u = N_{mNV}(u)$, with an explicit expression for $(4/3)N_{mNV}(u)$ in terms of $u$ and its derivatives. This solidifies the inverse-scattering approach for DS II and its mNV member, ensuring rigorous consistency of the solution framework and providing a corrected, explicit nonlinear term essential for well-posedness and further analysis.
Abstract
This paper corrects several errors in the author's previous papers (Journal of Spectral Theory 2016, Analysis and PDE 2014) on the Davey-Stewartson II (DS II) and modified Novikov-Veselov (mNV) equations. In each of these papers a proof was given that the solution by inverse scattering yields a classical solution to the PDE. The mNV equation lies in the integrable hierarchy of the DS II equation, so the same scattering transform may be used in both cases. In the 2014 paper, an incorrect formula is given for the nonlinearity in the mNV equation. Here we correct errors in the proof and obtain a correct statement of the mNV equation as solved by inverse scattering.