The Novikov–Veselov equation and the inverse scattering method, Part I: Analysis

The Novikov–Veselov (NV) equation is a (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg–de Vries (KdV) equation. The solution of the NV equation using the inverse scattering method has been discussed in the literature, but only formally (or with smallness assumptions in the case of nonzero energy) because of the possibility of exceptional points, or singularities in the scattering data. In this work, absence of exceptional points is proved at zero energy for evolutions with compactly supported, smooth and rotationally symmetric initial data of the conductivity type: q0=γ−1/2Δγ1/2q0=γ−1/2Δγ1/2 with a strictly positive function γγ. The inverse scattering evolution is shown to be well-defined, real-valued, and preserving conductivity-type. There is no smallness assumption on the initial data.

► The (2+1)-dimensional Novikov–Veselov equation is a generalization of the KdV equation.
► The first class of admissible high-contrast initial data for the NV equation is presented.
► The solution preserves the form qr(z)=μr(z)−1Δμr(z)qr(z)=μr(z)−1Δμr(z) for a strictly positive function μμ.
► The nonlinear evolution of μrμr may have some (so far unknown) physical significance.