Wave Phenomena

Miura maps and inverse scattering for the Novikov–Veselov equation

“…In this paper we will use inverse scattering methods to solve the Novikov-Veselov (NV) equation, a completely integrable, dispersive nonlinear equation in two space and one time (2 + 1) dimensions, for the class of conductivity type initial data that we define below. Our results solve a problem posed by Lassas, Mueller, Siltanen, and Stahel [40] in their analytical study of the inverse scattering method for the NV equation.…”

“…Let us describe the direct scattering transform T and inverse scattering transform Q for the Schrödinger operator at zero energy in more detail (see Nachman [45] and Lassas, Mueller, Siltanen, and Stahel [40] for details and references). To define the direct scattering map T on potentials q ∈ C ∞ 0 (R 2 ), we seek complex geometric optics (CGO) solutions ψ = ψ(z, k) of…”

Exceptional circles of radial potentials

“…If k ∈ C \ 0 is not an exceptional point of q λ , then uniqueness of the CGO solution ψ(z, k) shows that all k ∈ C with |k | = |k| are non-exceptional for q λ as well. Furthermore, we can argue as in [14,Section 4.1] and find out that the scattering transform satisfies t λ (k) = t λ (|k|) and t λ (k) = t λ (k).…”

“…Note that the positive function ψ above solves (−∆ + q) ψ = 0. This terminology arose when Schrödinger scattering theory was used to analyze the inverse conductivity problem in Nachman [18], and was also needed in [14,19]. In those works q is not necessarily compactly supported, but a condition implying lim |z|→∞ ψ(z) = 1 is crucial.…”

“…The scattering transform T : q → t plays an important role not only in Nachman's solution of the inverse conductivity problem, but also in the solution of the Novikov-Veselov (NV) equation by the method of inverse scattering (see Lassas, Mueller, Siltanen and Stahel [14] and Perry [19] for details and further references). Thus, a clear understanding of the singularities of T is important both for Schrödinger inverse scattering and for understanding the dynamics of the NV equation and other completely integrable equations in the NV hierarchy.…”