The inverse scattering transform for weak Wigner–von Neumann type potentials ^*

Abstract

In the context of the Cauchy problem for the Korteweg–de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner–von Neumann type potentials with small coupling constants. Our arguments are based on the theory of Hankel operators.

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

“…This can be thought of as the Fourier transform representation of the formulation we gave in Section 2. Recently this context has been used to prove interesting integrability results/connections for the cubic Szegö equation, see Pocovnicu [85], Grellier and Gerard [51] and Gerard and Pushnitski [49], and to extend regularity results for the Korteweg-de Vries equation, see Grudsky and Rybkin [52,53,54]. There is a natural decomposition L 2 (R) = H + ⊕ H − and thus an immediate direction to pursue would be to consider our Marchenko equation and Fredholm Grassmannian flow in this context and establish a connection to the results of, for example, Grellier and Gerard [51] and Grudsky and Rybkin [54].…”

“…The role of Hankel operators in integrable systems first explored by Pöppe, has recently re-emerged as an active and fruitful research direction. In particular, relevant to our results herein are Blower and Newsham [15], Blower and Doust [14], Grudsky and Rybkin [52,53,54], Grellier and Gerard [51] and Gerard and Pushnitski [49]. The combinatorial algebraic approach we consider herein was introduced in Malham [69] for the simpler non-commutative potential Korteweg-de Vries equation; also see Doikoi et al [31].…”