Tau Functions Associated with Linear Systems

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

“…In the KdV context, the Fredholm determinant of I + H appears to be studied for the first time in [30]. In recent [7,10] some ideas of [30] were extended far beyond the KdV case.…”

“…It is important that three out of four conditions in this characterization follow from the Zakharov-Faddeev trace formula (4.7) below, which is extended in [20] to L 2 potentials, all conditions being very mild. On the other hand, detailed descriptions of the Jost solution, the Weyl m-function, and the spectral measure 7 of the half-line (Dirichlet) Schrodinger operator with WvN type potentials (and their sums, finite or even infinite) are given in [17,22] (see section 4). This way we construct two spectral measures ρ ± (E) with a desirable behavior for E > ε > 0.…”

The inverse scattering transform for weak Wigner-von Neumann type potentials

“…In the KdV context, the Fredholm determinant of I + H appears to be studied for the first time in [33]. In recent [6,9] some ideas of [33] were extended far beyond the KdV case.…”

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

“…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].…”