Direct and Inverse Scattering on the Line

Abstract

The theory of valuations, O. F. G. Schilling 5 The kernel function and conformal mapping, S. Bergman 6 Introduction to the theory of algebraic functions of one variable, C. C. Chevalley 7.1 The algebraic theory of semigroups, Volume I, A. H. Clifford and G. B. Preston 7.2 The algebraic theory of semigroups, Volume II, A. H. Clifford and G. B. Preston 8 Discontinuous groups and automorphic functions, J. Lehner 9 Linear approximation, Arthur Sard 10 An introduction to the analytic theory of numbers, R. Ayoub 11 Fixed points and topological degree in nonlinear analysis, J. Cronin 12 Uniform spaces, J. R. Isbell

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

“…As indicated in [16], the distinctness of c~j is not an essential restriction 9 It has been proved that the inverse problem has a unique solution within a certain class of potentials for an open and dense set of scattering data. The solution of the inverse scattering problem for such linear systems is useful in solving the Cauchy problem for various nonlinear evolution equations; for details and further references, we refer the interested reader to [2,12,16] and the references therein. 1 …”

The KdV Hierarchy: Universality and a Painlevé Transcendent

“…This construction is well understood [10,1,38,3,16], but we summarize the main lines here for the convenience of the reader. For negative u, the Schrödinger operator has no point spectrum.…”

“…Using (2.4) and the asymptotics for the Jost solutions as λ → ∞, one shows that M solves a RH problem, see [1,38,3]:…”

“…Note that the functions ∆ (1) and ∆ (2) are meromorphic functions in U with simple poles at u c , and that ∆ (3) is meromorphic in U with a double pole at u c . Observe that the x-derivative of v R is not close to I on ∂U as ǫ → 0, since the x-derivatives of U and Q cause multiplication with ǫ −6/7 by (3.72)-(3.73).…”

Minimum energy pulse synthesis via the inverse scattering transform

“…There are several different approaches to solving the inverse scattering problem stated at the end of the previous section, see [1], [2], [4] or [8]. In this section we describe the solution of this problem via the Marchenko equations.…”