Direct and inverse scattering transforms with arbitrary spectral singularities

Abstract

IntroductionThe direct and inverse scattering problems for the n X n system (generalized the AKNS system) d -m + z [ J , m ] = q ( x ) m dx were solved by Beals and Coifman [3] for J diagonal with distinct eigenvalues and for potentials q, off-diagonal and generic. Partial solutions to the problems may be found in [13] and [7]. Certain bounded solutions m are piecewise meromorphic on the Riemann sphere relative to a contour 2, consisting of finitely many straight lines passing through the origin determined by J. Generic potentials are defined such that m extends continuously to the boundary from each component of C\Z (this implies that m has only finitely many poles) and only has simple poles of a special type. The Beals-Coifman scattering data are then defined by the multiplicative jump of m across I: and certain normalization constants at the poles. Following [8], we call z' E Z a spectral singularity of the scattering problem of q if the limit of m as z -+ z' from one of the components of C\X does not exist. Even for the potentials of Schwartz class, the poles may accumulate to spectral singularities and the spectral singularities themselves may accumulate (see Example 3.3.16). These problems (after the significant progress made by [3]) constitute the major obstacle to the complete solution of the direct and inverse scattering problems for first-order systems (including the 2 X 2 systems).When the inverse scattering method is applied in solving the Cauchy problem for the nonlinear evolution equations, the time evolution can be obtained by inverse scattering transform only if the initial q happens to be generic. Indeed, an a priori determination of all the generic potentials has not been obtained. This unpleasant situation apparently should not be ascribed to the evolution equations but to a deficiency of the existing theory of direct and inverse scattering transforms.In this paper, we find a way to overcome the deficiency by dropping the boundedness requirement for m. In our method, the bounded solutions of (1.1) are only required in a neighborhood of z = 00 (which is the only pole of zJ).According to [3], for q E L', the bounded solutions always exist there. Away

Higher‐Order Solitons in the N‐Wave System

“…For instance, in the case of involution (2.4) the Riemann-Hilbert problem has zero index because the zeros appear in complex conjugate pairs: k j = k * j . It is known [19][20][21][22][23][24] (see also Ref. [14]) that in the generic case the spectral data include simple (distinct) zeros k 1 , .…”

Asymptotics of Recurrence Relation Coefficients, Hankel Determinant Ratios, and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

“…(2) a contour D which is the finite union of piecewise-smooth, simple curves (as closed sets) is said to be orientable if its complement, C \ D, can always be divided into two, possibly disconnected, disjoint open sets + + + and − − − , either of which has finitely many components, such that D admits an orientation so that it can either be viewed as a positively oriented boundary D + for + + + or as a negatively oriented boundary D − for − − − [71], that is, the (possibly disconnected) components of C \ D can be coloured by + or by − in such a way that the + regions do not share boundary with the − regions, except, possibly, at finitely many points [72]; (3) for each segment of an oriented contour D, according to the given orientation, the "+" side is to the left and the "−" side is to the right as one traverses the contour in the direction of orientation; so, for a matrix-valued function A(z), A ± (z) denote the nontangential limits A ± (z) := lim z →z…”

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

“…1 0 are, respectively, the raising and lowering matrices, 0 = 0 0 0 0 , R ± := {x ∈ R; ±x > 0}, C ± := {z ∈ C; ±Im(z) > 0}, and sgn(x) := 0 if x = 0 and x|x| −1 if x 0; (2) for a scalar ω and a 2×2 matrix Υ, ω ad(σ 3 ) Υ := ω σ 3 Υω −σ 3 ; (3) a contour D which is the finite union of piecewise-smooth, simple curves (as closed sets) is said to be orientable if its complement C \ D can always be divided into two, possibly disconnected, disjoint open sets ℧ + and ℧ − , either of which has finitely many components, such that D admits an orientation so that it can either be viewed as a positively oriented boundary D + for ℧ + or as a negatively oriented boundary D − for ℧ − [85], that is, the (possibly disconnected) components of C \ D can be coloured by + or − in such a way that the + regions do not share boundary with the − regions, except, possibly, at finitely many points [86]; (4) for each segment of an oriented contour D, according to the given orientation, the "+" side is to the left and the "-" side is to the right as one traverses the contour in the direction of orientation, that is, for a matrix…”

“…♯ can be coloured by the two colours ±, C λ o ± are complementary projections [2,85,102,103], that is, (C − are complementary, the contour Γ ♯ can always be oriented in such a way that the ± regions lie on the ± sides of the contour, respectively.) The solution of the above (normalised at λ o ) RHP is given by the following integral representation.…”

“…In Section 2, necessary facts from the theory of compact Riemann surfaces are given, the respective 'even degree' and 'odd degree' RHPs on R are stated and the corresponding variational problems for the associated equilibrium measures are discussed, and the main results of this work, namely, asymptotics (as n → ∞) of π π π 2n (z) (in C), and ξ (2n) n and φ 2n (z) (in C) are stated in Theorems 2.3.1 and 2.3.2, respectively. In Section 3, the detailed analysis of the 'even degree' variational problem and the associated equilibrium measure is undertaken, including the construction of the so-called g-function, and the RHP formulated in Section 2 is reformulated as an equivalent, auxiliary RHP, which, in Sections 4 and 5, is augmented, by means of a sequence of contour deformations and transformations à la Deift-Venakides-Zhou, into simpler, 'model' (matrix) RHPs which, as n → ∞, and in conjunction with the Beals-Coifman construction [84] (see, also, the extension of Zhou [85]) for the integral representation of the solution of a matrix RHP on an oriented contour, are solved explicitly (in closed form) in terms of Riemann theta functions (associated with the underlying finite-genus hyperelliptic Riemann surface) and Airy functions, from which the final asymptotic (as n → ∞) results stated in Theorems 2.3.1 and 2.3.2 are proved. The paper concludes with an Appendix.…”