Direct and inverse scattering problems for the first‐order discrete system associated with the derivative NLS system

Abstract

The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semidiscrete version of the derivative nonlinear Schrödinger (NLS) system. The Jost solutions, the scattering coefficients, the bound-state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi-discrete version of the NLS system. The bound-state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to solve the inverse problem to recover the potential values in the first-order discrete system. One of these methods uses a newly derived, standard discrete Marchenko system using as input the scattering data directly coming from the first-order discrete system. This new Marchenko method is presented in a way that it is generalizable to other first-order systems both in the discrete and continuous cases for which a Marchenko system and a Marchenko theory are not yet available. Finally, using the time-evolved scattering data set, the inverse scattering transform is applied on the corresponding semi-discrete derivative NLS system, and in

The generalized Marchenko method in the inverse scattering problem for a first-order linear system

“…As the bound-state norming constants, we use the double-indexed quantities c jk for 1 ≤ j ≤ N and 0 ≤ k ≤ (m j − 1) and the double-indexed quantities cjk for 1 ≤ j ≤ N and 0 ≤ k ≤ ( mj − 1). The construction of the bound-state norming constants c jk from the transmission coefficient T (ζ) and the Jost solutions φ(ζ, x) and ψ(ζ, x) and the construction of the bound-state norming constants cjk from the transmission coefficient T (ζ) and the Jost solutions φ(ζ, x) and ψ(ζ, x) are analogous to the constructions presented for the discrete version of (1.1), and we refer the reader to [9] for the details. Such a construction involves the determination of the double-indexed "residues" t jk with 1 ≤ j ≤ N and 1 ≤ k ≤ m j and the the double-indexed "residues" tjk with 1 ≤ j ≤ N and 1 ≤ k ≤ mj , respectively, by using the expansions of the transmission coefficients at the bound-state poles, which are given by…”

“…We then recursively obtain (3.3). For the details of the procedure, we refer the reader to [9]. Similarly, the double-indexed dependency constants γjk with 1 ≤ j ≤ N and 0 ≤ k ≤ ( mj − 1) appear in the coefficients when we express at λ = λj the value of each…”

“…, and this procedure is explained in the proof of Theorem 4.2 and it is similar to the procedure described in Theorem 15 of [9]. In a similar manner, the norming constants cjk are formed by using the set of residues { tjk } mj k=1 and the set of dependency constants {γ jk } mj −1 k=0 .…”

“…For simplicity and clarity, we outline the main steps of the procedure by omitting the details. We refer the reader to [9] where the details of the procedure are presented for the discrete version of (1.1). The steps presented in [9] are general enough to apply to (1.1) and other linear systems.…”

“…We present the Marchenko method for (1.1) in such a way that the method can be applied on other linear systems and also on their discrete versions. We have already developed [9] the Marchenko method for the discrete analog of the linear system (1.1), and hence our emphasis in this paper is the development of the Marchenko method for the linear system (1.1).…”