In recent years there have been important and far reaching developments in the study of nonlinear waves and a class of nonlinear wave equations which arise frequently in applications. The wide interest in this field comes from the understanding of special waves called 'solitons' and the associated development of a method of solution to a class of nonlinear wave equations termed the inverse scattering transform (IST). Before these developments, very little was known about the solutions to such 'soliton equations'. The IST technique applies to both continuous and discrete nonlinear Schrödinger equations of scalar and vector type. Also included is the IST for the Toda lattice and nonlinear ladder network, which are well-known discrete systems. This book, first published in 2003, presents the detailed mathematical analysis of the scattering theory; soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed. Much of the material is not available in the previously-published literature.

The inverse scattering transform for the vector defocusing nonlinear Schrödinger ͑NLS͒ equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros ͑poles͒ of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert ͑RH͒ problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system.Downloaded 18 Sep 2013 to 128.112.200.107. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions q͑x,t͒ → q ± ͑t͒ = q 0 e 2iq 0 2 t±i␣ as x → ± ϱ and appear as localized dips of intensity q 0 2 sin 2 ␣ on the background field q 0 . While the IST for the scalar NLS equation was developed many years ago, both with vanishing and nonvanishing boundary conditions, the basic formulation of IST has not been fully developed for the vector nonlinear Schrödinger ͑VNLS͒ equationt͒ is, in general, an M-component vector and ʈ·ʈ is the standard Euclidean norm. The focusing case ͑ =−1͒ with vanishing boundary conditions in two components was developed by Manakov in 1974. 12 However, the IST for the VNLS with nonzero boundary conditions has been open for over 30 years ͑partial results can be found in Ref. 13͒. It is worth noting that Ref. 14 provides an elegant direct and inverse scattering theory for decaying potentials on the real line. The extension to nondecaying potentials, however, is not straightforward and therefore here we employ a different approach. We should also remark that direct methods have been applied to VNLS as a way to derive explicit bright and dark soliton solutions, see for instance Refs. 17-20 and the review article Ref. 21.In this work we present the IST for the two-component defocusing VNLS equation ͓namely, Eq. ͑1.3͒ with = 1 and M =2͔ with nonvanishing boundary conditions as x → ± ϱ. In Sec. II we discuss the direct scattering problem. Section II A is devoted to the study of the analyticity o...

The inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed. This problem had been previously studied, and many key results had been established. Here, a suitable transformation of the scattering problem is introduced in order to address the open issue of analyticity of eigenfunctions and scattering data. Moreover, the inverse problem is formulated as a Riemann-Hilbert problem on the unit circle, and a modification of the standard procedure is required in order to deal with the dependence of asymptotics of the eigenfunctions on the potentials. The discrete analog of Gel'fand-Levitan-Marchenko equations is also derived. Finally, soliton solutions and solutions in the small-amplitude limit are obtained and the continuum limit is discussed.