The Inverse Scattering Transform for the Defocusing Nonlinear Schrödinger Equations with Nonzero Boundary Conditions

Abstract

A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values q ± ≡ q 0 e iθ ± as x → ±∞ is presented. The direct problem is shown to be well posed for potentials q such that q − q ± ∈ L 1,2 (R ± ), for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann-Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

“…Most research involves solutions vanishing as x → ±∞ [5,6,16,26,31,33]. Recently there has been much interest in solutions nonvanishing as x → ±∞ [8,10,11]. In this article we study the IST for the focusing NLS equation which is discrete in position and continuous in time, obtained by applying forward differencing to the focusing Zakharov-Shabat system…”

On the Location of the Discrete Eigenvalues for Defocusing Zakharov-Shabat Systems having Potentials with Nonvanishing Boundary Conditions

“…The analyticity and continuity properties of the scattering coefficients a(k),b(k), b(k),ā(k) follow from the analyticity and continuity properties of the Jost solutions using their Wronskian representations [6]. It is well known that the scattering data associated to the ZS system (1.3) are (see [1,2,20,22]…”

“…In this section we study the direct scattering problem for (1.3) by using the same notations adopted in [6] to which we refer the interested reader for details. Moreover, we discuss a significant example which shows that in the spectral gap (−q 0 , q 0 ) there may exist a discrete eigenvalue if q + = q − .…”

“…However, it is important to remark that for Eq. (1.1) with NZBCs, the discrete eigenvalues belong to the spectral gap (−q 0 , q 0 ) and are simple (proved in [7]) and, under the hypothesis q − q ± ∈ L 1,4 (R ± ), are finite in number (established in [6]). We conclude this section analyzing an explicit example which confirms the results obtained in [5,Sec.…”

“…In particular, we focus our attention on some aspects which arise when the Inverse Scattering Transform (IST) (see [ In fact, the IST for (1.1) with NZBCs (see [3,4,7,12,13,17,22]) is much more complicated than for (1.1) with decaying potentials, in particular with regard to the analyticity properties of the eigenfunctions of the scattering problem (1.3) and the corresponding scattering data. A step forward in that direction was recently made in [6] where it was proved that the direct scattering problem is well defined for potentials q such that q − q ± belongs to the functional class…”