1,721,018 research outputs found
Discrete solitons of the Ablowitz-Ladik equation with nonzero boundary conditions via inverse scattering
No full textSoliton solutions of the focusing Ablowitz-Ladik (AL) equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform (IST). After reviewing the relevant aspects of the direct and inverse problems, explicit soliton solutions are obtained which are the discrete analog of the Tajiri-Watanabe and Kutznetsov-Ma solutions to the focusing NLS equation on a finite background. Then, by performing suitable limits of the above solutions, discrete analog of the celebrated Akhmediev and Peregrine solutions are also presented. The latter, which can be thought of as a discrete “rogue” wave, is expressed as a family of rational functions of the discrete spatial variable n ∈ Z and time t ∈ R, parametrically depending on the amplitudeQo of the background. These solutions, which had been recently derived by direct methods, are obtained for the first time within the framework of the IST, thus also providing a spectral characterization of the solutions and a description of the singular limit process
On some nondecaying potentials and related Jost solutions for the heat conduction equation
No full textOn the spectrum of the Dirac operator and the existence of discrete eigenvalues for the defocusing nonlinear Schrödinger equation
No full textWe revisit the scattering problem for the defocusing nonlinear Schrodinger
equation with constant, nonzero boundary conditions at infinity, i.e., the
eigenvalue problem for the Dirac operator with nonzero rest mass. By
considering a specific kind of piecewise constant potentials we address and
clarify two issues, concerning: (i) the (non)existence of an area theorem relating the presence/absence of discrete eigenvalues to an appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase difference of the potential from the continuous spectrum
Inverse scattering transform for the focusing Ablowitz-Ladik system with nonzero boundary conditions
No full textThe purpose of this paper is to construct the inverse scattering transform for the focusing Ablowitz-Ladik equation with nonzero boundary conditions at infinity. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann-Hilbert problem on a doubly connected curve in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz-Ladik potential
Determinant and Pfaffian solutions of the strong coupling limit of integrable discrete NLS systems
No full textThe strong coupling limits of the integrable semi-discrete and fully discrete nonlinear Schrödinger systems are studied by using the Hirota bilinear method. The determinant solutions (in both infinite and finite lattice cases) for the strong coupling limits of semi-discrete and fully discrete nonlinear Schrödinger systems are obtained using a determinant technique. The vector generalizations of the strong coupling limits of semi-discrete and fully discrete nonlinear Schrödinger systems are also presented. The Pfaffian solutions for vector systems are obtained using the Pfaffian technique
Solvability of the direct and inverse problems for the nonlinear Schrodinger equation
No full textIn this paper we study rigorous spectral theory and solvability for both the direct and inverse problems of the Dirac operator associated with the nonlinear Schrödinger equation. We review known results and techniques, as well as incorporating new ones, in a comprehensive, unified framework. We identify functional spaces in which both direct and inverse problems are well posed, have a unique solution and the corresponding direct and inverse maps are one to one
Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation with a One-Sided Non-Zero Boundary Condition
No full textThe inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrodinger (NLS) equation with one-sided non-zero boundary value as x → +∞ is presented. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that [q(x, t) − qr(t)θ(x)] ∈ L1,1(R) [here and in the following θ(x) denotes the Heaviside function] with respect to x ∈ R for all t ≥ 0, for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both
via (left and right) Marchenko integral equations and as a Riemann-Hilbert problem on a single sheet of the scattering variables , where k is the usual complex scattering parameter in the IST. The direct and inverse problems are also formulated in terms of a suitable uniformization variable that maps the two-sheeted Riemann surface for k into a single copy of the complex plane. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with the same amplitude as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of physically relevant NLS solutions with nontrivial boundary conditions, either via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations
Inverse scattering transform for the multicomponent nonlinear Schrödinger equation with nonzero boundary conditions at infinity
No full textThe Inverse Scattering Transform (IST) for the defocusing vector nonlinear Schrodinger equations (NLS), with an arbitrary number of components and nonvanishing boundary conditions at space infinities, is formulated by adapting and generalizing the approach used by Beals, Deift, and Tomei in the development of the IST for the N-wave interaction equations. Specifically, a complete set of sectionally meromorphic eigenfunctions is obtained from a family of analytic forms that are constructed for this purpose. As in the scalar and two-component defocusing NLS, the direct and inverse problems are formulated on a two-sheeted, genus-zero Riemann surface, which is then
transformed into the complex plane by means of an appropriate uniformization variable. The inverse problem is formulated as a matrix Riemann-Hilbert problem with prescribed poles, jumps, and symmetry conditions. In contrast to traditional formulations of the IST, the analytic forms and eigenfunctions are first defined for complex values of the scattering parameter, and extended to the continuous spectrum a posteriori
- …
