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

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.     

On the connection between the inverse transform method and the exact quantum eigenstates

The “inverse scattering transformation”, which has been used to solve certain non-linear field theories classically, is discussed in the context of the quantized version of these theories. In particular we consider the non-linear Schrödinger equation and the massive Thirring model. We find that certain Jost functions of the associated scattering problem lead already, in quantizing the theory, to creation operators for the exact eigenstates of the corresponding Hamiltonians.     

Exact Solutions of Ablowitz-Ladik Hierarchy with Self-Consistent Sources Revisited and Reductions

In the paper, Ablowitz-Ladik hierarchy with new self-consistent sources is investigated. First the source in the hierarchy is described as φ_nφ_n+1, where φ_n is related to the Ablowitz-Ladik spectral problem, instead of the corresponding adjoint spectral problem. Then by means of the inverse scattering transform, the multi-soliton solutions for the hierarchy are obtained. Two reductions to the discrete mKdV and nonlinear Schrödinger hierarchies with self-consistent sources are considered by using the uniqueness of the Jost functions, as well as their N-soliton solutions.     

Inverse Scattering Transform and Solitons for Square Matrix Nonlinear Schrödinger Equations with Mixed Sign Reductions and Nonzero Boundary Conditions

The inverse scattering transform (IST) with nonzero boundary conditions at infinity is developed for a class of 2 × 2 matrix nonlinear Schrödinger-type systems whose reductions include two equations that model certain hyperfine spin F = 1 spinor Bose-Einstein condensates, and two novel equations that were recently shown to be integrable, and that have applications in nonlinear optics and four-component fermionic condensates. In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows us to develop the IST on the standard complex plane instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity, symmetries and asymptotics of the scattering eigenfunctions and scattering data are derived, and properties of the discrete spectrum are analyzed in detail. In addition, the general behavior of the soliton solutions for all four reductions is discussed, and some novel soliton solutions are presented.     

On- and off-shell Jost functions and their integral representations

By judicious exploitation of the transpose operator relation in conjunction with the differential equations of special functions of mathematical physics, integral representations of the on- and off-shell Jost functions are derived from the particular integrals of the inhomogeneous Schrödinger equation. Using the particular integral of the inhomogeneous Schrödinger equation, exact analytical expressions for the Coulomb and Coulomb plus Yamaguchi off-shell Jost solutions are constructed in the maximal reduced form. As a case study, the limiting behaviours and the on-shell discontinuities of the Coulomb plus Yamaguchi Jost solutions are verified numerically.     

Some piecewise-analytical solutions to the one-dimensional Schrödinger equation II

The inverse approach method is further developed with the purpose to construct more complex analytically integrable one-dimensional Schrödinger-type equations based on simpler ones. The centrifugal and Coulomb potentials are taken into consideration. The analysis is presented by the example of the problem of particle scattering by the nuclear optical potential. The dependence of the constructed potential on the energy and angular momentum appears negligible.     

Inverse Spectral-Scattering Problems on the Half-Line with the Knowledge of the Potential on a Finite Interval

The inverse spectral and scattering problems for the radial Schrödinger equation on the half-line \({[0,\infty)}\) are considered for a real-valued, integrable potential having a finite first moment. It is shown that the potential is uniquely determined in terms of the mixed spectral or scattering data which consist of the partial knowledge of the potential given on the finite interval \({[0,\varepsilon]}\) for some \({\varepsilon > 0}\) and either the amplitude or phase (being equivalent to scattering function) of the Jost function, without bound state data.     

Inverse scattering transforms of the inhomogeneous fifth-order nonlinear Schrödinger equation with zero/nonzero boundary conditions

The present work studies the inverse scattering transforms (IST) of the inhomogeneous fifth-order nonlinear Schrödinger (NLS) equation with zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs). Firstly, the bound-state solitons of the inhomogeneous fifth-order NLS equation with ZBCs are derived by the residue theorem and the Laurent’s series for the first time. Then, by combining with the robust IST, the Riemann-Hilbert (RH) problem of the inhomogeneous fifth-order NLS equation with NZBCs is revealed. Furthermore, based on the resulting RH problem, some new rogue wave solutions of the inhomogeneous fifth-order NLS equation are found by the Darboux transformation. Finally, some corresponding graphs are given by selecting appropriate parameters to further analyze the unreported dynamic characteristics of the corresponding solutions.     

Inverse spectral problem for delta potentials

The scattering problem for the linear Schrödinger equation on the entire axis has been considered. Conditions under which the knowledge of the discrete spectrum of the Schrödinger operator is sufficient for the reconstruction of the potential have been determined. The main difference from the soliton sector is the self-similarity of the problem under consideration with respect to the extension of the spectral parameter λ. This makes it possible to reduce the inverse scattering problem to the study of the singularity of the Green’s function at λ = 0.     

Finite-difference approximation for inverse scattering problem

For the first time potentials are reconstructed in a finite-difference approximation using a genuine inverse scattering method instead of multiple repeated solutions of a direct problem with iterative fitting of scattering data. Up to now a fundamental difference between spectral properties of the Schrödinger operator and its discrete analog hindered from doing this.     

CLOSED FORM SOLUTIONS TO THE INTEGRABLE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using a matrix triplet (A, B, C). Here A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N-soliton and breather solutions as special cases. We also prove that these solutions reduce to known continuous matrix NLS solutions as the discretization step vanishes.     

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

We construct the formal solution to the Cauchy problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrödinger operator for a quantum particle in a time-dependent potential.     

Exact solutions to a class of nonlinear Schrödinger-type equations

A class of nonlinear Schrödinger-type equations, including the Rangwala-Rao equation, the Gerdjikov-Ivanov equation, the Chen-Lee-Lin equation and the Ablowitz-Ramani-Segur equation are investigated, and the exact solutions are derived with the aid of the homogeneous balance principle, and a set of subsidiary higher order ordinary differential equations (sub-ODEs for short).     

Wronskian analysis of resonance tunnelling reactions

A Wronskian formalism is developed for resonance tunnelling reactions. It is shown how the S matrix can be written in terms of Wronskians between solutions of the Schrödinger equation representing incoming and outgoing waves. The method is an adaptation of the Jost function approach to elastic two-body scattering. Formal expressions are derived for the imaginary parts of the energy eigenvalues that arise from the application of complex boundary conditions used in a previous semiclassical analysis. When the imaginary parts are small, so that the resonance is sharp, the S matrix can be written in a Breit-Wigner resonance form. The theory unifies and extends the semiclassical analysis of resonance tunnelling reactions given previously.     

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

The continuous states of the l-wave Schrödinger equation for the diatomic molecule represented by the hyperbolical function potential are carried out by a proper approximation scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the hyperbolical function potential are presented and the corresponding calculation formula of phase shifts is derived. Also, we interestingly obtain the corresponding bound state energy levels by analyzing analytical properties of scattering amplitude.     

New approach to the solution of the scattering of spinless particles by central potentials

The explicit forms of the regular solutions, of the Jost solutions and functions for the radial Schrödinger equation, which describe the scattering of spinless particles by central potentials, are found. The regular solutions are derived from the iterative solution of the integral equation which their suitably modified Laplace transforms fulfil. Two general classes of potentials are used each of them being expressed by the corresponding inverse Laplace transform. As such forms of the regular solutions are related to those of the Jost solutions, the Jost solutions (along with the Jost functions) are written directly. The regions of the complex angular momenta and wave numbers, to which they can be analytically continued, are specified. Some testing relations are also derived.Dedicated to Academician Václav Votruba on the occasion of his seventieth birthday.     

Cusp Interaction in Minimal Length Quantum Mechanics

The modified Schrödinger equation with a minimal length is considered under a Cusp potential which includes the exponential interaction. Next, exact analytical solutions of the problem are reported and thereby the scattering states as well as the corresponding transmission and reflection coefficients are reported.     

Inverse scattering problem for gratings with deep modulation

The similarity between one-dimensional Schrödinger and Helmholtz equations is discussed. The Helmholtz equation in optical coordinate is shown to reduce to the Schrödinger equation with an effective potential. Two examples of scattering problem are considered: sinusoidal Bragg grating with deep modulation and smooth hyperbolic secant layer. The inverse scattering problem is solved numerically for both cases. For the layer an analytical solution is presented as well. The analysis of the effective potential allows one to qualitatively predict some properties of the reflection spectrum.     

Completeness of the Jost Solutions in the Case of the NLS Equation Under the Nonvanishing Boundary Condition

Completeness of the Jost solutions in the case of the NLS equation under the nonvanishing boundary condition is proved based upon the inverse scattering transform in terms of the auxiliary spectral parameter.     

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.