Marchenko Equation Research Articles

On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition

An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line with a linear spectral parameter in the boundary condition. The scattering data of the problem are defined and a new fundamental equation is derived, which is different from the classical Marchenko equation. With help of this fundamental equation, in terms of the scattering data, the potential is recovered uniquely.

A unified approach to Darboux transformations

We analyze a certain class of integral equations related to Marchenko equations and Gel'fand–Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution. We show how this result provides a unified approach to derive Darboux transformations associated with various systems of ordinary differential operators. We illustrate our theory by deriving the Darboux transformation for the Zakharov–Shabat system and show how the potential and wavefunction change when a discrete eigenvalue is added to the spectrum.

The fundamental problem of inverse scattering method in solving the DNLS equation

This is a basic work on the inverse scattering transform that constructs the complete orthonormal system of the free Jost solution. In order to establish the system for the DNLS equation, we introduce a new transform for the spectral parameter and a gauge transform. The Marchenko's equation and the Zakharov-Shabat's equation of inverse scattering are derived based on the new spectral parameter.

SOLUTION OF THE INVERSE SCATTERING PROBLEM AT FIXED ENERGY FOR POTENTIALS BEING ZERO BEYOND A FIXED RADIUS

Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states.

Inverse Scattering Transform in Squared Spectral Parameter for DNLS Equation under Vanishing Boundary Conditions

After a transformation, the inverse scattering transform for the derivative nonlinear Schrödinger (DNLS) equation is developed in terms of squared spectral parameter. Following this approach, we obtain the orthogonality and completeness relations of free Jost solutions, which is impossibly constructed with usual spectral parameter in the previous works. With the help these relations, the Zakharov–Shabat equations as well as Marchenko equations of IST are derived in the standard way.

Integral equations and long-time asymptotics for finite-temperature Ising chain correlationfunctions

This work concerns the dynamical two-point spin correlation functions of the transverseIsing quantum chain at finite (non-zero) temperature, in the universal region near thequantum critical point. They are correlation functions of twist fields in the massiveMajorana fermion quantum field theory. At finite temperature, these are known to satisfy aset of integrable partial differential equations, including the sinh–Gordon equation. Weapply the classical inverse scattering method to study them, finding that the ‘initialscattering data’ corresponding to the correlation functions are simply related to theone-particle finite-temperature form factors calculated recently by one of the persentauthors. The set of linear integral equations (Gelfand–Levitan–Marchenko equations)associated with the inverse scattering problem then gives, in principle, the two-pointfunctions at all space and time separations, and all temperatures. From these, we evaluatethe long-time asymptotic expansion ‘near the light cone’, in the region where thedifference between the space and time separations is of the order of the correlationlength.

Inverse scattering for an AKNS problem with rational reflection coefficients

We study the AKNS (Ablowitz–Kaup–Newell–Segur) inverse scattering problem for rational reflection coefficients on a semi-infinite interval. We demonstrate that the Marchenko integral equations for the AKNS version on such an interval can be solved in a direct and straightforward way by algebraic methods for any set of rational reflection coefficients, vanishing at infinity. The general AKNS scattering problem for this case, as well as the usual symmetry reductions, are discussed. The connection to an alternative procedure by Rourke and Morris (1992 Phys. Rev. A 46 3631) is pointed out. Our procedure is built around a constant matrix, M, which can be constructed from the poles and residues of the rational reflection coefficients. Under certain conditions, which we define as minimal symmetry and which then suitably constrains the distribution of eigenvalues, we show that it is always possible to represent the resulting potentials as truncated N-soliton potentials. This procedure is of interest for solving initial-boundary value problems of integrable hyperbolic systems by the inverse scattering transform (IST) applied to a semi-infinite or finite interval.

Marchenko Equations and Norming Constants of the Matrix Zakharov-Shabat System

In this article we derive the Marchenko integral equations for solving the inverse scattering problem for the matrix Zakharov-Shabat system with a potential without symmetry properties and having L 1 entries under a technical hypothesis preventing the accumulation of discrete eigenvalues onthe continuous spectrum. Wederive additional symmetryproperties in the focusing case. The norming constant matrices appearing as parameter matrices in the Marchenko integral kernels are defined and studied without making any assumptions on the Jordan structure of the matrix Zakharov-Shabat Hamiltonian at the discrete eigenvalues.

Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions

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.

Structured matrix algorithms for inverse scattering on the line

In this article the Marchenko integral equations leading to the solution of the inverse scattering problem for the 1-D Schrodinger equation on the line are solved numerically. The linear system obtained by discretization has a structured matrix which allows one to apply FFT based techniques to solve the inverse scattering problem with minimal computational complexity. The numerical results agree with exact solutions when available. A proof of the convergence of the discretization scheme is given. Keywords Structured matrix systems, 1-D inverse scattering, Marchenko integral equation

Scattering theory for Jacobi operators with a steplike quasi-periodic background

We develop direct and inverse scattering theory for Jacobi operators with a steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand–Levitan–Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.

Marchenko Equation for the Derivative Nonlinear SchrödingerEquation

A simple derivation of the Marchenko equation is given for thederivative nonlinear Schrödinger equation. The kernel of theMarchenko equation is demanded to satisfy the conditions given by thecompatibility equations. The soliton solutions to theMarchenko equation are verified. The derivation is not concerned withthe revisions of Kaup and Newell.

Relativistic optical model on the basis of the Moscow potential and lower phase shifts for nucleon-nucleon scattering at laboratory energies of up to 3 GeV

Data of a partial-wave analysis of nucleon-nucleon scattering at energies of up to Elab = 3 GeV (lower partial waves) and the properties of the deuteron are described within the relativistic optical model based on deep attractive quasipotentials involving forbidden states (as exemplified by the Moscow potential). Partial-wave potentials are derived by the inverse-scattering-problem method based on the Marchenko equation by using present-day data from the partial-wave analysis of nucleon-nucleon scattering at energies of up to 3 GeV. Channel coupling is taken into account. The imaginary parts of the potentials are deduced from the phase equation of the variable-phase approach. The general situation around the manifestation of quark effects in nucleon-nucleon interaction is discussed.

Reconstruction of the optical potential from scattering data

We propose a method for reconstructing the optical potential (OP) from scattering data. The algorithm is a two-step, procedure. In the first step, the real part of the potential is determined analytically via solution of the Marchenko equation. At this point, we use a rational function fit of the corresponding unitary S matrix. In the second step, the imaginary part of the potential is determined via the phase equation of the variable phase approach. We assume that the real and imaginary parts of the OP are proportional (the Lax-type interaction). We use the phase equation to calculate the proportionality coefficient. A numerical algorithm is developed for a single and for coupled partial waves. The developed procedure is applied to analyze the ${}^{1}{S}_{0}\mathit{NN},{}^{3}{\mathit{SD}}_{1}\mathit{NN}$, and $P31\phantom{\rule{0.3em}{0ex}}{\ensuremath{\pi}}^{\ensuremath{-}}N$ data. For the NN states, we constructed partial potentials with forbidden states (Moscow potential). We examine the ${\ensuremath{\pi}}^{\ensuremath{-}}N$ and NN partial-wave analysis data and demonstrate that the Lax-type interaction may be valid for these systems at the considered energies.

Toda lattice mass transport in Lagrangian mechanics and in a two-dimensional system

This paper explores the connection between the hydrodynamic mass transport description and the thermodynamic description for a nonlinear range of the Toda lattices. Particular attention is paid to the broken isotropy in the KdV and Burgers equations. The flow variable representation is established from the Lagrangian mechanics for hydrodynamic mass transport. Based on the inverse scattering transform, the Gel’fand–Levitan–Marchenko (GLM) equation is formulated from the KdV equation expressed by the flow variable representation. We found that a kernel of the GLM equation is given by the concentration variable Q ( x , t ) . A Lagrangian is formulated for the KdV equation in state space ( Q ( x , t ) , K ( x , t ) ) . Next, an extension of the flow variable representation is sought in a two-dimensional system. The LHS of the Kadomtsev–Petviashvili (KP) equation takes the same form as in the second formalism of the KdV equation. By setting up the flow variable representation of the KP equation, the Burgers equation in two dimensions is formulated. These results contribute to an understanding of the broken isotropy for the nonlinear mass transport equations. These results provide physical insight into various consequences of the generalized form of the Kawasaki–Ohta equation from the viewpoint of mass transport.

Inverse scattering on matrices with boundary conditions

We describe inverse scattering for the matrix Schrodinger operator with general self-adjoint boundary conditions at the origin using the Marchenko equation. Our approach allows the recovery of the potential as well as the boundary conditions. It is easily specialized to inverse scattering on star-shaped graphs with boundary conditions at the node.

Practical pulse synthesis via the discrete inverse scattering transform

This paper provides the practical details required to use the inverse scattering (IST) approach to design selective RF-pulses. As in the Shinnar–Le Roux (SLR) approach, we use a hard pulse approximation to actually design the pulse. Unlike SLR, the pulse is designed using the full inverse scattering data (the reflection coefficient and the bound states) rather than the flip angle profile. We explain how to approximate the reflection coefficient to obtain a pulse with a prescribed rephasing time. In contrast to the SLR approach, we retain direct control on the phase of the magnetization profile throughout the design process. We give explicit recursive algorithms for computing the hard pulse from the inverse scattering data. These algorithms are quite different from the SLR recursion, being essentially discretizations of the Marchenko equations. We call our approach the discrete inverse scattering transform or DIST. Overall, it is as fast as the SLR approach. When bound states are present, we use both the left and right Marchenko equations to improve the numerical stability of the algorithm. We compute a variety of examples and consider the effect of amplitude errors on the magnetization profile.

Characteristic Properties of the Scattering Data for the mKdV Equation on the Half-Line

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip or in the quarter plane (0<x<∞)×[0,T), T≤∞. We suppose that this solution has a C∞ initial function vanishing as x→∞, and C∞ boundary values, vanishing as t→∞ when T=∞. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for x≥0 and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result of the paper is the collection A–B–C of characteristic properties of the scattering functions given below.

Green's Function Method for Perturbed sine-Gordon Equation**The project supported by National Natural Science Foundation of China under Grant Nos. 10375041 and 10174054

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.

Minimum energy pulse synthesis via the inverse scattering transform

This paper considers a variety of problems in the design of selective RF-pulses. We apply a formula of Zakharov and Manakov to directly relate the energy of an RF-envelope to the magnetization profile and certain auxiliary parameters used in the inverse scattering transform (IST) approach to RF-pulse synthesis. This allows a determination of the minimum possible energy for a given magnetization profile. We give an algorithm to construct both the minimum energy RF-envelope as well as any other envelope that produces a given magnetization profile. This includes an algorithm for solving the Gel’fand–Levitan–Marchenko equations with bound states. The SLR method is analyzed in terms of traditional scattering data, and shown to be a special (singular) case of the IST approach. RF-envelopes are computed for a variety of examples.