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Approximate direct and inverse scattering for the AKNS system

Vladislav V. Kravchenko

TL;DR

This work addresses the direct and inverse scattering problems for the AKNS system with complex, rapidly decaying potentials by introducing spectral-parameter power series (SPPS) representations for the Jost solutions. A Möbius transform $z=(\tfrac{1}{2}+i\rho)/(\tfrac{1}{2}-i\rho)$ yields unit-disk convergence, enabling a recurrent integration procedure to compute SPPS coefficients from initial data at $\rho=\tfrac{i}{2}$ and $-\tfrac{i}{2}$. The direct problem reduces to evaluating truncated SPPS polynomials on the real axis to obtain the scattering matrix and locate eigenvalues via zeros inside the unit disk, while the inverse problem becomes solving two linear systems for the SPPS coefficients to recover the potentials from the first coefficients. The method is demonstrated with numerical examples showing high accuracy (up to 1e-14 in some cases) and fast computation, highlighting its usefulness for nonlinear PDE solvers such as the nonlinear Schrödinger equation. Overall, the SPPS framework provides a simple, robust, and scalable approach to AKNS scattering, with clear pathways to implementation and application in integrable systems analysis.

Abstract

We study the direct and inverse scattering problems for the AKNS (Ablowitz-Kaup-Newell-Segur) system. New representations for the Jost solutions are obtained in the form of the power series in terms of a transformed spectral parameter. In terms of that parameter, the Jost solutions are convergent power series in corresponding unit disks. For the coefficients of the series simple recurrent integration procedures are devised. Solution of the direct scattering problem reduces to computing the coefficients and locating zeros of corresponding analytic functions in the interior of the unit disk. Solution of the inverse scattering problem reduces to the solution of two systems of linear algebraic equations for the power series coefficients, while the potentials are recovered from the first coefficients. The overall approach leads to a simple and efficient method for the numerical solution of both direct and inverse scattering problems, which is illustrated by numerical examples.

Approximate direct and inverse scattering for the AKNS system

TL;DR

This work addresses the direct and inverse scattering problems for the AKNS system with complex, rapidly decaying potentials by introducing spectral-parameter power series (SPPS) representations for the Jost solutions. A Möbius transform yields unit-disk convergence, enabling a recurrent integration procedure to compute SPPS coefficients from initial data at and . The direct problem reduces to evaluating truncated SPPS polynomials on the real axis to obtain the scattering matrix and locate eigenvalues via zeros inside the unit disk, while the inverse problem becomes solving two linear systems for the SPPS coefficients to recover the potentials from the first coefficients. The method is demonstrated with numerical examples showing high accuracy (up to 1e-14 in some cases) and fast computation, highlighting its usefulness for nonlinear PDE solvers such as the nonlinear Schrödinger equation. Overall, the SPPS framework provides a simple, robust, and scalable approach to AKNS scattering, with clear pathways to implementation and application in integrable systems analysis.

Abstract

We study the direct and inverse scattering problems for the AKNS (Ablowitz-Kaup-Newell-Segur) system. New representations for the Jost solutions are obtained in the form of the power series in terms of a transformed spectral parameter. In terms of that parameter, the Jost solutions are convergent power series in corresponding unit disks. For the coefficients of the series simple recurrent integration procedures are devised. Solution of the direct scattering problem reduces to computing the coefficients and locating zeros of corresponding analytic functions in the interior of the unit disk. Solution of the inverse scattering problem reduces to the solution of two systems of linear algebraic equations for the power series coefficients, while the potentials are recovered from the first coefficients. The overall approach leads to a simple and efficient method for the numerical solution of both direct and inverse scattering problems, which is illustrated by numerical examples.

Paper Structure

This paper contains 19 sections, 3 theorems, 168 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Direct and inverse problems, preliminaries
  3. Series representations for Jost solutions
  4. Integral representations, series expansions of integral kernels
  5. Spectral parameter power series for Jost solutions
  6. Recurrent integration procedure for the coefficients
  7. First coefficients
  8. Coefficients $a_{n}(x)$
  9. Coefficients $\widetilde{a}_{n}(x)$
  10. Coefficients $b_{n}(x)$
  11. Coefficients $\widetilde{b}_{n}(x)$
  12. Asymptotics of the coefficients
  13. Summary of the recurrent procedure
  14. Solution of the direct problem
  15. Solution of the inverse problem
  16. ...and 4 more sections

Key Result

Theorem 3.1

Let $q(x)$ and $r(x)$ belong to the class $\mathcal{P}$. Then the integral kernels in (phi Levin)-(psitil Levin) admit the following series representations where $L_{n}$ stands for the Laguerre polynomial of order $n$. Each coefficient is a vector function, for example, For any $x$ fixed the series converge in mean square.

Figures (11)

  • Figure 1: Decay of the computed coefficients $a_{1,n}(0)$ and $a_{2,n}(0)$ corresponding to Example 1, when $n$ increases.
  • Figure 2: Absolute error of the computed $\mathbf{a}(\rho)$ (left) and $\mathbf{b}(\rho)$ (right). The maximum absolute error of $\mathbf{a}(\rho)$ resulted in $4.9\cdot10^{-14}$, and that of $\mathbf{b}(\rho)$ in $1.09\cdot10^{-13}$.
  • Figure 3: Decay of the computed coefficients $a_{1,n}(0)$ and $a_{2,n}(0)$ corresponding to Example 2, when $n$ increases.
  • Figure 4: Equality (\ref{['aatil']}) is used as an indicator of the accuracy of the computed entries of the matrix $S(\rho)$.
  • Figure 5: Decay of the computed coefficients $a_{1,n}(0)$ and $a_{2,n}(0)$ corresponding to Example 3, when $n$ increases.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 4.1