arXiv subjects

Vladislav V. Kravchenko

Publications and source records attributed to Vladislav V. Kravchenko.

At least 19 recordsLinked to original sources

arXiv2025

Approximate direct and inverse scattering for the AKNS system

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.

math.CA↗

arXiv2025

Implementation of the inverse scattering transform method for the nonlinear Schrödinger equation

We study the initial-value problem for the nonlinear Schrödinger equation. Application of the inverse scattering transform method involves solving direct and inverse scattering problems for the Zakharov-Shabat system with complex potentials. We solve these problems by using new series representations for the Jost solutions of the Zakharov-Shabat system. The representations have the form of power series with respect to a transformed spectral parameter. In terms of the representations, solution of the direct scattering problem reduces to computing the series coefficients following a simple recurrent integration procedure, computation of the scattering coefficients by multiplying corresponding pairs of polynomials (partial sums of the series representations) and locating zeros of a polynomial inside the unit disk. Solution of the inverse scattering problem reduces to the solution of a system of linear algebraic equations for the power series coefficients, while the potential is recovered from the first coefficients. The system is obtained directly from the scattering relations. Thus, unlike other existing techniques, the method does not involve solving the Gelfand-Levitan-Marchenko equation or the matrix Riemann-Hilbert problem. The overall approach leads to a simple and efficient algorithm for the numerical solution of the initial-value problem for the nonlinear Schrödinger equation, which is illustrated by numerical examples.

math.AP↗

arXiv2025

Neumann series of Bessel functions in direct and inverse spherically symmetric transmission eigenvalue problems

The transmission eigenvalue problem (TEP) plays a central role in inverse scattering theory. Despite substantial theoretical progress, the numerical solution of direct and inverse TEP in spherically symmetric domains with variable refractive index covering real and complex eigenvalues remains challenging. This study introduces a novel Neumann Series of Bessel Functions (NSBF) methodology to address this challenge. After reformulating the TEP as a Sturm-Liouville equation via a Liouville transformation, we expand its characteristic function in an NSBF whose coefficients are computed by simple recursive integration. In the direct problem, eigenvalues real or complex are found by root finding on a truncated NSBF partial sum, yielding high accuracy with a few coefficients, as demonstrated with various examples. For the inverse problem, we develop a two-step approach: first, recovering the transformed interval length $δ$ from spectral data via a new NSBF-based algorithm, and second, reconstructing the refractive index $n(r)$ by solving a linear system for the first NSBF coefficients. A spectrum completion technique is also implemented to complete the spectrum and solve the corresponding inverse problem when eigenvalue data is limited. Numerical examples confirm the method's robustness and accuracy across a wide range of refractive indices, with no a priori assumptions on $δ$ or the sign of the contrast $1-n(r)$.

math.NA↗

arXiv2025

Reconstruction techniques for inverse Sturm-Liouville problems with complex coefficients

A variety of inverse Sturm-Liouville problems is considered, including the two-spectrum inverse problem, the problem of recovering the potential from the Weyl function, as well as the recovery from the spectral function. In all cases the potential in the Sturm-Liouville equation is assumed to be complex valued. A unified approach for the approximate solution of the inverse Sturm-Liouville problems is developed, based on Neumann series of Bessel functions (NSBF) representations for solutions and their derivatives. Unlike most existing approaches, it allows one to recover not only the complex-valued potential but also the boundary conditions of the Sturm-Liouville problem. Efficient accuracy control is implemented. The numerical method is direct. It involves only solving linear systems of algebraic equations for the coefficients of the NSBF representations, while eventually the knowledge only of the first NSBF coefficients leads to the recovery of the Sturm-Liouville problem. Numerical efficiency is illustrated by several test examples.

math.CA↗

arXiv2025

Spectral parameter power series for Zakharov-Shabat direct and inverse scattering problems

We study the direct and inverse scattering problems for the Zakharov-Shabat system. 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 the unit disk. The coefficients of the series are computed following a simple recurrent integration procedure. This essentially reduces the solution of the direct scattering problem to the computation of the coefficients and location of zeros of an analytic function inside of the unit disk. The solution of the inverse scattering problem reduces to the solution of a system of linear algebraic equations for the power series coefficients, while the potential is recovered from the first coefficient. 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.

math-ph↗

arXiv2025

Recovery of the rod cross section shape

A direct method for solving the inverse problem of determining the shape of the cross section of a rod is proposed. The method is based on Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. The first coefficient of the representation is sufficient for the recovery of the unknown function. A system of linear algebraic equations for finding this coefficient is obtained. The proposed method leads to an efficient numerical algorithm.

math.NA↗

arXiv2024

Schrödinger equation with finitely many $δ$-interactions: closed form, integral and series representations for solutions

A closed form solution for the one-dimensional Schrödinger equation with a finite number of $δ$-interactions \[ \mathbf{L}_{q,\mathfrak{I}_{N}}y:=-y^{\prime\prime}+\left( q(x)+\sum _{k=1}^{N}α_{k}δ(x-x_{k})\right) y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] is presented in terms of the solution of the unperturbed equation \[ \mathbf{L}_{q}y:=-y^{\prime\prime}+q(x)y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] and a corresponding transmutation operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ is obtained in the form of a Volterra integral operator. With the aid of the spectral parameter power series method, a practical construction of the image of the transmutation operator on a dense set is presented, and it is proved that the operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ transmutes the second derivative into the Schrödinger operator $\mathbf{L}_{q,\mathfrak{I}_{N}}$ on a Sobolev space $H^{2}$. A Fourier-Legendre series representation for the integral transmutation kernel is developed, from which a new representation for the solutions and their derivatives, in the form of a Neumann series of Bessel functions, is derived.

math.CA↗

arXiv2024

Sinc method in spectrum completion and inverse Sturm-Liouville problems

Cardinal series representations for solutions of the Sturm-Liouville equation $-y''+q(x)y=ρ^{2}y$, $x\in(0,L)$ with a complex valued potential $q(x)$ are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to $ρ$ in any strip $\left|\text{Im}ρ\right|<C$ of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential $q(x)$ as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two-spectra Sturm-Liouville problem and show its numerical efficiency.

math.CA↗

arXiv2023

Reconstruction techniques for complex potentials

An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y=λy with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectra Sturm-Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

math.CA↗

arXiv2023

Synthesis of a quantum tree Weyl matrix

A method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann map) of an arbitrary quantum tree is proposed. It allows one, starting from one boundary edge, to compute the Weyl matrix of a whole quantum graph by adding on new edges and solving elementary systems of linear algebraic equations in each step.

math.SP↗

arXiv2023

Reconstruction techniques for quantum trees

The inverse problem of recovery of a potential on a quantum tree graph from Weyl's matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two-spectra inverse problems on separate edges and reduce them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

math.CA↗

arXiv2022

Recovery of a potential on a quantum star graph from Weyl's matrix

The problem of recovery of a potential on a quantum star graph from Weyl's matrix given at a finite number of points is considered. A method for its approximate solution is proposed. It consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge with its posterior solution. The overall approach is based on Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations, and, in fact, the solution of the inverse problem on the quantum graph reduces to dealing with the NSBF coefficients. The NSBF representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the NSBF representation alone is sufficient for the recovery of the potential. The knowledge of the Weyl matrix at a set of points allows one to calculate a number of the NSBF coefficients at the end point of each edge, which leads to approximation of characteristic functions of two Sturm-Liouville problems and allows one to compute the Dirichlet-Dirichlet and Neumann-Dirichlet spectra on each edge. In turn, for solving this two-spectra inverse Sturm-Liouville problem a system of linear algebraic equations is derived for computing the first NSBF coefficient and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.

math.CA↗

arXiv2022

Spectrum completion and inverse Sturm-Liouville problems

Given a finite set of eigenvalues of a regular Sturm-Liouville problem for the equation -y{\prime}{\prime}+q(x)y=λy, the potential q(x) of which is unknown. We show the possibility to compute more eigenvalues without any additional information on the potential q(x). Moreover, considering the Sturm-Liouville problem with the boundary conditions y{\prime}(0)-hy(0)=0 and y{\prime}(π)+Hy(π)=0, where h, H are some constants, we complete its spectrum without additional information neither on the potential q(x) nor on the constants h and H. The eigenvalues are computed with a uniform absolute accuracy. Based on this result we propose a new method for numerical solution of the inverse Sturm-Liouville problem of recovering the potential from two spectra. The method includes the completion of the spectra in the first step and reduction to a system of linear algebraic equations in the second. The potential q(x) is recovered from the first component of the solution vector. The approach is based on special Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations possessing remarkable properties and leads to an efficient numerical algorithm for solving inverse Sturm-Liouville problems.

math.CA↗

arXiv2022

Method for solving inverse spectral problems on quantum star graphs

A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. The representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse spectral problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the representation is sufficient for the recovery of the potential. The method for solving the inverse spectral problem on the graph consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge. Then a system of linear algebraic equations is derived for computing the first coefficient of the series representation for the solution on each edge and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.

math.CA↗

arXiv2022

Realization of the inverse scattering transform method for the Korteweg-de Vries equation

A method for practical realization of the inverse scattering transform method for the Korteweg-de Vries equation is proposed. It is based on analytical representations for Jost solutions and for integral kernels of transformation operators obtained recently by the authors. The representations have the form of functional series in which the first coefficient plays a crucial role both in solving the direct scattering and the inverse scattering problems. The direct scattering problem reduces to computation of a number of the coefficients following a simple recurrent integration procedure with a posterior calculation of scattering data by well known formulas. The inverse scattering problem reduces to a system of linear algebraic equations from which the first component of the solution vector leads to the recovery of the potential. We prove the applicability of the finite section method to the system of linear algebraic equations and discuss numerical aspects of the proposed method. Numerical examples are given, which reveal the accuracy and speed of the method.

math.NA↗

arXiv2022

Generalized exponential basis for efficient solving of homogeneous diffusion free boundary problems: Russian option pricing

This paper develops a method for solving free boundary problems for time-homogeneous diffusions. We combine the complete exponential system of solutions for the heat equation, transmutation operators and recently discovered Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations to construct a complete system of solutions for the considered partial differential equations. The conceptual algorithm for the application of the method is presented. The valuation of Russian options with finite horizon is used as a numerical illustration. The solution under different horizons is computed and compared to the results that appear in the literature.

math.AP↗

arXiv2021

A practical method for recovering Sturm-Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm-Liouville potential from the Weyl-Titchmarsh $m$-function given on a countable set of points. We show that using the Fourier-Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from arXiv:2010.15275 is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

math.CA↗

arXiv2020

A direct method for solving inverse Sturm-Liouville problems

We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand-Levitan equation. The method allows one to reduce the inverse Sturm-Liouville problem directly to a system of linear algebraic equations, such that the potential is recovered from the first element of the solution vector. We prove the stability of the method and show its numerical efficiency with several numerical examples.

math.NA↗