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Irina Vaseva

Publications and source records attributed to Irina Vaseva.

8 recordsLinked to original sources
arXiv2026

Fast eight-order Pade schemes based on Chebyshev polynomials for direct Zakharov-Shabat problem

In this work, we construct fast eighth-order Pade schemes for the direct Zakharov-Shabat scattering problem. The schemes are based on an eighth-order exponential integrator obtained from the Magnus expansion. A direct extension of the conventional fast Pade representation to the eighth-order case leads to insufficiently accurate fast variants in the considered tests. To overcome this difficulty, we reformulate the spectral dependence on a finite real interval using the Joukowski mapping and represent the local numerators and denominators in the Chebyshev polynomial basis. This makes it possible to construct the global transition matrix by a fast product tree while retaining a compact polynomial representation. Numerical experiments for chirped hyperbolic secant potentials with both signs of dispersion show that the proposed Chebyshev-based fast schemes substantially improve the accuracy of the corresponding direct fast variants and can be used for efficient computation of the continuous nonlinear spectrum.

math.NA↗
arXiv2024

High-Order Block Toeplitz Inner-Bordering method for solving the Gelfand-Levitan-Marchenko equation

We propose a high precision algorithm for solving the Gelfand-Levitan-Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson's type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations.

math.NA↗
arXiv2021

Processing of optical signals by "surgical" methods for the Gelfand-Levitan-Marchenko equation

We propose a new method for solving the Gelfand-Levitan-Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB) with an arbitrary point to start the calculation. This makes it possible to find solutions of the GLME at an arbitrary point with a cutoff of the matrix coefficient, which allows to avoid the occurrence of numerical instability and to perform calculations for soliton solutions spaced apart in the time domain. Using an example of two solitons, we demonstrate our method and its range of applicability. An example of eight solitons shows how the method can be applied to a more complex signal configuration.

math.NA↗
arXiv2020

Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system in optical applications

Based on the generalized Cayley transform, a family of conservative one-step schemes of the sixth order of accuracy for the Zakharov-Shabat system is constructed. The exponential integrator is a special case. Schemes based on rational approximation allow the use of fast algorithms to solve the initial problem for a large number of values of the spectral parameter.

math.NA↗
arXiv2020

Introducing phase jump tracking -- a fast method for eigenvalue evaluation of the direct Zakharov-Shabat problem

We propose a new method for finding discrete eigenvalues for the direct Zakharov-Shabat problem, based on moving in the complex plane along the argument jumps of the function $a(ζ)$, the localization of which does not require great accuracy. It allows to find all discrete eigenvalues taking into account their multiplicity faster than matrix methods and contour integrals. The method shows significant advantage over other methods when calculating a large discrete spectrum, both in speed and accuracy.

math.NA↗
arXiv2019

Fast Computation of the Direct Scattering Transform by Fourth Order Conservative Multi-Exponential Scheme

A fourth-order multi-exponential scheme is proposed for the Zakharov-Shabat system. The scheme represents a product of 13 exponential operators. The construction of the scheme is based on a fourth-order three-exponential scheme, which contains only one exponent with a spectral parameter. This exponent is factorized to the fourth-order with the Suzuki formula of 11 exponents. The obtained scheme allows the use of a fast algorithm in calculating the initial problem for a large number of spectral parameters and conserves the quadratic invariant exactly for real spectral parameters.

math.NA↗
arXiv2019

Exponential Fourth Order Schemes for Direct Zakharov-Shabat problem

We propose two finite-difference algorithms of fourth order of accuracy for solving the initial problem of the Zakharov-Shabat system. Both schemes have the exponential form and conserve quadratic invariant of Zakharov-Shabat system. The second scheme contains the spectral parameter in exponent only and allows to apply the fast computational algorithm.

math.NA↗