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Mathematics > Numerical Analysis

arXiv:2206.01317 (math)
[Submitted on 2 Jun 2022]

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

Authors:Sergei M. Grudsky, Vladislav V. Kravchenko, Sergii M. Torba
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Abstract: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.
Comments: 37 pages, 4 figures
Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
MSC classes: 35Q53, 34L25, 33C45, 47B35, 65L09, 65L15, 65M99, 65R20
Cite as: arXiv:2206.01317 [math.NA]
  (or arXiv:2206.01317v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2206.01317
Related DOI: https://doi.org/10.1002/mma.9049

Submission history

From: Sergii Torba M. [view email]
[v1] Thu, 2 Jun 2022 21:58:38 UTC (150 KB)
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