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Realization of the inverse scattering transform method for the Korteweg-de Vries equation
  • Sergei Grudsky,
  • Vladislav Kravchenko,
  • Sergii Torba
Sergei Grudsky
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Vladislav Kravchenko
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Sergii 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.
02 Jun 2022Submitted to Mathematical Methods in the Applied Sciences
03 Jun 2022Submission Checks Completed
03 Jun 2022Assigned to Editor
08 Jul 2022Reviewer(s) Assigned
30 Sep 2022Review(s) Completed, Editorial Evaluation Pending
02 Oct 2022Editorial Decision: Revise Major
11 Dec 20221st Revision Received
12 Dec 2022Submission Checks Completed
12 Dec 2022Assigned to Editor
12 Dec 2022Review(s) Completed, Editorial Evaluation Pending
13 Dec 2022Reviewer(s) Assigned
27 Dec 2022Editorial Decision: Accept