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Zh. Vychisl. Mat. Mat. Fiz., 2017, Volume 57, Number 7, Pages 1113–1125 (Mi zvmmf10585)  

This article is cited in 2 scientific papers (total in 2 papers)

The geometric series method for constructing exact solutions to nonlinear evolution equations

A. V. Bochkarev, A. I. Zemlyanukhin

Saratov Technical University, Saratov, Russia

Abstract: It is proved that, for the majority of integrable evolution equations, the perturbation series constructed based on the exponential solution of the linearized problem is geometric or becomes geometric as a result of changing the variable in the equation or after a transformation of the series. Using this property, a method for constructing exact solutions to a wide class of nonintegrable equations is proposed; this method is based on the requirement for the perturbation series to be geometric and on the imposition of constraints on the values of the coefficients and parameters of the equation under which the sum of the series is the solution to be found. The effectiveness of using the diagonal Padé approximants the minimal order of which is determined by the order of the pole of the solution to the equation is demonstrated.

Key words: geometric series, perturbation method, evolution equation, exact solution, Padé approximant.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00176_а


DOI: https://doi.org/10.7868/S0044466917070079

Full text: PDF file (143 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2017, 57:7, 1111–1123

Bibliographic databases:

UDC: 517.957:517.537.3
Received: 03.02.2016
Revised: 10.01.2017

Citation: A. V. Bochkarev, A. I. Zemlyanukhin, “The geometric series method for constructing exact solutions to nonlinear evolution equations”, Zh. Vychisl. Mat. Mat. Fiz., 57:7 (2017), 1113–1125; Comput. Math. Math. Phys., 57:7 (2017), 1111–1123

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. I. Zemlyanukhin, A. V. Bochkarev, “Perturbation method, Padé approximants and exact solutions of nonlinear mechanics equations”, Mater. Phys. Mech., 35:1 (2018), 181–189  crossref  isi
    2. C. Cattani, T. A. Sulaiman, H. M. Baskonus, H. Bulut, “Solitons in an inhomogeneous Murnaghan's rod”, Eur. Phys. J. Plus, 133:6 (2018), 228  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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