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Matematicheskie Zametki, 2002, Volume 71, Issue 3, Pages 373–389
DOI: https://doi.org/10.4213/mzm353
(Mi mzm353)
 

This article is cited in 10 scientific papers (total in 10 papers)

A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations

K. A. Volosov

Moscow State Institute of Electronics and Mathematics
References:
Abstract: By using the recently discovered new invariant properties of the ansatz of R. Hirota's method, we prove that the classes of linear fractional solutions to some nonlinear equations are closed. This allows us to construct new solutions for a chosen class of dissipative equations. This algorithm is similar to the method of dressing the solutions of integrable equations. The equations thus obtained imply a compatibility condition and are known as a nonlinear Lax pair with variable coefficients. So we propose a method for constructing such pairs. To construct solutions of a more complicated form, we propose to use the property of zero denominators and factorized brackets, which has been discovered experimentally. The expressions thus constructed are said to be quasi-invariant. They allow us to find true relations between the functions contained in the ansatz, to correct the ansatz, and to construct a solution. We present some examples of new solutions constructed following this approach. Such solutions can be used for majorizing in comparison theorems and for modeling phase processes and process in neurocomputers. A program for computing solutions by methods of computer algebra is written. These techniques supplement the classical methods for constructing solutions by using their group properties.
Received: 25.12.2000
English version:
Mathematical Notes, 2002, Volume 71, Issue 3, Pages 339–354
DOI: https://doi.org/10.1023/A:1014846823941
Bibliographic databases:
Document Type: Article
UDC: 517
Language: Russian
Citation: K. A. Volosov, “A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations”, Mat. Zametki, 71:3 (2002), 373–389; Math. Notes, 71:3 (2002), 339–354
Citation in format AMSBIB
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  • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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