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Mat. Zametki, 2002, Volume 71, Issue 3, Pages 373–389 (Mi mz353)  

This article is cited in 7 scientific papers (total in 7 papers)

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

K. A. Volosov

Moscow State Institute of Electronics and Mathematics

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.

DOI: https://doi.org/10.4213/mzm353

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English version:
Mathematical Notes, 2002, 71:3, 339–354

Bibliographic databases:

UDC: 517
Received: 25.12.2000

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Bratus, AS, “Exact solutions to the Hamilton–Jacobi–Bellman equation for optimal correction with an integral constraint on the total resource of control”, Doklady Mathematics, 66:1 (2002), 148  mathscinet  zmath  isi
    2. Volosov KA, “About one of the properties of the ansatz of Hirota's method for quasi-linear equations”, Nuovo Cimento Della Societa Italiana Di Fisica B-General Physics Relativity Astronomy and Mathematical Physics and Methods, 118:1 (2003), 1–15  mathscinet  isi
    3. K. A. Volosov, “Eigenfunctions of structures described by the “shallow water” model in a plane”, J. Math. Sci., 151:1 (2008), 2639–2650  mathnet  crossref  mathscinet  zmath  elib  elib
    4. Kutafina E.V., “TRANSFORMATION OF AUTO-BACKLUND TYPE FOR HYPERBOLIC GENERALIZATION OF BURGERS EQUATION”, Journal of Nonlinear Mathematical Physics, 16:4 (2009), 411–420  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    5. Zhang Sh., Tian Ch., Qian W.-Y., “Bilinearization and new multisoliton solutions for the (4+1)-dimensional Fokas equation”, Pramana-J. Phys., 86:6 (2016), 1259–1267  crossref  isi  scopus
    6. Zhang Sh., Zhang L., “Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients”, Open Phys., 14:1 (2016), 69–75  crossref  isi  scopus
    7. Zhang Sh., Gao X., “Exact
      $$\varvec{N}$$
      N -soliton solutions and dynamics of a new AKNS equation with time-dependent coefficients”, Nonlinear Dyn., 83:1-2 (2016), 1043–1052  crossref  mathscinet  zmath  isi  scopus
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