ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Zemlyanukhin A. I., Artamonov N. A., Bochkarev A. V., Bezlyudny V. I. Power series reversion and exact solutions of nonlinear mathematical physics equations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2025, vol. 33, iss. 6, pp. 929-942. DOI: 10.18500/0869-6632-003180, EDN: XEHQXM

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Language: 
Russian
Heading: 
Nonlinear Waves. Solitons. Autowaves. Self-Organization.
Article type: 
Article
UDC: 
530.182, 517.912, 517.929
DOI: 
10.18500/0869-6632-003180
EDN: 
XEHQXM

Power series reversion and exact solutions of nonlinear mathematical physics equations

Autors: 
Zemlyanukhin Aleksandr Isaevich, Yuri Gagarin State Technical University of Saratov
Artamonov Nikolay Aleksandrovich, Yuri Gagarin State Technical University of Saratov
Bochkarev Andrej Vladimirovich, Yuri Gagarin State Technical University of Saratov
Bezlyudny Vladimir Ilyich , Yuri Gagarin State Technical University of Saratov
Abstract: 

Purpose. Develop a new method for finding exact solutions to equations of nonlinear mathematical physics.

Methods. The partial sum of a perturbation series, written for the original nonlinear equation, is represented as a power series in powers of the exponential function, which is the solution of the linearized equation. The rational generating function of the sequence of coefficients of the power series represents the exact solution of the original equation. The method is based on the property that inverted power series for soliton-like solutions terminates at powers at least one greater than the order of the pole of the solution.

Results. The effectiveness of the method is demonstrated in constructing exact localized solutions of the nonintegrable Korteweg–de Vries–Burgers equation, as well as nonlinear integrable differential-difference equations.

Conclusion. The proposed method is applicable to solving integrable and non-integrable differential equations with constant coefficients, as well as integrable differential-difference equations.

Key words: 
power series reversion
generating function
nonlinear differential equations
differential-difference equations
exact solutions
Acknowledgments: 
This work was supported by the Russian Science Foundation (project No. 24-29-00071).
Reference: 
  1. Kudryashov N A. Methods of Nonlinear Mathematical Physics. Dolgoprudny: Intellect; 2010. 368 p.
  2. Conte R, Musette M. The Painlevé Method and Its Applications. Moscow-Izhevsk: Institute of Computer Research, Regular and Chaotic Dynamics; 2011. 315 p.
  3. Polyanin AD, Zaitsev VF, Zhurov A I. Nonlinear Equations of Mathematical Physics and Mechanics. Solution Methods: Textbook and Practice for Universities. 2nd ed., revised and expanded. Moscow: Yurait; 2025. 256 p.
  4. Yamilov R. Symmetries as integrability criteria for differential-difference equations. J. Phys. A: Math. Gen. 2006;39:541-623. DOI: 10.1088/0305-4470/39/45/R01.
  5. Andrianov I, Avreytsevich Y. Methods of Asymptotic Analysis and Synthesis in Nonlinear Dynamics and Mechanics of Deformable Solids. Moscow-Izhevsk: Institute of Computer Research; 2013. 276 p. (in Russian).
  6. Hirota R. Direct Methods in Soliton Theory. In: Bullough RK, Caudrey PJ, editors. Solitons. Topics in Current Physics. Vol. 17. Berlin: Springer; 1980. DOI: 10.1007/978-3-642-81448-8_5.
  7. Van Dyke M. Perturbation Methods in Fluid Mechanics. N.Y.: Academic Press; 1964. 229 p.
  8. Euler L. Foundations of Differential Calculus. N.Y.: Springer; 2000. 194 p. DOI: 10.1007/b97699.
  9. Vinogradov VN, Gai EV, Rabotnov N S. Analytical Approximation of Data in Nuclear and Neutron Physics. Moscow: Energoatomizdat; 1987. 128 p.
  10. Bochkarev AV, Zemlyanukhin A I. Geometric series method for finding exact solutions of nonlinear evolution equations. Comput. Math. Math. Phys. 2017;57(7):1111-1123. 10.1134/S096554251707006510.1134/S0965542517070065.
  11. Zemlyanukhin AI, Bochkarev AV, Orlova AA, Ratushny A V. Geometric series method and exact solutions of differential-difference equations. In: Abramian AK, Andrianov IV, Gaiko VA, editors. Nonlinear Dynamics of Discrete and Continuous Systems. Advanced Structured Materials. Vol. 139. Cham: Springer; 2021. P. 239-253. DOI: 10.1007/978-3-030-53006-8_15.
  12. Lando S K. Lectures on Generating Functions: Textbook. Moscow: MCCME, 2007. 144 p.
  13. Safonov K V. Conditions for the sum of a power series to be algebraic and rational. Math. Notes. 1987;41(3):325-332. .97.
  14. Yagmur T. New approach to Pell and Pell-Lucas sequences. Kyungpook Math. J. 2019;59(1):23-34. DOI: 10.5666/KMJ.2019.59.1.23.
  15. Ablowitz MJ, Segur H. Solitons and the Inverse Scattering Transform. SIAM; 1981. 425 p.
  16. Nakoryakov VE, Pokusaev BG, Shreiber I R. Wave Dynamics of Gas and Vapor-Liquid Media. Moscow: Energoatomizdat, 1990. 246 p.
  17. Kudryashov N A. Bäcklund transformation for a fourth-order partial differential equation with the Burgers-Korteweg-de Vries nonlinearity. Sov. Phys. Dokl. 1988;33(5):336-338.
  18. Garifullin RN, Yamilov R I. On the integrability of lattice equations with two continuum limits. J. Math. Sci. 2021;252:283-289. DOI: 10.1007/s10958-020-05160-x.
  19. Hinch E J. Perturbation Methods. Cambridge: Cambridge University Press; 1991. 160 p DOI: 10.1017/CBO9781139172189.
Received: 
22.04.2025
Accepted: 
22.05.2025
Available online: 
19.06.2025
Published: 
28.11.2025
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