Dynamical System for the Nonlinear Fourth-Order Differential Equation from the K2 Hierarchy
https://doi.org/10.1134/S2304487X2006005X
Abstract
A fourth-order nonlinear equation, which is the second member of the K2 hierarchy equations, has been considered. The K2 hierarchy, as well as the K1 hierarchy of equations, was introduced more than twenty years ago. A distinctive feature of these hierarchies is that all the equations belonging to this hierarchy do not have any first integrals in a polynomial form and, apparently, as well as the six Painlevé equations, they do not have solutions expressed in terms of classical functions. The equations of the K2 hierarchy, as well as the equations of the first and second Painlevé hierarchies, are used to describe physical processes, and their investigation is of interest in this regard. It has been shown that this fourth-order equation from the K2 hierarchy can be represented as a special dynamical system for which a formal representation of the general solution can be obtained.
Supporting agencies: This work was supported by the Russian Foundation for Basic Research (project no. 18-29-10025) and the state task of the Ministry of Higher schools and sciences of the Russian Federation No. 0723 2020-0036
About the Author
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
References
1. Painlevé P. Sur les equations differentielles du second ordre et d’ordre superieur dont l’integrale generale est uniforme // Acta Math. 1902. V. 25. P. 1–85.
2. Gambier B. Sur les équations différetielles dont l’integrate générale est uniforme // C. R. Acad. Sci. Paris. 1906. V. 142. P. 266–269, 1403–1406, 1497–1500.
3. Borisov A. V., Kudryashov N. A. Paul Painlevé and His Contribution to Science // Regular and Chaotic Dynamics. 2014. V. 19. № 1. P. 1–19.
4. Kudryashov N. A. Higher Painlevé transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies // Regular and Chaotic Dynamics. 2014. V. 19. № 1. P. 48–63.
5. Conte R. The Painleve property, one century later, CRM series in mathematical physics. New York: Springer–Verlag, 1999. P. 77–180.
6. Drazin P. G., Johnson R. S. Solitons: an introduction. Cambridge University Press, 1989.
7. Drazin P. G. Nonlinear systems. Cambridge University Press, 1992.
8. Ablowitz M. J., Clarkson P. A. Solitons Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, 1991.
9. Kudryashov N. A. The first and second Painlevé equations of higher order and some relations between them // Physics Letters, Section A: General, Atomic and Solid State Physics. 1997. V. 224. № 6. P. 353–360.
10. Kudryashov N. A. On new transcendents defined by nonlinear ordinary differential equations // Journal of Physics A: Mathematical and General. 1998. V. 31. № 6. P. L129–L137, cited by 32.
11. Kudryashov N. A. Transcendents defined by nonlinear fourth-order ordinary differential equations // Journal of Physics A: Mathematical and General. 1999. V. 32. № 6. P. 999–1013, cited by 31.
12. Kudryashov N. A. Special polynomials associated with some hierarchies // Physics Letters, Section A: General, Atomic and Solid State Physics. 2008. V. 372. № 12. P. 1945–1956.
13. Kudryashov N. A. One generalization of the second Painlevé hierarchy // Journal of Physics A: Mathematical and General. 2002. V. 35. № 1. P. 93–99.
14. Kudryashov N. A. Amalgamations of the Painlevé equations // Journal of Mathematical Physics. 2003. V. 44. № 12. P. 6160–6178.
15. Kawai T., Koike T., Nishikawa Y., Takei Y. On the Stokes geometry of higher order Painlevé equations // Analyse complexe, systemes dynamiques, sommabilite des series divergentes et theories galoisiennes. II Astrisque 2004. № 297. P. 117–166.
16. Shimomura S. Poles and α-poles of meromorphic Solutions of the First Painlevé Hierarchy // Bull. RIMS, Kyoto Univ. 2004. V. 40. P. 471–485.
17. Aoki T. Multiple-Scale Analysis for Higher-order Painlevé Equations // RIMS Kokyuroke Bessatsu B. 2008. V. 5. P. 89–98.
Review
For citations:
Kudryashov N.A. Dynamical System for the Nonlinear Fourth-Order Differential Equation from the K2 Hierarchy. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(6):517-520. (In Russ.) https://doi.org/10.1134/S2304487X2006005X
Views:
95
Email this article 