Polynomial dynamical systems and ordinary differential equations associated with the heat equation
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.
- E. Yu. Bunkova and V. M. Buchstaber, “Heat equations and families of two-dimensional sigma functions,” in: Trudy MIAN, vol. 266, MAIK, Moscow, 2009, 5–32; English transl.: Proc. Steklov Inst. Math., 266:1 (2009), 1–28.
- P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 2nd ed., New York, 1993.
- R. Conte and M. Musette, The Painlevé handbook, Springer-Verlag, Dordrecht, 2008.
- V. M. Buchstaber, D. V. Leikin, and M. V. Pavlov, “Egorov hydrodynamic chains, the Chazy equation, and SL(2,ℂ),” Funkts. Anal. Prilozhen., 37:4 (2003), 13–26; English transl.: Functional Anal. Appl., 37:4 (2003), 251–262.
- S. Chakravarty, M. J. Ablowitz, and P. A. Clarkson, “Reductions of self-dual Yang-Mills fields and classical systems,” Phys. Rev. Lett., 65:1 (1990), 1085–1087. CrossRef
- B. Dubrovin, “Geometry of 2D topological field theories,” in: Lecture Notes in Math., vol. 1620, Springer-Verlag, Berlin, 1996, 120–348.
- P. A. Clarkson and P. J. Olver, “Symmetry and the Chazy equation,” J. Differential Equations, 124:1 (1996), 225–246. CrossRef
- E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1996.
- “The correspondence of S. V. Kovalevskaya and G. Mittag-Leffler,” in: Scientific Heritage, vol. 7, Nauka, Moscow, 1984, Letter 57.
- S. Chakravarty and R. G. Halburd, “First integrals and gradient flow for a generalized Darboux-Halphen system,” in: Contemp. Math., vol. 301, Amer. Math. Soc., Providence, RI, 2002, 273–281.
- V. M. Buchstaber and D. V. Leikin, “Addition laws on Jacobian varieties of plane algebraic curves,” in: Trudy MIAN, vol. 251, Nauka, Moscow, 2005, 54–126; English transl.: Proc. Steklov Inst. Math., 251 (2005), 49–120.
- Polynomial dynamical systems and ordinary differential equations associated with the heat equation
Functional Analysis and Its Applications
Volume 46, Issue 3 , pp 173-190
- Cover Date
- Print ISSN
- Online ISSN
- SP MAIK Nauka/Interperiodica
- Additional Links
- polynomial dynamical systems
- heat equation
- Chazy equation
- Darboux-Halphen system
- Industry Sectors