
ВИДЕОТЕКА 
VII Международная конференция по дифференциальным и функциональнодифференциальным уравнениям (DFDE)



Dierentialalgebraic solutions of the heat equation V. M. Bukhshtaber^{}, E. Yu. Netay^{} 

Аннотация: We discuss solutions of the heat equation ${\partial \over \partial t} \psi(z, t) = {1 \over 2} {\partial^2 \over \partial z^2} \psi(z, t)$ in the ansatz $ \psi(z,t) = f(t) \exp\left({1\over 2} h(t) z^2\right) \Phi(z,t)$ with additional conditions on In our ansatz we have the following classical examples of the heat equation solutions: the flat wave solution with $ \theta_1 \left( z, t \right) = \sqrt{{\omega \over \pi}}\sqrt[8]{\Delta}\exp( 2 \omega \eta z^2)\sigma(\hat{z}, g_2, g_3), $ where By a differentialalgebraic solution of the heat equation, we mean a solution in our ansatz, satisfying the additional conditions that Consider the differential operator $ \mathcal{L} = {\partial \over \partial y_1}  \sum_{s=1}^\infty (s+1) s y_{s} {\partial \over \partial y_{s+1}}. $ A polynomial We prove that a differentialalgebraic solution of the heat equation is an Examples of ordinary differential equations obtained from admissible polynomials for small \begin{align*} &h' =  h^2, \quad h'' =  6 h h'  4 h^3, \quad h''' =  12 h h'' + 18h'^2 + c_3 (h' + h^2)^2, \\ &h'''' =  20 h h''' + 24 h' h''  96 h^2 h'' + 144 h h'^2 + c_4 (h' + h^2) (h'' + 6 h h' + 4 h^3), \end{align*} where As The fourthorder equation has the Painléve property only in the case where its general solution is rational (see [3]). It is shown in [3] that the next (fifthorder) equation has series of parameters satisfying the Painléve test. The work was partially supported by the presidium RAN program “Fundamental problems in nonlinear dynamics.” [1]. Bunkova E. Yu. and Buchstaber V. M. Heat equations and families of twodimensional sigma functions, Proc. Steklov Inst. Math., 266, 1–28 (2009). [2]. Buchstaber V. M. and Netay E. Yu. Differentialalgebraic solutions of the heat equation, arXiv: 1405.0926 [3]. Vinogradov A. V. The Painlevé test for the ordinary differential equations associated with the heat equation, Proc. Steklov Inst. Math., 286 (2014). Язык доклада: английский 