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International conference "Geometrical Methods in Mathematical Physics"
December 17, 2011 10:00–10:45, Moscow, Lomonosov Moscow State University

The solutions of the heat and Burgers equations in terms of elliptic sigma functions.

V. M. Buchstaber

Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: The algebra of differential operators along $z,g_{2}$ and $g_{3}$, which annihilate the Weierstrass function $\sigma (z,g_{2},g_{3})$, is extracted from classical works and solves the problem of differentiation of elliptic functions along parameters $g_{2},g_{3}$ and, correspondingly, the problem of differentiation of some important dynamical system solutions along initial data. Using the generators of this algebra, we get dynamics on C$^{3}$, and on this basis the family of solutions of the heat equation in terms of the $\sigma$-function. The dynamics are determined by a solution of the Chazy equation.
Using the Cole-Hopf transformation and our solutions of the heat equation, we obtain solutions of the Burgers equation in terms of Weierstrass functions. The explicit formulas for the differentiation of this solutions by the initial data are obtained.
We show that the function $\phi (z,\tau )=\sigma (z;g_{2}(\tau ),g_{3}(\tau ))$ is a solution of the equation
\begin{equation*} 8\dot{\phi}=4\phi ^{\prime \prime }+u(\tau )z^{2}\phi \end{equation*}
with $u(\tau )=\wp (\tau +d,0,b_{3})$.
The natural problem to describe solutions of the heat and Burgers equations in terms of solutions of the previous differential equation with $u(\tau )=\wp (\tau +d,b_{2},b_{3})$ arises. We came to an ordinary differential equation of order 5 with solutions that in the case $b_{2}=0$ are defined by the solutions of the Chazy equation.
Results presented in the talk were obtained in recent joint works with E.Yu. Bunkova. Main definitions will be introduced during the talk.

Materials: gmmp2011_vmbuchstaber.pdf (220.7 Kb)

Language: English

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