

International conference "Geometrical Methods in Mathematical Physics"
December 17, 2011 10:00, 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

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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 ColeHopf 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.
Language: English

