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The Seventh International Conference on Differential and Functional Differential Equations
August 23, 2014 16:00–16:45, Moscow, Conference Hall, 2nd Floor

Dierential-algebraic solutions of the heat equation

V. M. Bukhshtaber, E. Yu. Netay
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Abstract: 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(-{1\over 2} h(t) z^2) \Phi(z,t)$
with additional conditions on $\Phi(z,t)$ that reduce the heat equation to a homogeneous non-linear ordinary differential equation. The corresponding Burgers equation solutions are obtained via the Cole-Hopf transform.
In our ansatz we have the following classical examples of the heat equation solutions: the flat wave solution with $h(t) = 0$ and $ \Phi(z,t) = \Phi(z,0)$; the Gaussian normal distribution with the standard deviation $\sigma = \sqrt{2 a t}$, where $f(t) = (2 \pi \sigma^2)^{- 1/2}$, $h(t) = \sigma^{-2}$, and $\Phi(z,t) = 1$; the solution in terms of the elliptic theta-function described in the ansatz
$ \theta_1 ( z, t ) = \sqrt{{\omega \over \pi}}\sqrt[8]{\Delta}\exp(- 2 \omega \eta z^2)\sigma(\hat{z}, g_2, g_3), $
where $\sigma(\hat{z}, g_2, g_3)$ is the Weierstrass sigma-function, $ \omega t = \omega'$, and $\hat{z} = 2\omega z$. In the last case, our method gives (see [1]) the Chazy-3 equation
$ y"'(t) = 2y(t) y"(t) - 3 y'(t)^2. $
By a differential-algebraic solution of the heat equation, we mean a solution in our ansatz, satisfying the additional conditions that $\psi(z,t)$ is regular for $z =0$ and $\Phi(z,t)$ or $\Phi'(z,t)$ is an even function in $z$ such that the series decomposition coefficients $\Phi_k(t)$ of $z^{2k}$ are homogeneous polynomials of degree $-2k$ in $x_2, \ldots, x_k$, where $\deg x_q = - 2q$, $q = 2, 3, \ldots$. A differential-algebraic solution is called an $n$-ansatz solution of the heat equation if $\Phi_k(t)$ are homogeneous polynomials of $n$ variables $x_2(t), \ldots, x_{n+1}(t)$.
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 $P(y_1, \ldots, y_{n+2})$ is called admissible if it is a homogeneous polynomial with respect to the grading $\deg y_k = - 2 k$ and $\mathcal{L} P(y_1, \ldots, y_{n+2}) = 0$.
We prove that a differential-algebraic solution of the heat equation is an $n$-ansatz solution if and only if the function $h = h(t)$ is a solution of the ordinary differential equation $P(h, h', \ldots, h^{(n+1)}) = 0$ with admissible $P(y_1, \ldots, y_{n+2})$. Fixed such a function $h(t)$, we find an expression for $f(t)$ in terms of $h(t)$ and recurrent expressions for $\Phi_k(t)$, $k = 2, 3 \ldots$, as differential polynomials of $h(t)$, see [2].
Examples of ordinary differential equations obtained from admissible polynomials for small $n$ are
\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 $c_3$ and $c_4$ are constants.
As $c_3 = 0$, the third-order equation becomes the Chazy-3 equation after the substitution $y(t) = - 6 h(t)$. The values of $c_3$ for which this equation has the Painléve property were described in classical papers.
The fourth-order 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 (fifth-order) 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 two-dimensional sigma functions,
Proc. Steklov Inst. Math., 266, 1–28 (2009).
[2]. Buchstaber V. M. and Netay E. Yu. Differential-algebraic 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).

Language: English

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