

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






Dierentialalgebraic 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 nonlinear ordinary differential equation. The corresponding Burgers equation solutions are obtained via the ColeHopf 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 thetafunction 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 sigmafunction, $ \omega t = \omega'$, and $\hat{z} = 2\omega z$. In the last case, our method gives (see [1]) the Chazy3 equation
$ y"'(t) = 2y(t) y"(t)  3 y'(t)^2. $
By a differentialalgebraic 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 differentialalgebraic 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 differentialalgebraic 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 thirdorder equation becomes the Chazy3 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 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).
Language: English

