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Integrability of the Szekeres system
- Speaker(s)
- Anna Gierzkiewicz
- Affiliation
- UR Kraków
- Date
- Nov. 27, 2015, 10:15 a.m.
- Room
- room 5840
- Seminar
- Seminar of Dynamical Systems Group
The Szekeres system is a four-dimensional system of first-order, ordinary differential equations, with nonlinear, but polynomial (quadratic) right
hand side.
\[
\begin{cases}
\rho'=-\Theta\rho\\
\Theta'=-\frac13\Theta^2 - 6\sigma^2-\frac12\rho\\
\sigma'=\sigma^2-\frac23\Theta\sigma - E\\
E'=3E\sigma- \Theta E-\frac12\rho\sigma\\
\end{cases}.
\]
It was derived from Einstein equations as a cosmological model of early universe, inhomogeneous with no symmetries. It was also used in modelling the evolution of galaxy superclusters.
I have studied integrability of this system in the sense of finding its first integrals via Darboux polynomials method (two independent integrals) and Jacobi's last multiplier method. Therefore, the Szekeres system is completely integrable.
hand side.
\[
\begin{cases}
\rho'=-\Theta\rho\\
\Theta'=-\frac13\Theta^2 - 6\sigma^2-\frac12\rho\\
\sigma'=\sigma^2-\frac23\Theta\sigma - E\\
E'=3E\sigma- \Theta E-\frac12\rho\sigma\\
\end{cases}.
\]
It was derived from Einstein equations as a cosmological model of early universe, inhomogeneous with no symmetries. It was also used in modelling the evolution of galaxy superclusters.
I have studied integrability of this system in the sense of finding its first integrals via Darboux polynomials method (two independent integrals) and Jacobi's last multiplier method. Therefore, the Szekeres system is completely integrable.
