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

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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.

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