Integrability of Truncated Hořava–Lifshitz Mixmaster Model in Rosenhain Functions

Abstract

The mixmaster Hořava–Lifshitz model belongs to generalized Euclidean Toda chains with 28 vectors. The longest three vectors of the spectrum play a dominant role in studying its dynamics. The truncated cosmological model is presented as a periodic three-particle Toda chain. It is associated with an affine Kac–Moody Lie algebra \(A_{2}^{+}\). According to the Adler–van Moerbeke criterion, the truncated Hamiltonian system is algebraically completely integrable. The phase curves wrap a torus of genus 2. The Jacobi problem of inversion of ultraelliptic integrals is solved by using theta-functions of two variables. The solutions of the dynamical problem are expressed in rational functions of Rosenhain theta-functions. They are four-periodic functions.