The Bianchi IX cosmological model in vacuum can be represented by several six-dimensional dynamical systems. In one of them we present a new closed form solution expressed by a third Painleve' function.
Deep Dive into A closed-form solution in a dynamical system related to Bianchi IX.
The Bianchi IX cosmological model in vacuum can be represented by several six-dimensional dynamical systems. In one of them we present a new closed form solution expressed by a third Painleve’ function.
The Bianchi IX cosmological model in vacuum can be defined by the metric [4]
in which e a α are the components of the three frame vectors, and σ 2 = ±1 according as the metric is Minkovskian or Euclidean. Introducing the logarithmic time τ by the hodograph transformation
this gives rise to the six-dimensional system of three second order ODEs
or equivalently
under the change of variables
If one introduces the six variables
log(BC), and cyclically, (7) the dynamical system (4) can be alternatively represented by [2] dy j dτ = -y j (z j -z k -z l ), dz j dτ = -y j (y j -y k -y l ), (8) in which (j, k, l) is any permutation of (1,2,3). This system admits the first integral
All the single valued solutions of (4) are known in closed form [1,8], except a four-parameter solution [5] which would extrapolate the three-parameter elliptic solution [1]
in which ℘, g 2 , g 3 , e j is the classical notation of Weierstrass,
and g 2 , g 3 , τ 0 are arbitrary. The solution (10) also represents the motion of a rigid body around its center of mass (Euler, 1750), σω ′ 1 = ω 2 ω 3 , and cyclically.
(
Any hint to find the above mentioned missing four-parameter solution would be welcome, and some indications can be found in Ref. [5]. In the present Letter, we present such a hint, as a five-parameter solution of (8). Despite its lack of physical meaning, it could share some analytic structure with the unknown solution and therefore provide a useful insight.
When one coordinate y i vanishes, say y 1 = 0, the correspondence (3) between the physical time t and the logarithmic time τ breaks down, but the system (8), whose investigation was then started in Ref. [6], can be integrated in closed form.
Taking account of the two additional first integrals [6],
the system reduces to
For c = 0, the system for z 2 -z 3 , y 2 + y 3 , y 2 -y 3 is another Euler top, whose general solution is
in which g 2 , g 3 , τ 1 , z 0 are the four arbitrary constants. For c = 0, the elimination of (y 3 , z 2 , z 3 ) between the two first integrals and the original system yields the general solution
and the second order ordinary differential equation for y 2 (or for y 3 as well) is a third Painlevé equation [7], with the correspondence
In the generic case cK 2 = 0, this solution is a meromorphic function of τ , with a transcendental dependence on the two constants of integration other than (c, K 1 , K 2 ). What is remarkable is that the unknown four-parameter solution of (4) and the Painlevé III solution (16) of ( 8) are both extrapolations of an Euler top. This suggests looking for another possible three-dimensional Euler top in the six-dimensional physical system (4). Such a threedimensional subsystem would necessarily correspond to a non self-dual curvature [3].
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