Singular dynamics of aq-difference Painlevé equation in its initial-value space

Abstract

We construct the initial-value space of a q-discrete first Painlevé equation explicitly and describe the behaviours of its solutions w(n) in this space as n → ∞, with particular attention paid to neighbourhoods of exceptional lines and irreducible components of the anti-canonical divisor. These results show that trajectories starting in domains bounded away from the origin in initial value space are repelled away from such singular lines. However, the dynamical behaviours in neighbourhoods containing the origin are complicated by the merger of two simple base points at the origin in the limit. We show that these lead to a saddle-point-type behaviour in a punctured neighbourhood of the origin.Dedicated to Rodney J. Baxter on the occasion of his 75th birthday.

Deautonomization by singularity confinement: an algebro-geometric justification

“…This can of course be done using the coordinate charts (7) for n + 1, after which the singularity of ϕ n+1 at (s n+1 , y n+1 ) = (0, 0) is fully resolved. Furthermore, since S n is the base point for the blow-up of Q, it is interesting to look at the consequence of this blow-up on the relation expressed in (11).…”

“…the very pattern obtained from singularity confinement. Moreover, the full map (12) can be used to study the asymptotic behaviour of the solutions of ϕ n , for arbitrary initial conditions, as shown in [11].…”

Elliptic asymptotics in $q$-discrete Painlevé equations

“…For q-P I , unstable solutions termed quicksilver solutions were identified [11]. The geometric description of the space of initial values in the asymptotic limit |ξ| → ∞ was given in [16]. In the latter study, the invariant for the autonomous leading-order equation was also considered.…”

Nonlinear q-Stokes phenomena for q-Painlevé I

“…The asymptotic study of variants of the sixth q-Painlevé equation have been investigated by Mano [41] and Joshi and Roffelson [33] using the q-analogue of the isomonodromic deformation approach. The first q-Painlevé equation has also been investigated in [28,30]. In particular, Joshi [28] proves the existence of true solutions of (1), which are asymptotic to a divergent asymptotic power series in the limit |x| → ∞.…”

“…More generally, equation ( 1) is known as a q-difference equation since the evolution of the independent variable x takes the form x = x 0 q n for some initial x 0 . Motivated by Boutroux's study of the first Painlevé equation [7], the asymptotic behaviour of (1) in the limit |x| → ∞ has been considered in [28,30]. Joshi [28] showed that there exists a true solution satisfying w → 0 as |x| → ∞, which is asymptotic to a divergent series.…”