Monday, June 24, 2013

1306.5045 (Nalini Joshi)

Quicksilver Solutions of a q-difference first Painlevé equation    [PDF]

Nalini Joshi
We prove the existence and asymptotic properties of a family of true solutions of a $q$-difference Painlev\'e equation. For definiteness, we focus on a discrete version of the first Painlev\'e equation (q$P_I$), which is a dynamical system iterated on a rational surface of type $A_7^{(1)}$. We overcome limitations in the treatment of non-linear $q$-difference equations in the literature to show that there exist true analytic solutions asymptotic to a power series near infinity in large domains. The method, while demonstrated for q$P_I$, is also applicable to other $q$-difference Painlev\'e equations. The resulting unstable transcendental solutions, called "quicksilver" solutions here, share a characteristic property.
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