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On critical expansions of solutions of the discrete Painleve equation $q$$P(A_1)$ and corresponding monodromy Roffelsen, Pieter
Description
In a recent paper [1], we classified the critical behaviour of solutions of the discrete Painleve equation $qP(A_1)$ around the origin and infinity. Even though this equation is ranked higher than qPVI and hence PVI in Sakai's classification scheme [2], the critical behaviours seem to resemble those found for the sixth Painleve equation. The generic expansions near the two critical points each contain two arbitrary qconstants, and relating them explicitly constitutes the $qP(A_1)$ connection problem. A well known method of attack is given by the isomonodromic deformation method, which we apply to an associated linear system formulated by Yamada [3]. It turns out that the connection problem of the Yamada system factorises asymptotically, both as the PainlevВґe variable approaches the origin and infinity, into two copies of a linear system associated with the continuous dual qHahn polynomials. The connection matrix of the limiting system can be calculated explicitly, which allows us to solve the direct monodromy problem for the critical expansions of solutions of $qP(A_1)$. In particular this reduces the $qP(A_1)$ connection problem to solving an equation involving qelliptic functions. References [1] N. Joshi and P. Roffelsen, Analytic solutions of $qP(A_1)$ near its critical points, arXiv:1510.07433 [nlin.SI]. [2] H. Sakai, Rational Surfaces Associated with Affine Root Systems and Geometry of the Painleve Equations, Commun. Math. Phys. 220 (2001). [3] Y. Yamada, Lax Formalism for qPainleve Equations with Affine Weyl Group Symmetry of Type $E_n^{(1)}$ , Int. Math. Res. Not. IMRN 2011 (2011).
Item Metadata
Title 
On critical expansions of solutions of the discrete Painleve equation $q$$P(A_1)$ and corresponding monodromy

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20161006T15:48

Description 
In a recent paper [1], we classified the critical behaviour of solutions of the discrete Painleve equation $qP(A_1)$ around the origin and infinity. Even though this equation is ranked higher than qPVI and hence PVI in Sakai's classification scheme [2], the critical behaviours seem to resemble those found for the sixth Painleve equation. The generic expansions near the two critical points each contain two arbitrary qconstants, and relating them explicitly constitutes the $qP(A_1)$ connection problem. A well known method of attack is given by the isomonodromic deformation method, which we apply to an associated linear system formulated by Yamada [3]. It turns out that the connection problem of the Yamada system factorises asymptotically, both as the PainlevВґe variable approaches the origin and infinity, into two copies of a linear system associated with the continuous dual qHahn polynomials. The connection matrix of the limiting system can be calculated explicitly, which allows us to solve the direct monodromy problem for the critical expansions of solutions of $qP(A_1)$. In particular this reduces the $qP(A_1)$ connection problem to solving an equation involving qelliptic functions.
References
[1] N. Joshi and P. Roffelsen, Analytic solutions of $qP(A_1)$ near its critical points, arXiv:1510.07433 [nlin.SI].
[2] H. Sakai, Rational Surfaces Associated with Affine Root Systems and Geometry of the Painleve Equations, Commun. Math. Phys. 220 (2001).
[3] Y. Yamada, Lax Formalism for qPainleve Equations with Affine Weyl Group Symmetry of Type $E_n^{(1)}$ , Int. Math. Res. Not. IMRN 2011 (2011).

Extent 
23 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: The University of Sydney

Series  
Date Available 
20170407

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0343501

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International