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Stokes phenomena in discrete Painlevé II
N Joshi1, C J Lustri2, S Luu1
1School of Mathematics and Statistics, F07 , The University of Sydney , New South Wales 2006, Australia.
Proceedings. Mathematical, Physical, and Engineering Sciences
|March 16, 2017
Summary
Researchers analyzed the second discrete Painlevé equation
Area of Science:
- Mathematical Physics
- Complex Analysis
- Differential Equations
Background:
- The second discrete Painlevé equation is a complex mathematical model.
- Understanding its behavior for large independent variables is crucial.
- Classical methods struggle with certain asymptotic phenomena.
Purpose of the Study:
- To investigate the asymptotic behavior of the second discrete Painlevé equation.
- To identify and describe phenomena invisible to traditional power series methods.
- To compare these behaviors with solutions of the continuous Painlevé equation.
Main Methods:
- Asymptotic power series analysis.
- Exponential asymptotic techniques.
- Investigation of Stokes phenomena and Stokes curves.
Main Results:
- Identified asymptotically pole-free solutions in specific complex plane regions.
- Mathematically described rapid switching behavior associated with Stokes curves.
- Determined regions where power series accurately describe asymptotic behavior.
Conclusions:
- Asymptotic solutions of the discrete Painlevé equation exhibit complex phenomena.
- Exponential asymptotics reveal behaviors missed by classical methods.
- Shared features exist between discrete and continuous Painlevé equation solutions.
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