On the algebraic solutions of the Painleve-III (D7) equation
Robert J. Buckingham, Peter D. Miller
TL;DR
This paper investigates algebraic solutions of the Painleve-III (D7) equation, providing a Riemann-Hilbert framework and analyzing their asymptotic behavior for large parameters.
Contribution
It introduces a Riemann-Hilbert representation for algebraic solutions of the Painleve-III (D7) equation using the isomonodromy method.
Findings
Solutions are rational functions of x^{1/3} for certain parameters.
The Riemann-Hilbert representation facilitates rigorous asymptotic analysis.
Behavior of solutions in the large parameter limit is characterized.
Abstract
The D7 degeneration of the Painleve-III equation has solutions that are rational functions of for certain parameter values. We apply the isomonodromy method to obtain a Riemann-Hilbert representation of these solutions. We demonstrate the utility of this representation by analyzing rigorously the behavior of the solutions in the large parameter limit.
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Taxonomy
TopicsNonlinear Waves and Solitons · Numerical methods for differential equations · Algebraic structures and combinatorial models