| Abstract: |
| Most solutions of the famous Painlev\`e differential equations are highly transcendental, yet all but the Painlev\`e-I equation admit particular solutions that are elementary rational functions. These particular solutions are important in diverse applications, including the description of equilibrium patterns of fluid vortices, universal phenomena in nonlinear wave theory, electrochemistry, and string theory. The generic Painlev\`e-III equation has two free parameters and admits a rational solution whenever at least one of them is an integer. This talk will describe recent asymptotic results on the properties of rational solutions for large integer parameter. |
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