Painlevé IV: A numerical study of the fundamental domain and beyond

• First comprehensive study of the solution space to the fourth Painlevé equation.
• Extensions to parameter regimes that were unreachable by previous approaches.
• Previously known solutions found to be ‘non-typical’ of the general case.
• Solutions featuring singularity-free half-planes explored.

The six Painlevé equations were introduced over a century ago, motivated by rather theoretical considerations. Over the last several decades, these equations and their solutions, known as the Painlevé transcendents, have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast ‘pole field solver’ (Fornberg and Weideman, 2011). The fourth Painlevé equation has two free parameters in its coefficients, as well as two free initial conditions. After summarizing key analytical results for PIV, the present study applies this new computational tool to the fundamental domain and a surrounding region of the parameter space. We confirm existing analytic and asymptotic knowledge about the equation and also explore solution regimes which have not been described in the previous literature.