The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). For numerical studies of ${\text{P}}_{\text{I}}$ see Holmes and Spence (1984), Noonburg (1995), and Fornberg and Weideman (2011). For numerical studies of ${\text{P}}_{\text{II}}$ see Rosales (1978), Miles (1978, 1980), Kashevarov (1998, 2004), and S. Olver (2011). For numerical studies of ${\text{P}}_{\text{IV}}$ see Bassom et al. (1993).