The Painlevé equations arise as reductions of the soliton equations such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation and so on. In this study, we are concerned with numerical approximation of the asymptotics of solutions of the second Painlevé equation on pole-free intervals along the real axis. Classical integrators such as high order Runge-Kutta schemes might be expensive to simulate oscillation, decay and blow-up behaviours depending on initial conditions. However, a lower order functional fitting method catches all kinds of solutions even for relatively large step sizes.
Erdoǧan, U., & Koçak, H. (2013). Numerical study of the asymptotics of the second Painlevé equation by a functional fitting method. Mathematical Methods in the Applied Sciences, 36(17), 2347-2352. https://doi.org/10.1002/mma.2757