Linearization and Special Asymptotic Behaviour of the Second Painlevé Equation

Linearization of the initial value problem associated with the special second Painlevé equation is discussed using a different isomonodromic spectral problem than the one used in [1]. Further properties of the monodromy data [2, 3] are detected and these properties are used to reduce the problem to a linear singular integral equation via a Riemann–Hilbert boundary value problem on an imaginary line. A special asymptotic increasing solution of the Painlevé equation is constructed forx→−∞ from the above integral equation. Moreover, failure to extract asymptotics forx→+∞ is also mentioned.