A separating surface for the Painlevé differential equation \(x''= x^ 2-t\)
The author is concerned with the Painlevé equation (1) \(x'' = x^2 - t\). The equation (1) has exactly three different types of solutions for \(t > 0\): (A) a 2-parameter family of solutions that oscillate infinitely often and slowly approach \(- \sqrt t\) as \(t \to \infty\); (B) a 1- parameter family of solutions asymptotic to \(\sqrt t\) as \(t \to \infty\); and (C) a family of solutions, each of which tends to \(+ \infty\) at some finite value of \(t\). It is shown that the solutions of type (B) form a surface \(\mathbb{S} \subset \mathbb{R}^3\) separating the solutions of type (A) from these of type (C). Every solution to type (B) must enter the interior of the parabola \({\mathcal P} = \{(t,x) : x^2 - t = 0\}\) for the final time through one of the points \(P_\tau : (t,x) = (|\tau |, \text{sign} (\tau) \sqrt {|\tau |})\) with positive and uniquely determined slope \(a^* (\tau)\), where \(\tau\) is a real parameter. On the other hand, for each point \(P_\tau\) on \({\mathcal P}\) there exists a unique positive slope \(a^* (\tau)\) such that the solution \(x_{a^*} (t)\) of (1) throughout \(P_\tau\) with the slope \(a^*\) is asymptotic to \(+ \sqrt t\) as \(t \to \infty\). The function \(a^*\) is proved to be continuous and differentiable everywhere except at \(\tau = 0\). Several approximate values of \(a^* (\tau)\) and the graphs of \(a^* (\tau)\) and of the surface \(\mathbb{S}\) are given.
scientific article; zbMATH DE number 811206