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Zbl 0835.34035
Noonburg, V.Anne
A separating surface for the PainlevĂ© differential equation $x''= x\sp 2-t$.
(English)
[J] J. Math. Anal. Appl. 193, No.3, 817-831 (1995). ISSN 0022-247X

The author is concerned with the PainlevĂ© 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 $t$.\par It is shown that the solutions of type (B) form a surface $\bbfS \subset \bbfR^3$ separating the solutions of type (A) from these of type (C). Every solution to type (B) must enter the interior of the parabola ${\cal P} = \{(t,x) : x^2 - t = 0\}$ 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 ${\cal P}$ 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 $\bbfS$ are given.
[J.Kalas (Brno)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: location of integral curves; manifolds of solutions; oscillating solutions; separating surface; PainlevĂ© equation

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