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Note on the uniqueness of a global positive solution to the second Painlevé equation. (English) Zbl 0983.34082

Summary: The purpose of this note is to study the uniqueness of solutions to \( u'' -u^3 + (t-c)u = 0\), for \( t \in (0,+\infty)\) with Neumann condition at \(0\). Assuming a certain conditon at infinity, B. Helfer and F. B. Weissler [Eur. J. Appl. Math. 9, No. 3, 223-243 (1998; Zbl 0920.34051)]have found a unique solution. The author shows that, without any assumption at infinity, this problem has exactly one global positive solution. Moreover, the solution behaves like \(\sqrt{t}\) as \(t\) approaches infinity.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34B15 Nonlinear boundary value problems for ordinary differential equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
82D55 Statistical mechanics of superconductors

Citations:

Zbl 0920.34051
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