Nonlinear differential equations of second Painlevé type with the quasi-Painlevé property. (English) Zbl 1166.34057
The author considers a class of nonlinear differential equations of second Painlevé equation \(E_k\) of the form
\[ y''=\frac{k+1}{k^2}y^{2k+1}+xy+\alpha,\;k\in\mathbb{N},\;\alpha\in\mathbb{C}. \]
For \(k=1,\) equation \(E_1\) is the second Painlevé equation. All equations in this class \(E_k\), with a single exception when \(k=2\), admit the quasi-Painlevé property along a rectifiable curve, that is, for general solutions every movable singularity along a rectifiable curve is at most an algebraic branch point. The author discusses a global multi-valuedness of solutions. In addition, it is shown by the value distribution theory that equations \(E_k\) do not admit nontrivial entire solutions and for \(k\geq 2\) there are also no nontrivial meromorphic solutions. For the exceptional equation the method of successive approximations is used to construct a general solution having a movable logarithmic branch point.
\[ y''=\frac{k+1}{k^2}y^{2k+1}+xy+\alpha,\;k\in\mathbb{N},\;\alpha\in\mathbb{C}. \]
For \(k=1,\) equation \(E_1\) is the second Painlevé equation. All equations in this class \(E_k\), with a single exception when \(k=2\), admit the quasi-Painlevé property along a rectifiable curve, that is, for general solutions every movable singularity along a rectifiable curve is at most an algebraic branch point. The author discusses a global multi-valuedness of solutions. In addition, it is shown by the value distribution theory that equations \(E_k\) do not admit nontrivial entire solutions and for \(k\geq 2\) there are also no nontrivial meromorphic solutions. For the exceptional equation the method of successive approximations is used to construct a general solution having a movable logarithmic branch point.
Reviewer: Galina Filipuk (Warszawa)
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |
Keywords:
nonlinear differential equation; quasi-Painlevé property; Painlevé equation; hyperelliptic integral; movable singularityReferences:
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