Higher-Order Algebraic Differential Equations

  • Norbert Steinmetz
Part of the Universitext book series (UTX)


In this chapter we will extend the investigations of the previous chapter to second-order algebraic differential equations and two-dimensional Hamiltonian systems whose solutions are meromorphic functions. Having established the Painlevé property for distinguished equations and systems, we will draw a comprehensive picture of the solutions. This includes detecting and describing the distribution of zeros and poles, zero- and pole-free regions, and asymptotic expansions on pole-free regions, and characterising the so-called sub-normal solutions. As in the preceding chapter, a crucial role is played by the method of Yosida Re-scaling. It establishes the central discovery that the first, second, and fourth Painlevé transcendents belong to the Yosida classes \(\mathfrak{Y}_{\frac{1} {2},\frac{1} {4} }\), \(\mathfrak{Y}_{\frac{1} {2},\frac{1} {2} }\), \(\mathfrak{Y}_{1,1}\), respectively.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Norbert Steinmetz
    • 1
  1. 1.Fakultät für MathematikTU DortmundDortmundGermany

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