Higher-Order Algebraic Differential Equations

Abstract

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 Painleve 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 Painleve 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.