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Application of uniform asymptotics method to analyzing the asymptotic behaviour of the general fourth Painlevé transcendent. (English) Zbl 1090.34072
The authors use the uniform asymptotic method proposed by A. P. Bassom, P. A. Clarkson, C. K. Law and J. B. McLeod [Arch. Rat. Mech. Anal., 143, 241–271 (1998; Zbl 0912.34007)] to study the general solution of the fourth Painlevé equation
\[ y''=\frac{y^{\prime 2}}{2y}+ \frac{2}{3}y^3+ 4xy^2+2(x^2-\alpha)y+ \frac{\beta}{y}.\tag{P\(_{\text{IV}}\)} \]
At the present time, there are not many results about the asymptotics of fourth Painlevê equation (see the second author [Int. J. Math. Math. Sci. 2003, No. 13, 845–851 (2003; Zbl 1025.34091)]). The authors study the behaviour of the real solutions of (P\(_{\text{IV}}\)) when \(\beta>0\) and \(\alpha>0\), and obtain the following result on the asymptotics of its real solutions.
Theorem: If \(\beta>0\), then the solutions of (P\(_{\text{IV}}\)) cannot cross the \(x\)-axis. Furthermore, if \(\alpha>0\), then the only negative solution of Painlevé equation (P\(_{\text{IV}}\)) that does not blow up at any finite point when \(x\) goes to positive infinity is oscillating as \(x\rightarrow +\infty\) and it satisfies the following relations:
As \(x\rightarrow +\infty\), \[ \begin{aligned} & y=-\frac{2}{3}x\pm d\cos\phi + O(x^{-1}),\quad x\rightarrow +\infty\\ & y'=\frac{2\sqrt{3}x}{3}d\sin\phi + O(x^{-1}),\quad x\to+\infty,\end{aligned}\tag{1} \] where \(\phi=(\sqrt{3}/3)x^2-(\sqrt{3}/4)d^2\log x+\phi_{0}+O(x^{-1}), d\) and \(\phi_{0}\) are real parameters.
(2)As \(x\rightarrow -\infty\), \(y\) blows up at a finite point of \(x\).

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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