Application of uniform asymptotics method to analyzing the asymptotic behaviour of the general fourth Painlevé transcendent (Q2575948)

The authors use the uniform asymptotic method proposed by \textit{A. P. Bassom}, \textit{P. A. Clarkson}, \textit{C. K. Law} and \textit{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 \(x\).