an:02236256
Zbl 1090.34072
Qin, Huizeng; Lu, Youmin
Application of uniform asymptotics method to analyzing the asymptotic behaviour of the general fourth Painlev?? transcendent
EN
Int. J. Math. Math. Sci. 2005, No. 9, 1421-1434 (2005).
00120887
2005
j
34M55
Painlev?? equations; asymptotics
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 Painlev?? 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 Painlev?? 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 Painlev?? 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 \(x\).
Nikolay Vasilye Grigorenko (Ky??v)
Zbl 0912.34007; Zbl 1025.34091