The title is slightly misleading: in the first three sections of
this four-section text the authors study just the ordinary
differential Cauchy problem $y'(x) = cos[\pi xy(x)]$, with emphasis
upon its numerical and asymptotic analysis. The authors feel guided
by a resemblance of solutions $y(x)$ (and, in particular, of their
separatrices) to harmonic-oscillator bound-state wave functions and,
in particular, to their Sturm-Liouville oscillation-theorem-based
qualitative features. In this picture the role of the growth of the
harmonic-oscillator-like eigenenergies $E=E_n, n = 0, 1, \ldots$ is
transferred to the growth of the initial-value quantities $y(0)=a_n,
n = 0, 1, \ldots$. The nonlinear-problem parallel to the WKB $n \gg
1$ approximation is then developed as a hidden linearization
yielding the leading-order formula $a_n \sim 2^{1/3} \sqrt{2n}
\approx 1.781797436 \sqrt{n}$ (accompanied, incidentally, by the
conjecture of coincidence of the number $1.781797436\ldots$ with a
Hayman's universal power-series constant of ref. [7]). Finally, the
acquired methodical experience is fructified in a proposal of the
author's future project covering the first Painlev\'e transcendent
equation. A few supportive numerical experiments are performed
yielding, {\em mutatis mutandis}, the Painlev\'e-related
leading-order formula $a_n \sim C {n}^{3/5}$ where $C \approx
4.284$.


MR3216779 (Sent 2014-08-22) Bender, Carl M.; Fring, Andreas;
Komijani, Javad Nonlinear eigenvalue problems. J. Phys. A 47 (2014),
no. 23, 235204, 15 pp. 34L30 (34E20 37K10)