Nonlinear eigenvalue problems

This paper presents an asymptotic study of the differential equation y' (x) = cos[pi xy(x)] subject to the initial condition y(0) = a. While this differential equation is nonlinear, the solutions to the initial-value problem bear a striking resemblance to solutions to the linear time-independent Schrodinger eigenvalue problem. As x increases from 0, y(x) oscillates and thus resembles a quantum wave function in a classically allowed region. At a critical value x = x(crit), where x(crit) depends on a, the solution y(x) undergoes a transition; the oscillations abruptly cease and y(x) decays to 0 monotonically as x -> infinity. This transition resembles the transition in a wave function at a turning point as one enters the classically forbidden region. Furthermore, the initial condition a falls into discrete classes; in the nth class of initial conditions a(n-1) < a < a(n) (n = 1, 2, 3, ... ), y(x) exhibits exactly n maxima in the oscillatory region. The boundaries a(n) of these classes are the analogues of quantum-mechanical eigenvalues. An asymptotic calculation of a(n) for large n is analogous to a high-energy semiclassical (WKB) calculation of eigenvalues in quantum mechanics. The principal result of this paper is that as n -> infinity, a(n) similar to A root n, where A = 2(5/6). Numerical analysis reveals that the first Painleve transcendent has an eigenvalue structure that is quite similar to that of the equation y' (x) = cos[pi xy(x)] and that the nth eigenvalue grows with n like a constant times n(3/5) as n -> infinity. Finally, it is noted that the constant A is numerically very close to the lower bound on the power-series constant P in the theory of complex variables, which is associated with the asymptotic behavior of zeros of partial sums of Taylor series.