Nonlinear eigenvalue problems

Abstract

This paper presents an asymptotic study of the differential equation y′(x) = cos [πxy(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 Schrödinger 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 = xcrit, where xcrit depends on a, the solution y(x) undergoes a transition; the oscillations abruptly cease and y(x) decays to 0 monotonically as x → ∞. 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 an − 1 < a < an (n = 1, 2, 3, …), y(x) exhibits exactly n maxima in the oscillatory region. The boundaries an of these classes are the analogues of quantum-mechanical eigenvalues. An asymptotic calculation of an 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 → ∞, , where A = 25/6. Numerical analysis reveals that the first Painlevé transcendent has an eigenvalue structure that is quite similar to that of the equation y′(x) = cos [πxy(x)] and that the nth eigenvalue grows with n like a constant times n3/5 as n → ∞. 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.