Nonlinear eigenvalue problems

• Published in: arXiv.org
• Main Authors: Bender, Carl M, Fring, Andreas, Komijani, Javad
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.01.2014
• Subjects: Asymptotic properties; Complex variables; Eigenvalues; Initial conditions; Lower bounds; Mathematics - Mathematical Physics; Nonlinear differential equations; Numerical analysis; Physics - Chaotic Dynamics; Physics - Mathematical Physics; Physics - Quantum Physics; Quantum mechanics; Taylor series
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1401.6161

This paper presents a detailed asymptotic study of the nonlinear differential equation y'(x)=\cos[\pi xy(x)] subject to the initial condition y(0)=a. Although the differential equation is nonlinear, the solutions to this initial-value problem bear a striking resemblance to solutions to the time-independent Schroedinger eigenvalue problem. As x increases from x=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-->\infty. This transition resembles the transition in a wave function that occurs 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 analogs 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-->\infty, a_n~A\sqrt{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-->\infty. 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.