Nonlinear eigenvalue problems

Fring, A., Bender, C. & Komijani, J. (2014). Nonlinear eigenvalue problems. Journal of Physics A: Mathematical and Theoretical, 47(23), p. 235204. doi: 10.1088/1751-8113/47/23/235204

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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 → ∞, $a_n\sim A\sqrt{n}$, 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.

Item Type: Article
Additional Information: This is an author-created, un-copyedited version of an article published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at
Subjects: Q Science > QA Mathematics
Divisions: School of Engineering & Mathematical Sciences > Department of Mathematical Science

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