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arXiv · 1401.6161

Nonlinear eigenvalue problems

Carl M. Bender·Andreas Fring·Javad Komijani
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Abstract

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} \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.

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BibTeXRIS

Carl M. Bender, Andreas Fring, Javad Komijani. 2014-01-23. Nonlinear eigenvalue problems. https://doi.org/10.1088/1751-8113/47/23/235204

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