Abstract
The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental equation can be integrated once to a first order nonlinear equation, e.g., the Ricatti equation. It is shown that the nonlinear eigenvalue problems of these semi-transcendental equations are equivalent to linear eigenvalue problems. They share the exactly same eigenvalues. The eigensolutions in the two problems are closely related. The nonlinear eigenvalue problem equivalent to the (half) harmonic oscillator in quantum mechanics is solved exactly. This is the first solvable nonlinear eigenvalue problem. The nonlinear eigenvalue problems of some extended equations are also studied.
Highlights
The nonlinear eigenvalue problem was first proposed in 2014 by Bender et al as an initialvalue problem for the first order nonlinear differential equation [1]y′(x) = cos[πxy(x)]. (1)For large x, the leading asymptotic behaviour for the solution of the above equation is y(x) ∼ m + /x, x → ∞, (2)where m is an integer
/x, a slightly different initial condition y(0) = a + δ will give a solution with the same asymptotic behaviour when δ is sufficiently small
We study the nonlinear eigenvalue problem of another second order differential equation with the form d2W (x) dx2
Summary
The nonlinear eigenvalue problem was first proposed in 2014 by Bender et al as an initialvalue problem for the first order nonlinear differential equation [1]. For these second order differential equations, two types of nonlinear eigenvalue problems were studied. The nonlinear eigenvalues follow power laws in the leading asymptotic behaviour. The asymptotic behaviours of large nonlinear eigenvalues exhibit similar power laws as in the first two Painleve equations [6, 7]. Unlike all the asymptotic results on the nonlinear eigenvalue problems studied so far, this equivalent relation is exact This is the primary result of this paper. The second example is a numerical study on the nonlinear eigenvalue problem which is equivalent to a half quartic anharmonic oscillator. The asymptotic behaviour of large eigenvalues are found to follow simple power laws, just like the first two Painleve equations and the generalised Painleve equations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
