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Nonlinear eigenvalue problems for generalized Painlevé equations.

Authors :
Carl M Bender
Javad Komijani
Qing-hai Wang
Source :
Journal of Physics A: Mathematical & Theoretical. 8/2/2019, Vol. 52 Issue 31, p1-1. 1p.
Publication Year :
2019

Abstract

Eigenvalue problems for linear differential equations, such as time-independent Schrödinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. Previously studied examples of nonlinear differential equations that possess discrete eigenvalue spectra are the first-order equation and the first, second, and fourth Painlevé transcendents. It is shown here that the differential equations for the first and second Painlevé transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra. The large-eigenvalue behavior is studied in detail, both analytically and numerically, and remarkable new features, such as hyperfine splitting of eigenvalues, are described quantitatively. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
17518113
Volume :
52
Issue :
31
Database :
Academic Search Index
Journal :
Journal of Physics A: Mathematical & Theoretical
Publication Type :
Academic Journal
Accession number :
137426745
Full Text :
https://doi.org/10.1088/1751-8121/ab2bcc