Comment on "Nonlinear eigenvalue problems"
TL;DR: In this paper, an alternative derivation of the asymptotic results that looks at the solutions in the regions x y, and confirms the behaviour found previously for larger values of a. This method uses the small amplitude and high frequency of the oscillatory behaviour in the region x <y.
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Abstract: The asymptotic behaviour of solutions to y'(x)=cos[πxy(x)] was investigated by Bender et al (2014 J. Phys. A: Math. Theor. 47 235204). They found, for example, a relation between the initial value y(0)=a and the number of maxima that the solution exhibited. We present an alternative derivation of the asymptotic results that looks at the solutions in the regions x y, and confirms the behaviour found previously for larger values of a. This method uses the small amplitude and high frequency of the oscillatory behaviour in the region x<y.
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Citations
Nonlinear eigenvalue problems
TL;DR: In this article, it was argued that in the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that spawn the separatrix play the role as an Eigenvalue.
Nonlinear eigenvalue problems for generalized Painlev\'e equations
TL;DR: In this paper, it was shown that the differential equations for the first and second Painleve transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra.
9
Nonlinear eigenvalue problems and PT-symmetric quantum mechanics
Carl M. Bender
- 27 Jul 2017
Abstract: Semiclassical (WKB) techniques are commonly used to find the large-energy behavior of the eigenvalues of linear time-independent Schrödinger equations. In this talk we generalize the concept of an eigenvalue problem to nonlinear differential equations. The role of an eigenfunction is now played by a separatrix curve, and the special initial condition that gives rise to the separatrix curve is the eigenvalue. The Painlevé transcendents are examples of nonlinear eigenvalue problems, and semiclassical techniques are devised to calculate the behavior of the large eigenvalues. This behavior is found by reducing the Painlevé equation to the linear Schrödinger equation associated with a non-Hermitian PT-symmetric Hamiltonian. The concept of a nonlinear eigenvalue problem extends far beyond the Painlevé equations to huge classes of nonlinear differential equations.
9
Nonlinear eigenvalue problems
TL;DR: In this paper, a detailed asymptotic study of the nonlinear differential equation y'(x)=\cos[\pi xy(x)] subject to the initial condition y(0)=a is presented.
References
•Book
Advanced mathematical methods for scientists and engineers
Carl M. Bender,Steven A. Orszag +1 more
- 01 Jan 1978
TL;DR: A self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations is given in this paper.
5.8K
A fracture mechanics and mechanistic approach to the failure of cortical bone
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161
From data to dynamical systems
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17
Nonlinear eigenvalue problems
TL;DR: In this article, it was argued that in the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that spawn the separatrix play the role as an Eigenvalue.
Nonlinear eigenvalue problems
TL;DR: In this paper, a detailed asymptotic study of the nonlinear differential equation y'(x)=\cos[\pi xy(x)] subject to the initial condition y(0)=a is presented.