How is the large-eigenvalue separatrix curve found?

For linear eigenvalue problems this limit is accessible by using WKB theory but for the nonlinear eigenvalue problem studied here the large-eigenvalue limit is accessible because the problem becomes linear in this limit; indeed, the large-eigenvalue separatrix curve was found by reducing the problem to a linear random walk problem that can be solved exactly.

How do the authors show that the solution z(t) to the scaled equation (14)?

The authors then show numerically that for large λ the solution z(t) to the scaled equation (14) that satisfies the initial condition z(0) = 21/3 is oscillatory until t = 1, at which point it decays smoothly like z(t) ∼ 1/t as t→∞.

What is the first eigenvalue of the Taylor series?

The first few Taylor coefficients areb0 = y(0) = a, b1 = 1, b2 = 0, b3 = −16π 2a2, b4 = −14π 2a, b5 = 1120π 4a4 − 110π 2, b6 = 118π 4a3, b7 = − 15040π 6a6 + 221π 4a2, b8 = − 1180π 6a5 + 31480π 4a, b9 = 1362880π 8a8 − 1616480π 6a4 + 171080π 4. (16)The authors then observe that the authors can reorganize and regroup the terms in the Taylor series.

What is the partial summation of the Fourier series?

1. The partial summation suggests that a and y are both of order √ n and motivates the changes of variable (12) and (13), which give the scaled differential equation (14).

What is the infimum limit of the sequence of numbers n(f)?

the authors define the infimum limit ρ(f) of the sequence of numbers ρn(f):ρ(f) ≡ lim inf n→∞ ρn(f) = lim n→∞ [ inf k>n ρk(f) ] . (39)Finally, the authors define the power series constant P as the supremum of ρ(f) over all functions f in F :P ≡ sup f∈F ρ(f).

What is the difference between the two curves in Fig. 1?

Ten separatrix curves are shown in Fig. 2.While the separatrix curves are unstable for increasing x, they are stable for decreasing x and thus it is numerically easy to trace these curves backward from large values of x down to x = 0.

Open AccessJournal Article10.1088/1751-8113/47/23/235204

Nonlinear eigenvalue problems

Carl M. Bender, +3 more

- 23 May 2014

- Journal of Physics A

- Vol. 47, Iss: 23, pp 235204

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.

Abstract: This paper begins with a review of earlier work on extending the notion of an eigenvalue problem for a linear differential equation to an eigenvalue problem for a nonlinear differential equation. In previous work 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 of an eigenvalue. Previously discussed nonlinear differential equations that have discrete eigenvalue structure include the first-order equation y′ = cos(πxy) and the Painleve transcendents P-I and P-II. In new work it is shown here that the concept of a nonlinear eigenvalue problem extends to huge classes of nonlinear differential equations. Numerical and analytical results on the eigenvalue behavior for some of these new differential equations are presented.

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Most frequently asked questions

1. What is the key feature of the separatrix curves?

The key feature of these separatrix curves is that after passing through n double poles, they approach the curve + √ −x exponentially fast as x → −∞.

2. What is the main thrust of the analysis in this paper?

(11)The principal thrust of the analysis in this paper is an asymptotic study of the separatrices, which for large x are approximated by the formula in (5) with m odd.

3. What is the strategy of transforming a nonlinear problem to a linear integral?

The strategy of transforming a nonlinear problem to an equivalent linear problem is reminiscent of the Hopf-Cole substitution that reduces the nonlinear Burgers equation to the linear diffusion equation, the inverse-scattering analysis that reduces the nonlinear Korteweg-de Vries equation to a linear integral equation, of the Bäcklund transformation that linearizes some integrable nonlinear wave equations.

4. What is the simplest way to describe the convergence of a Fourier series?

The convergence of z(t) (which is rapidly oscillatory when 0 ≤ t ≤ 1) to Z(t) (which is smooth and nonoscillatory) as λ → ∞ strongly resembles the convergence of a Fourier series.

FIG. 8: Plot of ρ50(fτ ) as a function of τ . At the optimal value of the parameter τ the maximum of the curve is close to 1.7818.

FIG. 5: Convergence of the N = 5, 20, and 80 partial sums in (17) of the Fourier sine series for f(x) = 1. The partial sums of the Fourier series converge to 1 as N → ∞ in much the same way that z(t) converges to Z(t) as λ→∞. Like the behaviors in Figs. 3 and 4, as N increases, the frequency of oscillation increases and the amplitude of oscillation approaches zero.

FIG. 7: Non-eigenvalue solutions to the first Painlevé transcendent. If y(0) = 1 but y′(0) is not one of the eigenvalues an, the curve either oscillates about and approaches the stable asymptotic curve − √ −x as in the left panel or else it lies above the unstable asymptotic curve + √ −x and passes through an infinite number of double poles as in the right panel.

FIG. 6: Eigencurve solutions to the first Painlevé trenscendent. The eigencurves pass through y(0) = 1 and the slopes of the curves at x = 0 are the eigenvalues an. As x → −∞, the eigencurves approach + √ −x exponentially rapidly. Left panel: first two eigencurves corresponding to the eigenvalues a1 = 0.231955 and a2 = 3.980669. The a1 curve approaches + √ −x from above and the a2 curve approaches + √ −x from below. Right panel: The next two eigencurves for the Painlevé transcendent corresponding to the eigenvalues a3 = 6.257998 and a4 = 8.075911. Note that the second pair of eigenvalues passes through one double pole before approaching the curve + √ −x.

FIG. 3: Upper panel: Numerical plots of the first four separatrix solutions z(t) (eigensolutions) to (14) (blue, cyan, magenta, and green in the electronic version). These solutions have one, two, three, and four maxima. As λ increases, these curves approach the solution to (14) for λ = ∞ (dashed curve) (red in the electronic version). [The λ =∞ curve is called Z(t) and satisfies the differential equation (31).] Lower panel: A plot of the differences between the solid curves and the dashed curve.

FIG. 1: Numerical solutions y(x) to (1) for 0 ≤ x ≤ 24 with initial conditions y(0) = 0.2k for k = 1, 2, 3, . . . , 50. The solutions initially oscillate but abruptly become smoothly and monotonically decaying. In the decaying regime the solutions merge into discrete quantized bundles.

Citations

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- 03 Mar 2016

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Nonlinear eigenvalue problems for generalized Painlev\'e equations

Carl M. Bender, +2 more

- 26 Mar 2019

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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.

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Comment on "Nonlinear eigenvalue problems"

Oliver S. Kerr

- 12 Sep 2014

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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math> -symmetric quantum mechanics

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References

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Geometry of the string equations

Gregory W. Moore

- 01 Oct 1990

- Communications in Mathematical Physics

TL;DR: In this article, it was shown that the partition function of the matrix model is the tau function for isomonodromic deformation of an equation with an irregular singularity.

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•Journal Article•10.1307/MMJ/1029000105

Zeros of partial sums of power series.

J. D. Buckholtz

- 01 Dec 1968

- Michigan Mathematical Journal

Journal Article•10.1093/QJMAM/37.4.525

On a painlevé-type boundary-value problem

Philip Holmes, +1 more

- 01 Nov 1984

- Quarterly Journal of Mechanics and Appli...

TL;DR: In this article, the authors discuss the existence and unicite des solutions in 2 points limites y =y 2 −x avec y(0)=0 and y(x)∼+(x) 1/2 quand x→∞.

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Journal Article•10.1088/1751-8113/48/47/475202

Painlevé transcendents and -symmetric Hamiltonians

Carl M. Bender, +2 more

- 28 Oct 2015

- Journal of Physics A

TL;DR: In this article, the first and second Painleve transcendents are studied both numerically and analytically, and the constants and coefficients in these asymptotic behaviors are determined numerically.

...read moreread less

Monograph•10.1090/MMONO/227

Algebraic Analysis of Singular Perturbation Theory

Takahiro Kawai, +1 more

- 30 Nov 2005

...

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Nonlinear eigenvalue problems

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Most frequently asked questions

1. What is the key feature of the separatrix curves?

2. What is the main thrust of the analysis in this paper?

3. What is the strategy of transforming a nonlinear problem to a linear integral?

4. What is the simplest way to describe the convergence of a Fourier series?

5. How is the large-eigenvalue separatrix curve found?

6. How do the authors show that the solution z(t) to the scaled equation (14)?

7. What is the first eigenvalue of the Taylor series?

8. What is the partial summation of the Fourier series?

9. What is the infimum limit of the sequence of numbers n(f)?

10. What is the difference between the two curves in Fig. 1?

Figures

Citations

Spectroscopic factors, overlaps, and isospin symmetry from an R -matrix point of view

Generating asymptotics for factorially divergent sequences

Nonlinear eigenvalue problems for generalized Painlev\'e equations

Comment on "Nonlinear eigenvalue problems"

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math> -symmetric quantum mechanics

References

Geometry of the string equations

Zeros of partial sums of power series.

On a painlevé-type boundary-value problem

Painlevé transcendents and -symmetric Hamiltonians

Algebraic Analysis of Singular Perturbation Theory

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