Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations

Chapman, S. Jonathan; King, John R.; Adams, K. L.

doi:10.1098/rspa.1998.0278

Cited by 82 publications

(279 citation statements)

References 18 publications

(21 reference statements)

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“…Having determined the form of the late terms of the asymptotic series of φ, we can now truncate the series and study the behaviour of the remainder as in Daalhuis et al (1995); Chapman et al (1998). In this way we will observe the Stokes switching directly through a smoothing of the Stokes discontinuities.…”

Section: Stokes Smoothingmentioning

confidence: 99%

“…Since to identify the Stokes structure all we need is the asymptotic behaviour of φ (n) for large n, it is therefore natural to ask whether this can be obtained directly, that is, whether we can approximate the equation for φ (n) for large n and then solve, rather than solving exactly and then approximating. In this section, we examine the possibilities and difficulties of obtaining the late terms through a direct application of a factorial/power ansatz, as is possible for ordinary differential equations (see Chapman et al (1998)). Thus we suppose that…”

Section: Direct Application Of 'Factorial Over Power' Ansatzmentioning

confidence: 99%

“…Across the Stokes line an exponentially small term "beyond all orders" is switched on as shown by Chapman et al (1998). Although this term is exponentially small at the origin where it is turned on, it grows exponentially as s → ∞ and its presence means that the boundary conditions (1.2) cannot be satisfied.…”

Section: Introductionmentioning

confidence: 99%

“…In both linear (Dingle (1973)) and nonlinear (Chapman et al (1998)) singularly perturbed o.d.e. problems we find the n th term of the expansion to be a single term with the form of 'factorial over power'.…”

Section: (B) Finding the Late Termsmentioning

confidence: 99%

“…With this end in mind, in Section 5 we try to obtain the late terms using a WKBJ type ansatz of the form factorial/power, as has been done by Chapman et al (1998) for nonlinear ordinary differential equations. We will find that determining the "phase" of the approximation is possible (but not straightforward, since the boundary data may be given on a characteristic of the resulting eikonal equation), but that determining the premultiplying amplitude involves solving an inner problem analogous to the inner diffraction problems that arise in the Geometrical Theory of Diffraction (see Keller (1958Keller ( , 1962).…”

Section: Introductionmentioning

confidence: 99%

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Exponential asymptotics and Stokes lines in a partial differential equation

Chapman¹,

Mortimer²

2005

Proc. R. Soc. A.

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A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The 'new' Stokes lines discussed by Berk et al. (1982) are second-generation Stokes lines, while the 'vanishing' Stokes lines discussed by Aoki et al. (1998) are switched off by a higher-order Stokes line.

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Section: Stokes Smoothingmentioning

confidence: 99%

Section: Direct Application Of 'Factorial Over Power' Ansatzmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: (B) Finding the Late Termsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exponential asymptotics and Stokes lines in a partial differential equation

Chapman¹,

Mortimer²

2005

Proc. R. Soc. A.

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Recurrence relationship of successive level singulants

Say

2021

Math Methods in App Sciences

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We study the asymptotic behaviour of the solution to a singularly perturbed model differential equation. The analytical solution is asymptotically represented by a formal power series in the perturbation parameter. This paper provides an explanation of the relation between the pre-factor functions P(z) and Q(z) and the singulants of the expansion of the general second-order singularly perturbed differential equation in higher levels of the asymptotic analysis. Exponential asymptotics typically relies upon the derivation of the singulant function as it is crucial to determine the location of the Stokes lines. By doing the careful analysis, we connect the subsequent order singulants with the previous order singulants and determine the asymptotics of the late coefficients. By this relation, one can predict and investigate the Stokes phenomenon and exponential smoothing of the equations in the form of the model differential equation.

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On the asymptotic behavior of a second‐order general differential equation

Say

2021

Numerical Methods Partial

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Studying ordinary or partial differential equations or integrals using traditional asymptotic analysis, unfortunately, fails to extract the exponentially small terms and fails to derive some of their asymptotic features. In this paper, we discuss how to characterize an asymptotic behavior of a singular linear differential equation by the methods in exponential asymptotics. This paper is particularly concerned with the formulation of the series representation of a general second‐order differential equation. It provides a detailed explanation of the asymptotic behavior of the differential equation and its relation between the prefactor functions and the singulant of the expansion of the equation. Through having this relationship, one can directly uncover and investigate invisible exponentially small terms and Stokes phenomenon without doing more work for the particular type of equations. Here, we demonstrate how these terms and form of the expansion can be computed straight‐away, and, in a manner, this can be extended to the derivation of the potential Stokes and anti‐Stokes lines.

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Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations

Cited by 82 publications

References 18 publications

Exponential asymptotics and Stokes lines in a partial differential equation

Exponential asymptotics and Stokes lines in a partial differential equation

Recurrence relationship of successive level singulants

On the asymptotic behavior of a second‐order general differential equation

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