Journal ArticleDOI
Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations
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In this article, a technique for calculating exponentially small terms beyond all orders in singularly perturbed ODEs is presented, based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines.Abstract:
A technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented. The approach is based on the application of a WKBJ–type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well–known Stokes line–smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples.read more
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Journal ArticleDOI
The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series
TL;DR: In this article, the authors use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a hyperasymptotic approximation.
Journal ArticleDOI
Subcritical transition in channel flows
TL;DR: In this article, through a formal asymptotic analysis of the Navier-Stokes equations, it was shown that for streamwise initial perturbations γ =−1 and −3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane PoISEUille flow), while for oblique initial perturbs γ < −5/4 and −7/4.
Journal ArticleDOI
Exponential asymptotics of localised patterns and snaking bifurcation diagrams
TL;DR: In this paper, localised patterns emerging from a subcritical modulation instability are analyzed by carrying the multiple-scales analysis beyond all orders, and the full bifurcation diagram is computed analytically.
Book ChapterDOI
Discrete Painlevé Equations
TL;DR: The Cargese summer school celebrated the 100th anniversary of the Painleve property, the property that was introduced byPainleve and subsequently by Gambier and their school to classify ordinary differential equations (ODEs) according to the singularity behavior of their solutions as mentioned in this paper.
Book ChapterDOI
Geometric integration and its applications
Chris Budd,Matthew D. Piggott +1 more
TL;DR: Geometric integration as discussed by the authors is a relatively new area of numerical analysis called geometric integration, which aims to preserve the qualitative (and geometrical) features of a differential equation when it is discretised.
References
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Journal ArticleDOI
Structural stability of the Korteweg-De Vries solitons under a singular perturbation
TL;DR: In this paper, the stability of a solitary wave solution of the Korteweg-de Vries equation was investigated when a fifth order spatial derivative term is added, and it was shown that the solution ceases to be strictly localized but develops an infinite oscillating tail.
Journal ArticleDOI
Asymptotics beyond all orders in a model of crystal growth
Martin D. Kruskal,Harvey Segur +1 more
Journal ArticleDOI
Asymptotics beyond all orders
Martin D. Kruskal,Harvey Segur +1 more
TL;DR: In this article, the Geometric Model of Crystal Growth (GMMG) is used to analyze the dynamics of Dendritic Growth. But the model is not suitable for the case of solitons.
Journal ArticleDOI
Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation
Roger Grimshaw,Nalini Joshi +1 more
TL;DR: Using the techniques of exponential asymptotics, it is established that solitary wave solutions of the Korteweg–de Vries equation form a one-parameter family characterized by the phase shift of the trailing oscillations.