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Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation

Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation
Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation
We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ordinary differential equation with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ordinary differential equation with a parameter. This paper provides an exponential asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study.
0951-7715
1559-1584
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Olde Daalhuis, A.B.
d2254863-03c9-4e12-aee7-2855b60dc933
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Olde Daalhuis, A.B.
d2254863-03c9-4e12-aee7-2855b60dc933

Howls, C.J. and Olde Daalhuis, A.B. (2012) Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation. Nonlinearity, 25 (6), 1559-1584. (doi:10.1088/0951-7715/25/6/1559).

Record type: Article

Abstract

We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ordinary differential equation with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ordinary differential equation with a parameter. This paper provides an exponential asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study.

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Submitted date: May 2011
Published date: 20 April 2012
Organisations: Applied Mathematics

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Local EPrints ID: 151867
URI: http://eprints.soton.ac.uk/id/eprint/151867
ISSN: 0951-7715
PURE UUID: b312e2ca-e010-4db9-b5cd-0c94ff83322b
ORCID for C.J. Howls: ORCID iD orcid.org/0000-0001-7989-7807

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Date deposited: 19 May 2010 10:18
Last modified: 09 Jan 2022 03:03

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Contributors

Author: C.J. Howls ORCID iD
Author: A.B. Olde Daalhuis

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