Exponential asymptotics of woodpile chain nanoptera using numerical analytic continuation

Traveling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading‐order behavior. For many...

Full description

Saved in:
Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) Vol. 150; no. 2; pp. 520 - 557
Main Authors: Deng, Guo, Lustri, Christopher J.
Format: Journal Article
Language:English
Published: Cambridge Blackwell Publishing Ltd 01.02.2023
Subjects:
AAA approximation
analytic continuation
Approximation
Asymptotic methods
Chains
Continuation methods
exponential asymptotics
Mass ratios
Mathematical analysis
nanoptera
Nonlinear systems
Numerical methods
Oscillations
Pade approximation
Production methods
Solitary waves
Traveling waves
ISSN:0022-2526, 1467-9590
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Traveling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading‐order behavior. For many nonlinear systems, such as granular woodpile chains, it is not possible to calculate the leading‐order solution explicitly. We show that accurate asymptotic approximations can be obtained using numerical approximation in place of the exact leading‐order behavior. We calculate the oscillation behavior for Toda woodpile chains, and compare the results to exponential asymptotics based on previous methods from the literature: long‐wave approximation and tanh‐fitting. We then use numerical analytic continuation methods based on Padé approximants and the adaptive Antoulas–Anderson (AAA) method. These methods are shown to produce accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish. Exponential asymptotics using an AAA approximation for the leading‐order behavior is then applied to study granular woodpile chains, including chains with Hertzian interactions—this method is able to calculate behavior that could not be accurately approximated in previous studies.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0022-2526
1467-9590
DOI:10.1111/sapm.12548