期刊
STUDIES IN APPLIED MATHEMATICS
卷 150, 期 2, 页码 520-557出版社
WILEY
DOI: 10.1111/sapm.12548
关键词
AAA approximation; analytic continuation; exponential asymptotics; nanoptera
In this study, the oscillation behavior in woodpile chains is investigated using numerical approximation and exponential asymptotics. Accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish are obtained. Furthermore, exponential asymptotics are applied to study granular woodpile chains with Hertzian interactions.
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 Pade 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.
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