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...
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| Vydané v: | Studies in applied mathematics (Cambridge) Ročník 150; číslo 2; s. 520 - 557 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Cambridge
Blackwell Publishing Ltd
01.02.2023
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| ISSN: | 0022-2526, 1467-9590 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| Author | Lustri, Christopher J. Deng, Guo |
| Author_xml | – sequence: 1 givenname: Guo surname: Deng fullname: Deng, Guo organization: Macquarie University – sequence: 2 givenname: Christopher J. orcidid: 0000-0001-9504-277X surname: Lustri fullname: Lustri, Christopher J. email: christopher.lustri@mq.edu.au organization: Macquarie University |
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| Snippet | Traveling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These... |
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| SubjectTerms | 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 |
| Title | Exponential asymptotics of woodpile chain nanoptera using numerical analytic continuation |
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