Nanoptera in nonlinear woodpile chains with zero precompression
arXiv:2105.02448 · doi:10.1016/j.physd.2021.133053
Abstract
We use exponential asymptotics to study travelling waves in woodpile systems modelled as singularly perturbed granular chains with zero precompression and small mass ratio. These systems are strongly nonlinear, and there is no analytic expression for their leading-order solution. We instead obtain an approximated leading-order solution using a hybrid numerical-analytic method. We show that travelling waves in these nonlinear woodpile systems are typically "nanoptera", or travelling waves with exponentially small but non-decaying oscillatory tails which appear as a Stokes curve is crossed. We demonstrate that travelling wave solutions in the zero precompression regime contain two Stokes curves, and hence two sets of tailing oscillations in the solution. We calculate the behaviour of these oscillations explicitly, and show that there exist system configurations which cause the oscillations to cancel entirely, producing solitary wave behaviour. We then study the behaviour of travelling waves in woodpile chains as precompression is increased, and show that there exists a value of the precompression above which the two Stokes curves coalesce into a single curve, meaning that cancellation of the tailing oscillations no longer occurs. This is consistent with previous studies, which showed that cancellation does not occur in chains with strong precompression.
arXiv comment21 pages, 9 figures
References in corpus (33)
- The AAA algorithm for rational approximation
- Tunability of solitary wave properties in one dimensional strongly nonlinear phononic crystals
- Energy trapping and shock disintegration in a composite granular medium
- Anomalous wave reflection from the interface of two strongly nonlinear granular media
- Discrete Breathers in One-Dimensional Diatomic Granular Crystals
- How Hertzian solitary waves interact with boundaries in a 1-D granular medium
- Highly Nonlinear Solitary Waves in Periodic Dimer Granular Chains
- Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures
- Intrinsic Energy Localization through Discrete Gap Breathers in One-Dimensional Diatomic Granular Crystals
- Translationally invariant discrete kinks from one-dimensional maps
- Shocks in non-loaded bead chains with impurities
- Wave propagation in single column woodpile phononic crystals: Formation of tunable band gaps
- Moving solitons in the discrete nonlinear Schrödinger equation
- Nonlinear Waves in Disordered Diatomic Granular Chains
- Scattering of solitary waves in granular media
- Travelling kinks in discrete phi^4 models
- Superdiffusive Transport and Energy Localization in Disordered Granular Crystals
- Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio
- Traveling Waves and their Tails in Locally Resonant Granular Systems
- Hysteresis Loops and Multi-stability: From Periodic Orbits to Chaotic Dynamics (and Back) in Diatomic Granular Crystals
- Nonlinear Low-to-High Frequency Energy Cascades in Diatomic Granular Crystals
- Delayed Scattering of Solitary Waves from Interfaces in a Granular Container
- Breathers in a locally resonant granular chain with precompression
- Direct measurement of superdiffusive and subdiffusive energy transport in disordered granular chains
- Nanoptera in a Period-2 Toda Chain
- Standing lattice solitons in the discrete NLS equation with saturation
- Scattering of Waves by Impurities in Precompressed Granular Chains
- Quasiperiodic granular chains and Hofstadter butterflies
- Nonlinear Excitations in Magnetic Lattices with Long-Range Interactions
- Granular chains with soft boundaries: Slowing the transition to quasi-equilibrium
- On the structure of parasitic gravity-capillary waves in the small surface tension limit
- The equilibrium phase in heterogeneous Hertzian chains
- Nanoptera in weakly nonlinear woodpile and diatomic granular chains