Metastability of solitary waves in diatomic FPUT lattices

Nickolas Giardetti; Amy Shapiro; Stephen Windle; J. Douglas Wright; Nickolas Giardetti; Amy Shapiro; Stephen Windle; J. Douglas Wright

doi:10.3934/mine.2019.3.419

Mathematics in Engineering

2019, Volume 1, Issue 3: 419-433. doi: 10.3934/mine.2019.3.419

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Metastability of solitary waves in diatomic FPUT lattices

Department of Mathematics, Drexel University, 3141 Chestnut St, Philadelphia, PA 19104

Received: 14 December 2018 Accepted: 25 February 2019 Published: 30 April 2019

It is known that long waves in spatially periodic polymer Fermi-Pasta-Ulam-Tsingou lattices are well-approximated for long, but not infinite, times by suitably scaled solutions of Korteweg-de Vries equations. It is also known that dimer FPUT lattices possess nanopteron solutions, i.e., traveling wave solutions which are the superposition of a KdV-like solitary wave and a very small amplitude ripple. Such solutions have infinite mechanical energy. In this article we investigate numerically what happens over very long time scales (longer than the time of validity for the KdV approximation) to solutions of diatomic FPUT which are initially suitably scaled (finite energy) KdV solitary waves. That is we omit the ripple. What we find is that the solitary wave continuously leaves behind a very small amplitude "oscillatory wake." This periodic tail saps energy from the solitary wave at a very slow (numerically sub-exponential) rate. We take this as evidence that the diatomic FPUT "solitary wave" is in fact quasi-stationary or metastable.
- diatomic Fermi-Pasta-Ulam-Tsingou lattices,
- metastability,
- Hamiltonian lattices,
- solitary waves,
- nonlinear waves
Citation: Nickolas Giardetti, Amy Shapiro, Stephen Windle, J. Douglas Wright. Metastability of solitary waves in diatomic FPUT lattices[J]. Mathematics in Engineering, 2019, 1(3): 419-433. doi: 10.3934/mine.2019.3.419

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Abstract

It is known that long waves in spatially periodic polymer Fermi-Pasta-Ulam-Tsingou lattices are well-approximated for long, but not infinite, times by suitably scaled solutions of Korteweg-de Vries equations. It is also known that dimer FPUT lattices possess nanopteron solutions, i.e., traveling wave solutions which are the superposition of a KdV-like solitary wave and a very small amplitude ripple. Such solutions have infinite mechanical energy. In this article we investigate numerically what happens over very long time scales (longer than the time of validity for the KdV approximation) to solutions of diatomic FPUT which are initially suitably scaled (finite energy) KdV solitary waves. That is we omit the ripple. What we find is that the solitary wave continuously leaves behind a very small amplitude "oscillatory wake." This periodic tail saps energy from the solitary wave at a very slow (numerically sub-exponential) rate. We take this as evidence that the diatomic FPUT "solitary wave" is in fact quasi-stationary or metastable.

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