Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients

Joshua A. McGinnis; J. Douglas Wright

doi:10.3934/dcdss.2021100

Article Contents

2022, Volume 15, Issue 9: 2581-2607. Doi: 10.3934/dcdss.2021100

This issue Previous Article Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems Next Article Large-amplitude modulation of periodic traveling waves

Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients

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

Received: March 2021

Revised: July 2021

Early access: August 2021

Published: September 2022

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Abstract

We consider a linear Fermi-Pasta-Ulam-Tsingou lattice with random spatially varying material coefficients. Using the methods of stochastic homogenization we show that solutions with long wave initial data converge in an appropriate sense to solutions of a wave equation. The convergence is strong and both almost sure and in expectation, but the rate is quite slow. The technique combines energy estimates with powerful classical results about random walks, specifically the law of the iterated logarithm.

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Mathematics Subject Classification: 37L55, 37L60, 74J05.

Citation:

Full Text(HTML)

Figure 1. Figure 1 is a log-log plot of the relative error $ \rho $ divided by $ \sqrt{\log\log(1/\epsilon)}. $

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Figure 2. Figure 2 is 10 box plots of 40 different realization of masses at 10 various epsilons. It is also log-log

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Figure 3. Figure 3 is a log-log plot of the relative error masses chosen periodically

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Figure 4. In Figure 4 masses are chosen so that $ \chi(j) $ will grow like $ \sqrt{j} $

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Figure 5. In Figure 5 $ \epsilon $ is fixed and small while $ \sigma $ is varied and the absolute error is measured. When $ \sigma $ is smallest, the data is concentrated near the error for the constant coefficient case

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