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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Joshua A. McGinnis ◽  
J. Douglas Wright
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Random Walks ◽  
Random Coefficients ◽  
Energy Estimates ◽  
Stochastic Homogenization ◽  
Iterated Logarithm ◽  
Long Wave ◽  
The Law ◽  
Spatially Varying

<p style='text-indent:20px;'>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.</p>

Propagation of classical solutions to the perturbed wave equation in a space of odd dimension

1984 ◽  
Vol 96 (3-4) ◽  
pp. 337-344
Author(s):  
Gustavo Perla Menzala
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Riemann Function ◽  
Energy Estimates ◽  
Classical Solutions ◽  
Large Space ◽  
Huygens Principle

SynopsisWe prove that classical solutions of the perturbed wave equation in ℝn × ℝ (n = odd ≧ 3) do not satisfy Huygens' principle in the presence of symmetries. The difficulties arising from the singularities of the Riemann function (for large space dimensions) are overcome by considering a class of potentials and initial data which are radial and smooth. Our method is elementary and based on energy estimates.

Novel Transparent Boundary Conditions for the Regularized Long Wave Equation

2019 ◽  
Vol 17 (06) ◽  
pp. 1950021
Author(s):  
M. Abounouh ◽  
H. Al-Moatassime ◽  
S. Kaouri
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Boundary Conditions ◽  
Transparent Boundary Conditions ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Space Discretization ◽  
Infinite Norm ◽  
Nicolson Scheme ◽  
Norm Error

In this paper, we present a new method to derive transparent boundary conditions for the Regularized Long Wave equation and its linearized equation. These boundary conditions have the advantage of being exact for both linearized and nonlinear equations. The resulting problems supplemented with initial data are approximated numerically using finite difference method in space discretization and Crank–Nicolson scheme in time for the linearized equation, while implicit scheme in time is used for the Regularized Long Wave equation. Numerical experiments are made for solitary initial data and various source terms to illustrate the transparency of the introduced boundary conditions. Results are compared with the use of usual boundary conditions, namely, Dirichlet and Neumann via infinite norm error on the boundary and also by displaying geophones and instantaneous figures.

scholarly journals Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data

2019 ◽  
Vol 21 (05) ◽  
pp. 1850035
Author(s):  
Motohiro Sobajima ◽  
Yuta Wakasugi
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Hypergeometric Functions ◽  
Smooth Boundary ◽  
Decay Rates ◽  
Energy Estimates ◽  
Damping Term ◽  
Confluent Hypergeometric Functions ◽  
Weighted Energy ◽  
Weighted Energy Estimates

This paper is concerned with weighted energy estimates for solutions to wave equation [Formula: see text] with space-dependent damping term [Formula: see text] [Formula: see text] in an exterior domain [Formula: see text] having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polynomials are given and these decay rates are almost sharp, even when the initial data do not have compact support in [Formula: see text]. The crucial idea is to use special solution of [Formula: see text] including Kummer’s confluent hypergeometric functions.

scholarly journals The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities

2019 ◽  
Vol 16 (02) ◽  
pp. 379-400 ◽  
Author(s):  
Artur Alho ◽  
Grigorios Fournodavlos ◽  
Anne T. Franzen
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Blow Up ◽  
Physical Space ◽  
Spacelike Hypersurface ◽  
Big Bang ◽  
Energy Estimates ◽  
Dirichlet Data ◽  
Open Set ◽  
The Big Bang

We consider the wave equation, [Formula: see text], in fixed flat Friedmann–Lemaître–Robertson–Walker and Kasner spacetimes with topology [Formula: see text]. We obtain generic blow up results for solutions to the wave equation toward the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to the solutions that blow up in an open set of the Big Bang hypersurface [Formula: see text]. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate [Formula: see text]-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the [Formula: see text] norms of the solutions blow up toward the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates, respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

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