Long-time asymptotics and the radiation condition for linear evolution equations on the half-line with time-periodic boundary conditions
Abstract
The large time asymptotics for scalar, constant coefficient,linear, third order, dispersive equations are obtained for asymptotically time-periodic Dirichlet boundary data and zero initial data on the half-line modeling a wavemaker acting upon an initially quiescent medium. The asymptotic Dirichlet-to-Neumann (D-N) map is constructed by expanding upon the recently developed -equation method. The D-N map is proven to be unique if and only if the radiation condition that selects the unique wavenumber branch of the dispersion relation for a sinusoidal, time-dependent boundary condition holds: (i) for frequencies in a finite interval, the wavenumber is real and corresponds to positive group velocity, (ii) for frequencies outside the interval, the wavenumber is complex with positive imaginary part. For fixed spatial location , the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time-periodic wave. Uniform-in- asymptotic solutions for the physical cases of the linearized Korteweg-de Vries and Benjamin-Bona-Mahony (BBM) equations are obtained via integral asymptotics. The linearized BBM asymptotics are found to quantitatively agree with viscous core-annular fluid experiments.
Keywords
Cite
@article{arxiv.2307.14670,
title = {Long-time asymptotics and the radiation condition for linear evolution equations on the half-line with time-periodic boundary conditions},
author = {Yifeng Mao and Dionyssios Mantzavinos and Mark A. Hoefer},
journal= {arXiv preprint arXiv:2307.14670},
year = {2023}
}