Long-time asymptotics and the radiation condition for linear evolution equations on the half-line with time-periodic boundary conditions

TL;DR

This paper analyzes the long-time behavior of linear dispersive equations on the half-line with time-periodic boundary conditions, establishing the radiation condition and deriving asymptotic solutions that match experimental results.

Contribution

It introduces a method to construct the Dirichlet-to-Neumann map for third-order dispersive equations with time-periodic data, proving its uniqueness under a specific radiation condition.

Findings

Asymptotic solutions are either traveling waves or spatially decaying waves.

The radiation condition determines the unique wavenumber branch for the asymptotics.

Quantitative agreement with fluid experiments for the BBM equation.

Abstract

The large time $t$ 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 $Q$-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 $x$, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time-periodic wave. Uniform-in-$x$ 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.