Yifeng Mao, Dionyssios Mantzavinos, Mark A. Hoefer
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Long-time asymptotics and the radiation condition with time-periodic boundary conditions for linear evolution equations on the half-line and experiment
The asymptotic Dirichlet-to-Neumann (D-N) map is constructed for a class of scalar, constant coefficient, linear, third-order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half-line, modeling a wavemaker acting upon an initially quiescent medium. The large time asymptotics for the special cases of the linear Korteweg-de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D-N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time-dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number 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. The linearized BBM asymptotics are found to quantitatively agree with viscous core-annular fluid experiments.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.