半直线上的KdV方程：时间周期与质量输运
The KdV equation on the half-line: Time-periodicity and mass transport
摘要：
这里提出的工作源于表面水波传播的实验观测所产生的问题。实验的特点是周期性移动造波机位于水槽的一端，产生的单向波的振幅相对较小，波长较长，与未受干扰的深度相比。据观察，通道中任何点的波剖面很快就会在时间上变得周期性，与造波机的周期相同。这里讨论的问题之一是，这种性质是否适用于这种波的模型方程。在目前的讨论中，这是审查的背景下Korteweg-de Vries方程使用最近开发的版本的逆散射理论的边界值问题提出的福卡斯和他的合作者。结果表明，Korteweg-de Vries方程确实具有这样的性质，即当在边界处周期性地施加力时，在通道下的固定点处的解具有时间上的渐近周期性。然而，与质量守恒有关的一个更微妙的问题在一个小参数（即典型的波幅除以未扰动深度）下无法保持在二阶。
英文摘要：
The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that generated unidirectional waves of relatively small amplitude and long wavelength when compared with the undisturbed depth. It was observed that the wave profile at any point down the channel very quickly became periodic in time with the same period as that of the wavemaker. One of the questions dealt with here is whether or not such a property holds for model equations for such waves. In the present discussion, this is examined in the context of the Korteweg-de Vries equation using the recently developed version of the inverse scattering theory for boundary value problems put forward by Fokas and his collaborators. It turns out that the Korteweg-de Vries equation does possess the properly that solutions at a fixed point down the channel have the property of asymptotic periodicity in time when forced periodically at the boundary. However, a more subtle issue to do with conservation of mass fails to hold at the second order in a small parameter which is the typical wave amplitude divided by the undisturbed depth.
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