水波的相对周期性弹性碰撞
Relative-Periodic Elastic Collisions of Water Waves
摘要：
我们计算的时间周期和相对周期的自由表面欧拉方程的解决方案，采取的形式超越碰撞的单向孤立波的不同振幅的周期域。作为一个开始的猜测，我们叠加两个斯托克斯波偏移一半的空间周期。使用超定射击方法，由斯托克斯波的碰撞产生的背景辐射被调谐为在每次碰撞之前和之后是相同的。在某些情况下，在这个过程中，辐射被有效地消除，产生光滑的类孤子的解决方案，弹性相互作用永远。我们发现了一些例子，在这些例子中，较大的波在每次碰撞时都会包含较小的波，而在另一些例子中，拖尾波会碰撞到前导波，在没有完全合并的情况下传递能量。尽管有相似之处，这些解决方案被发现定量位于Korteweg-de Vries制度之外。我们得出结论，当区域是周期的时候，准周期弹性碰撞对于可积模型水波方程不是唯一的。
英文摘要：
We compute time-periodic and relative-periodic solutions of the free-surface Euler equations that take the form of overtaking collisions of unidirectional solitary waves of different amplitude on a periodic domain. As a starting guess, we superpose two Stokes waves offset by half the spatial period. Using an overdetermined shooting method, the background radiation generated by collisions of the Stokes waves is tuned to be identical before and after each collision. In some cases, the radiation is effectively eliminated in this procedure, yielding smooth soliton-like solutions that interact elastically forever. We find examples in which the larger wave subsumes the smaller wave each time they collide, and others in which the trailing wave bumps into the leading wave, transferring energy without fully merging. Similarities notwithstanding, these solutions are found quantitatively to lie outside of the Korteweg-de Vries regime. We conclude that quasi-periodic elastic collisions are not unique to integrable model water wave equations when the domain is periodic.
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