| Abstract: |
| Extreme water-wave motion is investigated by considering soliton interactions on a horizontal plane. We determine numerically that soliton solutions of the unidirectional Kadomtsev-Petviashvili equation (KPE), with equal far-field individual amplitudes, survive well in the bidirectional and higher-order Benney-Luke and potential-flow equations (BLE and PFE). An exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed BLE and PFE at an initial time, verifying that the KPE-fourfold amplification approximately persists. More extremely, a known three-soliton solution of the KPE is analysed, in a combined geometric-analytical approach proving its ninefold amplification. This three-soliton solution leads to an extreme splash at one location in space and time. Subsequently, we seed BLE and PFE with that three-soliton solution at a suitable initial time prior to and establish its maximum numerical amplification: it is at least 7.6 to 9 for an exact KPE amplification of 9 (depending on the choice of small parameters). In our simulations, computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to (``time-periodic") cnoidal-wave solutions. Special geometric (finite-element) variational integrators in space and time have been used to avoid artificial numerical damping of wave amplitude. A larger goal, in progress, is to use these simulations for designs of suitable wave-tank experiments. Finally, we investigate whether an analytical solution of the KPE with four line-solitons of far-field amplitude A can yield amplitude 16A at the origin in space-time. |
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