Numerical Experiments on Extreme Waves Through Oblique–Soliton Interactions

Abstract

Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev–Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney–Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult-to-attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. Herein, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude.