Analysis and Simulations of Three-Soliton Interactions with Extreme Wave-Amplification in a Hierarchy of Water-Wave Models

  • Bokhove, Onno (Leeds Institute for Fluid Dynamics)

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Extreme water-wave motion is investigated analytically and numerically by considering two- 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 of the solitons, survive 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 verify that the KPE-fourfold amplification approximately persists. More extremely, a known three-soliton solution of the KPE is analysed, seemingly much further than before, in a combined geometric-analytical approach to assess its ninefold amplification, the latter which is shown to exist in a relevant limit after suitable adjustments. It avoids a singularity in the location of the maximum we had encountered earlier in our work (Choi et al. 2022). Note that this three-soliton solution leads to an extreme splash at one location in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time prior to, and to establish, its maximum numerical amplification: it is at least 7 to 7.8 for an exact KPE amplification of 8.41 (depending on the choice of a small parameter). In our simulations, the 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. 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. Novel simulations —currently in progress— will hopefully be shown for the BLE seeded by the KPE solution, well prior in time to it reaching its ninefold amplification, as well as analogous simulations with the full potential-flow water-wave equations instead of the BLE. A larger goal, not reached here, is to use and validate these simulations for designs and measurements of suitable wave-tank experiments, respectively. Choi, Bokhove, Kalogirou, Kelmanson 2022: Numerical experiments on extreme waves through oblique soliton interactions. Water Waves 4, 139–179.