| Abstract: |
| We investigate the weakly nonlinear isotropic bi-directional Benney-Luke (BL) equation, with a particular focus on soliton dynamics. The associated modulation equations are derived that describe the evolution of soliton amplitude and slope. By analyzing rarefaction waves and shock waves within these modulation equations, we derive the Riemann invariants and modified Rankine-Hugoniot conditions, which help characterize the Mach expansion and Mach reflection phenomena. We also derive analytical formulas for the critical angle and the Mach stem amplitude, showing that as the soliton speed is in the vicinity of unity, the results from the BL equation align closely with those of the Kadomtsev-Petviashvili (KP) equation. Furthermore, as a far-field approximation for the forced BL equation -- which models wave and flow interactions with local topography -- the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation. |
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