Two-dimensional interactions of solitons in a two-layer fluid of finite depth

Abstract

Two-dimensional (2D) interactions of two interfacial solitons in a two-layer fluid of finite depth are investigated under the assumption of a small but finite amplitude. When the angle ϑ between the wave normals of two solitons is not small, it is shown by a perturbation method that in the lowest order of approximation the solution is a superposition of two intermediate long wave (ILW) solitons and in the next order of approximation the effect of the interaction appears as position phase shifts and as an increase in amplitude at the interaction center of two solitons. When ϑ is small, it is shown that the interaction is described approximately by a nonlinear integro-partial differential equation that we call the two-dimensional ILW (2DILW) equation. By solving it numerically for a V-shaped initial wave that is an appropriate initial value for the oblique reflection of a soliton due to a rigid wall, it is shown that for a relatively large angle of incidence ϑi the reflection is regular, but for a relatively small ϑi the reflection is not regular and a new wave called stem is generated. The results are also compared with those of the Kadomtsev–Petviashvili (KP) equation and of the two-dimensional Benjamin–Ono (2DBO) equation.