A direct and inverse problem for wave crests modelled by interactions of two solitons

The paper addresses a new “inverse” problem for reconstructing the amplitudes of 2D surface waves from observation of the wave patterns (formed by wave crests). These patterns will depend on the amplitudes because of nonlinear effects. We show that the inverse problem can be solved when the waves are modelled by an equation that supports soliton solutions. Specifically, the explicit solution to the inverse problem is derived for two interacting solitons of the KP (Kadomtsev–Petviashvili) equation. As a prerequisite, the “direct” problem of two-soliton solutions is investigated, presented in such a way that generalizations to an arbitrary number of solitons can be done. In this investigation we give a new meaning to the concept of interaction soliton that makes it easier to write down the two-soliton solution and to describe the soliton interactions.