DROMIONS AND A BOUNDARY-VALUE PROBLEM FOR THE DAVEY-STEWARTSON-1 EQUATION

We solve an initial-boundary value problem for the Davey-Stewartson 1 equation, which is a two-dimensional generalization of the nonlinear Schrödinger equation. This equation, which describes the interaction of a surface wave envelope of amplitude q(x, y, t) with the mean flow, arises in a wide range of physical problems. We find that the energy from the mean flow can be transferred to the surface envelope and create focusing effects. Indeed, for generic non-zero boundary conditions on the mean flow, an arbitrary initial q(x, y, t) will form a number of two-dimensional, exponentially decaying in both x and y, localized structures. Furthermore, in contrast to the one-dimensional solitons, these solutions do not preserve their form upon interaction and hence can exchange energy. These coherent structures can be driven everywhere in the plane by choosing a suitable motion for the boundaries. We call these novel localized coherent structures dromions. © 1990.