Dromions and a boundary value problem for the Davey-Stewartson 1 equation. (English) Zbl 0707.35144

Localized solitons in the plane have been a hot topic since their discovery in 1988; see here M. Boiti, J. Leon, L. Martina, and F. Pempinelli (BLMP), Phys. Lett. A 132, 432-439 (1988), and Nonlinear evolution equations..., Manchester Univ. Press, 1990, pp. 249-262 (Proc. Workshop, Como, 1988).
The localized traveling solutions (dromions) studied here are more general; for special parameter values they reduce to the localized solitons of BLMP. We extract instead from the abstract: We solve an initial-boundary value problem for the Devey-Stewartson (DS) I equation, which is a 2 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 whole 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 envelope q(x,y,t) will form a number of 2 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.
Reviewer: R.Carroll


35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI


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