Dynamics of localized structures

We describe qualitative behaviour of solutions of the Gross-Pitaevskii equa tion in 2D in terms of motion of vortices and radiation. To this end we int roduce the notion of the intervortex energy. We develop a rather general ad iabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of t his theory. Finally we mention briefly two special situations where we are able to obtain a rather detailed picture of the vortex dynamics. Our approa ch is rather general and is applicable to a wide class of evolution nonline ar equation which exhibit localized, stable static solutions. It yields des cription of general time-dependent solutions in terms of dynamics of those static solutions "glued" together. (C) 1998 Elsevier Science B.V. All right s reserved.