On soliton dynamics in nonlinear schrödinger equations

In this paper we describe our results on dynamics of solitons in generalized nonlinear Schrödinger equations with external potentials (potNLS's) in all dimensions except for 2. We also outline some of the ideas of the proofs involved. The detailed discussion of the results as well as the proofs are presented in [GS2]. For a certain class of nonlinearities a potNLS has solutions which are periodic in time and exponentially decaying in space and which are centered near different critical points of the potential. We call those solutions which are centered near the minima and which minimize energy restricted to the ℒ²-unit sphere, trapped solitons or just solitons. Our results show that, under certain conditions on the potentials and initial conditions, trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position plus a small fluctuation. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.