Solitary-wave solutions to nonlinear Schrödinger equations

Abstract

We examine the solitary-wave behavior of eigenstate solutions to various nonlinear Schr\"odinger equations (NLSE's) in an arbitrary number of dimensions and with a general potential. These eigenstate solutions are the only wave functions that can rigorously preserve their shape. We show that solitary-wave motion is only possible if the nonlinearity is decoupled from the absolute position of the wave packet and if the potential in the moving frame differs by at most a linear term from that for the eigenstate problem. If these conditions are satisfied then the motion is along the fully classical trajectory, although the nonlinear term may introduce an additional acceleration. We comment on the implications of these results to the study of the behavior of Bose-Einstein condensed atoms in harmonic trapping potentials, for which the relevant NLSE is the Gross-Pitaevskii equation. Numerical simulations are presented for harmonic and anharmonic potentials in one dimension to illustrate our results.