Dark solitons as quasiparticles in trapped condensates
  We present a theory of dark soliton dynamics in trapped quasi-one-dimensional Bose-Einstein condensates, which is based on the local density approximation. The approach is applicable for arbitrary polynomial nonlinearities of the mean-field equation governing the system as well as to arbitrary polynomial traps. In particular, we derive a general formula for the frequency of the soliton oscillations in confining potentials. A special attention is dedicated to the study of the soliton dynamics in adiabatically varying traps. It is shown that the dependence of the amplitude of oscillations {\it vs} the trap frequency (strength) is given by the scaling law $X_0\propto\omega^{-\gamma}$ where the exponent $\gamma$ depends on the type of the two-body interactions, on the exponent of the polynomial confining potential, on the density of the condensate and on the initial soliton velocity. Analytical results obtained within the framework of the local density approximation are compared with the direct numerical simulations of the dynamics, showing remarkable match. Various limiting cases are addressed. In particular for the slow solitons we computed a general formula for the effective mass and for the frequency of oscillations. 
V.A. Brazhnyi, V.V. Konotop, L.P. Pitaevskii