Bright solitons in ultracold atoms

We review old and recent experimental and theoretical results on bright solitons in Bose–Einstein condensates made of alkali-metal atoms and under external optical confinement. First we deduce the three-dimensional Gross–Pitaevskii equation (3D GPE) from the Dirac–Frenkel action of interacting identical bosons within a time-dependent Hartree approximation. Then we discuss the dimensional reduction of the GPE from 3D to 1D, deriving the 1D GPE and also the 1D nonpolynomial Schrödinger equation (1D NPSE). Finally, we analyze the bright solition solutions of both 1D GPE and 1D NPSE and compare these theoretical predictions with the available experimental data.