  The review presents the methods of generation of nonlinear coherent
excitations in strongly nonequilibrium Bose-condensed systems of trapped atoms
and their properties. Non-ground-state Bose-Einstein condensates are
represented by nonlinear coherent modes. The principal difference of nonlinear
coherent modes from linear collective excitations is emphasized. Methods of
generating nonlinear modes and the properties of the latter are described.
Matter-wave interferometry with coherent modes is discussed, including such
effects as interference patterns, internal Josephson current, Rabi
oscillations, Ramsey fringes, harmonic generation, and parametric conversion.
Dynamic transition between mode-locked and mode-unlocked regimes is shown to be
analogous to a phase transition. Atomic squeezing and entanglement in a lattice
of condensed atomic clouds with coherent modes are considered. Nonequilibrium
states of trapped Bose-condensed systems, starting from weakly nonequilibrium
state, vortex state, vortex turbulence, droplet or grain turbulence, and wave
turbulence, are classified by means of effective Fresnel and Mach numbers. The
inverse Kibble-Zurek scenario is described. A method for the formation of
directed beams from atom lasers is reported.
