We analyze a class of multicomponent nonlinear Schrodinger equations (MNLS) related to the symmetric BD.I-type symmetric spaces and their reductions. We briefly outline the direct and the inverse scattering method for the relevant Lax operators and t
he soliton solutions. We use the Zakharov-Shabat dressing method to obtain the two-soliton solution and analyze the soliton interactions of the MNLS equations and some of their reductions.
A three- and five-component nonlinear Schrodinger-type models, which describe spinor Bose-Einstein condensates (BECs) with hyperfine structures F=1 and F=2 respectively, are studied. These models for particular values of the coupling constants are in
tegrable by the inverse scattering method. They are related to symmetric spaces of BD.I-type SO(2r+1)/(SO(2) x SO(2r-1)) for r=2 and r=3. Using conveniently modified Zakharov-Shabat dressing procedure we obtain different types of soliton solutions.
The two time-dependent Schrodinger equations in a potential V(s,u), $u$ denoting time, can be interpreted geometrically as a moving interacting curves whose Fermi-Walker phase density is given by -dV/ds. The Manakov model appears as two moving intera
cting curves using extended da Rios system and two Hasimoto transformations.
